Proceedings of the Royal Society of Edinburgh: Submitted Paper Paper 6 July 2001

Large-Time Behavior of Solutions of Burgers’ Equation

Daniel B. Dix

(MS received ; )

The large-time asymptotic behavior of real-valued solutions of the pure initial-value problem for Burgers’ equation u_{t} + uu_{x} − \frac{1}{2} u_{xx} = 0, is studied. The initial data satisfies u_{0}(x) ∼ \frac{n}{x} as |x| → ∞, where n ∈ ℝ. There are two constants of the motion which affect the large-time behavior: μ = lim_{A→∞} ∫_{−A}^{A} u_{0}(x) dx and n. Eberhard Hopf [20] considered the case n = 0 (i.e. u_{0} ∈ L^{1}(ℝ)), and the case ∥u_{0}(x) − \frac{nx}{1+x^{2}} ∥_{L^{1}(x)} + |n| sufficiently small was considered in [11]. Here we completely remove that smallness condition. When n < 1 we find an explicit function U(ξ), depending only on μ and n, such that t^{1/2}u(ξt^{1/2}, t) = U(ξ) + o((1 + |ξ|)^{−1}) as t → ∞, uniformly in ξ. When n ⩾ 1 there are two different functions U(ξ) which simultaneously “attract” the quantity t^{1/2}u(ξt^{1/2}, t), and each one “wins” in its own range of ξ. Thus we give an asymptotic description of the solution in different regions and compute its decay rate in L^{∞}. Sharp error estimates are proved.

1. Introduction

1.1. Statement of the Main Problem

This work studies in detail the large-time asymptotic behavior of solutions of the initial-value problem for Burgers’ equation

u_{t} + uu_{x} − \frac{1}{2}u_{xx} = 0,  x ∈ ℝ, t > 0,
u(x, 0) = u_{0}(x),  x ∈ ℝ,

where u(x, t) and u_{0}(x) are real-valued functions. We assume (at least) that

u_{0}(x) = u_{1}(x) + \frac{nx}{1 + x^{2}} , x ∈ ℝ,

where u_{1} ∈ L^{1}(ℝ) and n ∈ ℝ. This implies that in some average sense u_{0}(x) ∼ \frac{n}{x} as |x| → ∞. We have selected the function nx/(1 + x^{2}) to simplify our formulae and hypotheses as much as possible; for example, since nx/(1 + x^{2}) is odd we have that conserved quantity μ = lim_{A→∞} ∫_{−A}^{A} u_{0}(x) dx = ∫_{−∞}^{∞} u_{1}(x) dx is well-defined. See below for further remarks concerning our assumptions on u_{0}.

1.2. History

Especially prominent in the study of the large-time behavior of u(x, t) are the self-similar solutions, namely those of the form ũ(x, t) = t^{−1/2}U(xt^{−1/2}). The function U(ξ) is called the self-similar profile. Any solution of the ordinary differential 2 Daniel B. Dix equation

U(ξ) + ξU′(ξ) − 2U(ξ)U′(ξ) + U′′(ξ) = 0

gives rise to a self-similar solution of Burgers’ equation. Integrating this equation once we obtain (c.f. [32]) a Riccati equation:

ξU(ξ) − U(ξ)^{2} + U′(ξ) = n.  (R)

The constant of integration n in (R) is chosen so that the large |ξ| asymptotics of U(ξ) will coincide with those of u_{0}. One of the main motivations for considering initial data of the type we have is to explore the relation between the large-time asymptotic behavior of u(x, t) and self-similar solutions with profiles satisfying (R).

The origin of the connection between these two is the work of Eberhard Hopf [20] (see also [16], [23]). He considered the case n = 0, and analyzed the initial-value problem and the large-time behavior using a transformation which he rediscovered, and which is now called the Cole-Hopf Transformation (see also Forsythe [17], and Cole [8]). This transformation consists of the following. First one defines

v_{0}(x) = C exp (− ∫_{0}^{x} u_{0}(y) dy) , x ∈ ℝ, C > 0.

The constant C is arbitrary. Then one solves the initial-value problem for the heat equation

v_{t} − \frac{1}{2}v_{xx} = 0,  x ∈ ℝ, t > 0,
v(x, 0) = v_{0}(x),  x ∈ ℝ.

Then finally the solution of the initial-value problem for Burgers’ equation is given by the explicit formula

u(x, t) = −\frac{v_{x}(x, t)}{v(x, t)}, x ∈ ℝ, t ⩾ 0.

This transformation from solutions of the heat equation to solutions of Burgers’ equation, when specialized to similarity solutions, becomes the standard linearization procedure for solving the Riccati equation (R): if ṽ(x, t) = t^{−n/2}V (xt^{−1/2}) is a self-similar solution of the heat equation, so that V solves the linear ordinary differential equation

nV + ξV′(ξ) + V′′(ξ) = 0,  (L)

then U = −V′/V solves (R). When n = 0 we have ∫_{−∞}^{∞} u(x, t) dx = ∫_{−∞}^{∞} u_{0}(x) dx = μ for all t ⩾ 0, hence Hopf considered the solution U of (R) such that μ = ∫_{−∞}^{∞} U(ξ) dξ = − ln[V (∞)/V (−∞)]. The solutions V of (L) with this property can be found using an integrating factor, and written as

V (ξ) = V (−∞) [1 + \frac{(e^{−μ} − 1)}{\sqrt{2π}} ∫_{−∞}^{ξ} e^{−y^{2}/2} dy] ,

where V (−∞) is nonzero, but otherwise arbitrary. If U = −V′/V is defined in terms of this V, Hopf was able to show that

lim_{t→∞} ∥t^{1/2}u(ξt^{1/2}, t) − U(ξ)∥_{W^{r,∞}(ξ)} = 0

Figure 1.

for all integers r ⩾ 0, where W^{r,s}(ℝ) is the usual Sobolev space of functions with r derivatives in L^{s}(ℝ). A detailed discussion of the self-similar solution ũ(x, t) can be found in [35].

This result of Hopf was extended to include nonzero, but small, values of n in [11]. That result was proved for initial data of the form u_{0} = u_{1} + Hu_{2}, where u_{1}, u_{2} ∈ L^{1}(ℝ), and Hϕ(x) = \frac{1}{π} p. v. ∫_{−∞}^{∞} \frac{ϕ(x−y)}{y} dy is the Hilbert transform. (The decomposition of u_{0} as u_{1} +Hu_{2} is not unique, so for simplicity in the present work we can set u_{2}(x) = \frac{n}{1+x^{2}} so that (Hu_{2})(x) = \frac{nx}{1+x^{2}} . Hence our present assumption on u_{0} is a special case of the assumption u_{0} = u_{1} + Hu_{2} made in [11].) In that work we defined μ = ∫_{−∞}^{∞} u_{1}(x) dx and η = ∫_{−∞}^{∞} u_{2}(x) dx. It was shown that if ∥u_{1}∥_{L^{1}} + ∥u_{2}∥_{L^{1}} is sufficiently small then there exists a unique global solution u(x, t) of Burgers’ equation with initial data u_{0} = u_{1} + Hu_{2}, and there exists a unique solution U(ξ) of (R) (with n = η/π) such that

μ = \frac{Û(0^{+}) + Û(0^{−})}{2} = ∫_{−∞}^{∞} [U(ξ) − \frac{nξ}{1 + ξ^{2}} ] dξ = lim_{A→∞} ∫_{−A}^{A} U(ξ) dξ;

furthermore for this U it was shown that

lim_{t→∞} ∥t^{1/2}u(ξt^{1/2}, t) − U(ξ)∥_{W^{r,s}(ξ)} = 0

for all integers r ⩾ 0 and all 1 < s ⩽ ∞. If ln(2 + |x|)^{ϵ}u_{2}(x) is integrable over x ∈ ℝ for some ϵ > 0 (such as for the particular choice u_{2}(x) = \frac{n}{1+x^{2}} we consider in this work) the above estimate was also proved [11] to hold when s = 1. The profile U 4 Daniel B. Dix (see Figure 1 for some examples) was not computed explicitly in that work, nor was the Cole-Hopf transformation involved in the proofs, since the result was actually proved for a dispersive generalization of Burgers’ equation to which the Cole-Hopf transformation does not apply.

Our main goal for this present work will be to describe in detail the leading-order large-time behavior of solutions of Burgers’ equation arising from initial data of the above type without any size restrictions on ∥u_{1}∥_{L^{1}} or on n.

1.3. Motivations for our study

The problem of extending the above results to allow large initial data is interesting for reasons which extend beyond the context of Burgers’ equation by itself. The problem belongs to the wider study of the large time behavior of solutions of nonlinear dispersive-dissipative wave equations, which have been receiving increasing attention; see [2], [3], [4], [5], [6], [7], [11], [12], [13], [15], [16], [26], [27], [28], [29], [30], [37], [38], [39], [24], [25], and the references therein. In some of these works, e.g. [2], [11], [29], [39], [24], the large-time behavior of some more general equation is found to coincide with that of a solution of Burgers’ equation. Hence a deeper study of the large time behavior of solutions of Burgers’ equation can have ramifications for other equations as well. But also, according to the classification scheme discussed in [12], the solutions of Burgers’ equation of the type considered here are said to be in the balanced case, where nonlinear and dissipative effects are both supposed to make leading-order contributions to the large-time behavior. However, this classification scheme is based on the presumption of asymptotically self-similar behavior, which so far has only been proved to hold (generally) for small initial data. Only a relatively small fraction of the above work applies to the balanced case; it is more difficult because nonlinear effects cannot be treated as small perturbations, especially when the solutions are large. In the case of Burgers’ equation, its integrable structure makes the study of its large solutions feasible, thus providing a window into the type of behavior we might expect in other equations as well. This study will provide valuable clues as to whether or not the asymptotic balance in a solution can depend merely on the size of the initial data, the other parameters (p = q = 2, γ = 0) relevant to the classification (see [12]) remaining fixed.

Correction In [12] when describing our assumptions on the initial data u_{0} we effectively claim that if u_{0} ∈ L^{1} ∩ L^{2} then |k|^{−1/2}|û_{0}(k) − û_{0}(0)χ_{[−1,1]}(k)| is square integrable for k ∈ ℝ. This claim was used to support the assertion that our assumptions on u_{0} were general enough to include all initial data from L^{1} ∩ H^{∞}. Both the claim and the resulting assertion are false, but this does not affect the correctness of any other statement or argument in the paper.

1.4. Methods

Our strategy for generating the leading order term ũ(x, t) of the large time asymptotics of u(x, t) is to find a sufficiently accurate asymptotic approximation ṽ(x, t) to the solution v(x, t) of the heat equation, and then apply the Cole-Hopf transformation, i.e. ũ(x, t) = −\frac{ṽ_{x}(x,t}{ṽ(x,t)}.

Because of our assumption on the initial data u_{0} we have

v_{0}(x) = C exp (−\frac{n}{2} ln(1 + x^{2}) − ∫_{0}^{x} u_{1}(y) dy) .

Therefore we have

v_{0}(x) ∼ C|x|^{−n}e^{−μ_{±}},  as ±x → ∞,  μ_{±} = ∫_{0}^{±∞} u_{1}(x) dx.

Hence μ = μ_{+} − μ_{−}. We choose C = exp[\frac{μ_{+}+μ_{−}}{2} ] so that the leading-order asymptotics of v_{0}(x) as |x| → ∞ depend on u_{1}(x) only through the value of μ. Thus

v_{0}(x) = (1 + x^{2})^{−n/2} exp (\frac{μ_{+} + μ_{−}}{2} − ∫_{0}^{x} u_{1}(z) dz)

= (1 + x^{2})^{−n/2} exp (−\frac{1}{2} ∫_{−∞}^{x} u_{1}(z) dz + \frac{1}{2} ∫_{x}^{∞} u_{1}(z) dz) .

The question then arises about how the large-time behavior of the solution v(x, t) of the heat equation with initial data v_{0} of the above type for arbitrary values of n is to be extracted and the errors estimated. The leading-order large-time asymptotics of the solution v(x, t) of the initial-value problem for the heat equation has been studied from a different point of view in [9], [10], [14], and [33] (Theorem 1, page 195). But those results are not sufficent for our purposes. When n > 0, v̂_{0}(k) = ∫_{−∞}^{∞} e^{−ikx}v_{0}(x) dx is an ordinary function, and its asymptotic expansion as k → 0 is known to be the source of the different terms of the large-time asymptotic expansion of v(x, t). However, v̂_{0} makes sense only as a distribution when n ⩽ 0. Since distributions are unavoidable in this study, we find that their systematic use makes the study of the Fourier transform v̂_{0} unnecessary.

Define G(ξ) = e^{−ξ^{2}/2}/\sqrt{2π} and note that

v(ξe^{s}, e^{2s}) = ∫_{−∞}^{∞} G(ξ − ξ′)v_{0}(ξ′e^{s}) dξ′,

where t = e^{2s}. Define ϑ = x\frac{d}{dx}; we consider it to act on temperate distributions. Then (e^{ϑs}v_{0})(ξ) = v_{0}(ξe^{s}). Thus the large-time behavior of v(ξt^{1/2}, t) can be understood as the large s behavior of the distribution e^{ϑs}v_{0}. This is in turn understood by expanding v_{0} in terms of the (generalized) eigendistributions of ϑ, which are the (associated) homogeneous distributions [18]. A complete expansion of v_{0} is not needed (if even possible); we only require the components present in v_{0} from the (generalized) eigenspaces with the largest eigenvalues of ϑ.

For different values of n this results in different expansions of v_{0}. In preparation for these expansions we need some definitions. Define the homogeneous function

ṽ_{0}(x) = |x|^{−n}[cosh(\frac{μ}{2}) − sinh(\frac{μ}{2}) sgn(x)] = \begin{cases}|x|^{−n}e^{−μ/2} & x > 0, \\ |x|^{−n}e^{μ/2} & x < 0, \end{cases}

which has the same large |x| asymptotics as v_{0}(x). When n < 1 this is a locally integrable function and hence a distribution; clearly ϑṽ_{0} = −nṽ_{0}. ṽ_{0} can also be 6 Daniel B. Dix defined as a distribution when n ⩾ 1. Define N = [[n]] be the greatest integer less than or equal to n, and let ϕ be a temperate test function. Then if n > 1 is not an integer we define (see page 50 of [18])

⟨ṽ_{0}, ϕ⟩ = ∫_{−∞}^{∞} ṽ_{0}(y) [ϕ(y) − ∑_{j=0}^{N−1} \frac{ϕ^{(j)}(0)}{j!}y^{j}] dy.

This is an eigendistribution of ϑ with eigenvalue −n. If n ⩾ 1 is an integer we define (again following [18])

⟨ṽ_{0}, ϕ⟩ = ∫_{−∞}^{∞} ṽ_{0}(y) [ϕ(y) − ∑_{j=0}^{n−2} \frac{ϕ^{(j)}(0)}{j!}y^{j} − \frac{ϕ^{(n−1)}(0)}{(n − 1)!}y^{n−1}H(1 − y^{2})] dy,

where H is the Heaviside function, i.e. H(x) = 1 for x > 0 and H(x) = 0 for x < 0. This is a generalized eigendistribution of ϑ with eigenvalue −n, i.e. (ϑ + n)^{2}ṽ_{0} = 0. For all integers n ⩾ 1 the operator ϑ has the (ordinary) eigendistribution δ^{(n−1)} (the (n − 1)st distributional derivative of the Dirac delta) with eigenvalue −n.

It can be shown that v_{0} has the following asymptotic eigen-expansions.

1. If n < 1 we have v_{0} = ṽ_{0} + r.

2. If n > 1 is not an integer and N = [[n]] we have

v_{0} = ∑_{j=0}^{N−1} \frac{(−1)^{j}M_{j}}{j!} δ^{(j)} + ṽ_{0} + r, M_{j} = ∫_{−∞}^{∞} x^{j}v_{0}(x) dx.

3. If n ⩾ 1 is an integer, so that n = N, we have the same expansion for v_{0} as in case (2), where ṽ_{0} has a changed definition (see above), and M_{j} for j = n − 1 has the changed definition

M_{n−1} = ∫_{−∞}^{∞} x^{n−1}[v_{0}(x) − ṽ_{0}(x)H(x^{2} − 1)] dx.

In all three cases we have e^{ns}(e^{sϑ}r) → 0 in the sense of temperate distributions as s → ∞. These expansions are called quasi-asymptotic expansions in the terminology of Vladimirov, Drozzinov, and Zavialov [33]. Our most accurate asymptotic approximation ṽ to the solution v of the heat equation will be obtained by solving the heat equation with the initial data v_{0} − r, which can be done explicitly in all cases. Because of the structure of this solution ṽ it is natural to consider three cases: n < 1, n > 1 but n ∉ Z, and n ⩾ 1 an integer.

It will also be our goal to control the relative error between t^{1/2}u(ξt^{1/2}, t) and our leading-order asymptotic approximation t^{1/2}ũ(ξt^{1/2}, t) as t → ∞. This was already done in an average sense for small |n| in [11], since the error was controlled in L^{1} but U(ξ) decays too slowly to be in L^{1}. Here we intend to strengthen those results by obtaining weighted L^{∞} estimates on the error, where the weights exactly reflect the behavior of ũ(x, t). This will demonstrate that we have found the correct leadingorder term as t → ∞, regardless of how x varies. This will be possible because we

Figure 2.

will find that the distributional expressions ⟨r, ϕ⟩ have integral formulae which are amenable to careful estimation. Naturally, this estimation will entail some further assumptions on u_{1} beyond merely that u_{1} ∈ L^{1}. We succeed in obtaining optimal estimates except for the case n = 0, where U(ξ) decays exponentially as |ξ| → ∞, and even more stringent assumptions on u_{1} would probably be required in order to control the relative error. The issue of controlling relative errors has been studied for other equations in the works [29] and [13]. The results we present here are strictly a leading-order theory. We leave the higher-order theory for later work.

Since Burgers’ equation is nonlinear, large-time asymptotics of x derivatives of u do not automatically follow from the large-time asymptotics of u. Because of the strong parabolic smoothing effect we expect such results to hold without assuming anything about the derivatives of u_{0} (take note of this aspect in the earlier results quoted above). In this work we do not strive so hard to prove this, and are content to impose restrictions directly on the derivatives of u_{0} which are commensurate with the strength of the estimates proved. When n ⩾ 1 we also fail to control the relative error in the rarefaction region (defined below) for the x derivatives of u. Since u is a nonlinear function of v, estimating ∂_{x}^{r}u entails some elaborate measures (see section 2).

1.5. Main Results

When n < 1 the leading-order term for the asymptotics of u(x, t) will be found to be self-similar, ũ(x, t) = t^{−1/2}U(xt^{−1/2}), with an explicitly determined profile U(ξ), depending on μ and n (see Theorem 1.1 below for an exact statement). As

Figure 3.

discussed in [11], this result can be interpreted to mean that the profile U(ξ) is an asymptotically stable τ-independent solution of the rescaled Burgers’ equation

φ_{τ} − \frac{1}{2}∂_{ξ}[ξφ − φ^{2} + ∂_{ξ}φ] = 0,  φ(ξ, ln t) = t^{1/2}u(ξt^{1/2}, t),  τ = ln t,

restricted to hyperplanes of initial conditions where μ and n are constant. The shapes of the profiles U(ξ) for several values of n are shown in Figure 1.

The explicit formula we find for the profile U(ξ) also makes sense when n > 1, n ∉ Z, but it has at least one pole on the real ξ axis; see Figure 2. Since the solution u(x, t) of Burgers’ equation is smooth, we have reason to suspect that in this case the profile U(ξ) is no longer asymptotically stable. Indeed, we will show (see Theorem 1.2 below for an exact statement and section 6 for discussion) that the rescaled solution φ(ξ, τ) is an asymptotic composite of two different profiles

φ(ξ, τ) ≐ \begin{cases}ξ & |ξ| ≪ \sqrt{(n − 1)τ}, \\ U(ξ) & |ξ| ≫ \sqrt{(n − 1)τ}, \end{cases}

as τ → ∞, where U(ξ) is given by the same explicit formula that we found applied to the case n < 1. U_{1}(ξ) = ξ solves (R) with n = 1, and corresponds to a rarefaction solution of Burgers’ (and inviscid Burgers’) equation, [35]. It of course has the wrong large |ξ| asymptotic behavior to be compatible with initial data of the type we are considering. Hence in the region |ξ| ∼ \sqrt{(n − 1)τ} there is a rapid transition from U_{1}(ξ) to U(ξ); see Figure 3. We will call the region |ξ| ≪ \sqrt{(n − 1)τ} the rarefaction

Large-Time Behavior for Burgers’ Equation 9

region. Our picture applies whenever t is sufficiently large so that all the poles of U(ξ) are within the rarefaction region.

The case where n ⩾ 1 is an integer is similar to the case n > 1, n ∉ Z, although some transitional peculiarities appear in the case n = 1. This case is described in detail in Theorem 1.3 below.

Our first main result is the following.

Theorem 1.1. Suppose u_{0}(x) = u_{1}(x) + \frac{nx}{1+x^{2}} , where u_{1} ∈ L^{1}(ℝ). Then there is a unique global smooth solution u(x, t) of the Burgers’ equation with initial data u_{0}, i.e. for all y ∈ ℝ we have ∫_{0}^{x} u(z, t) dz → ∫_{0}^{y} u_{0}(z) dz as x → y, t → 0^{+} (see [20]). Suppose n < 1, μ = ∫_{−∞}^{∞} u_{1}(x) dx and U(ξ) = −V′(ξ)/V (ξ), where

V (ξ) = \frac{1}{\sqrt{2π}} \{cosh(\frac{μ}{2})2^{\frac{1−n}{2}} Γ(\frac{1−n}{2})M(\frac{n}{2}, \frac{1}{2}, −\frac{ξ^{2}}{2} )
− sinh(\frac{μ}{2})2^{1− \frac{n}{2}} Γ(1 − \frac{n}{2})ξM(\frac{1+n}{2}, \frac{3}{2}, −\frac{ξ^{2}}{2} )\}.

In the above, M(a, b, z) is the confluent hypergeometric function, [1]. Suppose r ⩾ 0 is an integer and lim sup_{|x|→∞} |x|^{1+γ}|u_{1}^{(γ)}(x)| = 0 for all 0 ⩽ γ ⩽ r. Then we have the asymptotic estimate

lim_{t→∞} ∥(1 + |ξ|)^{r+1}∂_{ξ}^{r}[t^{1/2}u(ξt^{1/2}, t) − U(ξ)]∥_{L^{∞}(ξ)} = 0.

Remarks The corresponding solution v(x, t) of the heat equation and the solution V (ξ) of (L) assume only positive real values, so the functions u(x, t) and U(ξ) are well-defined and smooth. For all integers r ⩾ 0 we have U^{(r)}(ξ) = (−1)^{r}r!ξ^{−1−r}[n + o(1)] as |ξ| → ∞. Weighted L^{∞} decay estimates on u(x, t) follow immediately from the above theorem:

lim_{t→∞} t^{(r+1)/2}∥(1 + |x|t^{−1/2})^{r+1}∂_{x}^{r}u(x, t)∥_{L^{∞}(x)} = ∥(1 + |ξ|)^{r+1}U^{(r)}(ξ)∥_{L^{∞}(ξ)} < ∞.

These estimates are clearly sharp whenever n ≠ 0. See also Theorem 1.4 below. Our next main result is the following.

Theorem 1.2. Suppose n > 1 and n ∉ Z. Suppose r ⩾ 0, u_{0}, u and μ are as in Theorem 1. Define G(ξ) = e^{−ξ^{2}/2}/\sqrt{2π} and ũ(x, t) = −ṽ_{x}(x, t)/ṽ(x, t), where

v_{0}(y) = (1 + y^{2})^{−n/2} exp (−\frac{1}{2} ∫_{−∞}^{y} u_{1}(z) dz + \frac{1}{2} ∫_{y}^{∞} u_{1}(z) dz) ,
ṽ(x, t) = t^{−1/2}G(xt^{−1/2}) ∫_{−∞}^{∞} v_{0}(y) dy + t^{−n/2}V (xt^{−1/2}),

and V is as in Theorem 1.1. Define the weight function

z_{r+1}(x, t) = \frac{1}{[t^{1/2}(1 + |ξ|)]^{r+1}} \frac{t^{(n−1)/2}e^{−ξ^{2}/2}(1 + |ξ|)^{n+2r+2} + 1}{t^{(n−1)/2}e^{−ξ^{2}/2}(1 + |ξ|)^{n} + 1},

where ξ = xt^{−1/2}. Then we have the asymptotic estimate

lim_{t→∞} ∥\frac{∂_{x}^{r}[u(x, t) − ũ(x, t)]}{z_{r+1}(x, t)}∥_{L^{∞}(x)} = 0.

10 Daniel B. Dix

Remarks The weight function z_{1} exactly reflects the behavior of ũ(x, t) and enables us to prove sharp L^{∞} decay estimates on u. See Theorem 4 below. The weight function z_{r+1} for r ⩾ 1 faithfully reflects the behavior of the quotient ∂_{x}^{r+1}ṽ/ṽ, and hence (see section 5) provides an upper bound for the behavior of ∂_{x}^{r}ũ. However, in the rarefaction region ∂_{x}^{r}ũ decays more rapidly than this upper bound, and hence we have not proved that ∂_{x}^{r}ũ is the leading-order description for ∂_{x}^{r}u in that region. This phenomenon is the result of certain cancellations which take place in the nonlinear combinations

ũ_{x} = −\frac{ṽ_{xx}}{ṽ} +(\frac{ṽ_{x}}{ṽ} )^{2} ,  ũ_{xx} = −\frac{ṽ_{xxx}}{ṽ} +3 (\frac{ṽ_{x}}{ṽ} ) (\frac{ṽ_{xx}}{ṽ} )−2 (\frac{ṽ_{x}}{ṽ} )^{3} ,  etc.

Leading-order descriptions of u_{x} and u_{xx}, etc. in the rarefaction region can be obtained by finding the higher-order terms of the large-time asymptotic expansion of u and by proving the error estimates appropriate for such more accurate asymptotic approximations. Such an enterprise is quite feasible along the lines of what we have presented here.

The third case is described in the following.

Theorem 1.3. Suppose r ⩾ 0, u_{0}, u, μ, v_{0} and G are as in Theorems 1.1 and 1.2. In addition to the assumptions on u_{1} made in Theorem 1.1 suppose that

∫_{|y|⩾1} \frac{1}{|y|} |∫_{y}^{sgn(y)∞} u_{1}(z) dz| dy < ∞, ∫_{|y|⩾1} |y|^{j}|u_{1}^{(j)}(y)| dy < ∞, 0 ⩽ j ⩽ r.

If n ⩾ 1 is an integer, define

L_{n} = (−1)^{n}e^{\frac{μ}{2}} − e^{− \frac{μ}{2}} = \begin{cases}2 sinh(\frac{μ}{2}) & n even \\ −2 cosh(\frac{μ}{2}) & n odd \end{cases} ,

N_{n} = (−1)^{n}e^{\frac{μ}{2}} + e^{− \frac{μ}{2}} = \begin{cases}2 cosh(\frac{μ}{2}) & n even \\ −2 sinh(\frac{μ}{2}) & n odd \end{cases} ,

V (ξ) = \frac{(−1)^{n−1}}{(n − 1)!} [L_{n}(ln |D| G^{(n−1)})(ξ) + \frac{π}{2} N_{n}(HG^{(n−1)})(ξ)].

In the above H is the Hilbert transform, and ln |D|G is defined in terms of its Fourier transform, i.e. (ln |D|G) (k) = e^{−k^{2}/2} ln |k|. Define ξ = xt^{−1/2}.

1. If n = 1 suppose that

M_{0} = ∫_{−∞}^{∞}\{v_{0}(x) − H(x^{2} − 1)|x|^{−1} [cosh (\frac{μ}{2}) − sinh (\frac{μ}{2}) sgn(x)]\} dx,
ṽ(x, t) = \frac{1}{t^{1/2}} \{[M_{0} + cosh(\frac{μ}{2})(−2γ + ln t)]G(ξ) + V (ξ)\},

z_{r+1}(x, t) = \frac{1}{[t^{1/2}(1 + |ξ|)]^{r+1}} \frac{ln(t)(1 + |ξ|)^{3+2r}e^{−ξ^{2}/2} + 1}{ln(t)(1 + |ξ|)e^{−ξ^{2}/2} + 1}.

In the above, H(x) is the Heaviside function, and γ is the Euler constant.

2. If n ⩾ 2 is an integer, suppose that

ṽ(x, t) = t^{−1/2}G(ξ) ∫_{−∞}^{∞} v_{0}(x) dx + t^{−n/2}V (ξ),

z_{r+1}(x, t) = \frac{1}{[t^{1/2}(1 + |ξ|)]^{r+1}} \frac{t^{(n−1)/2}e^{−ξ^{2}/2}(1 + |ξ|)^{n+2r+2} + 1}{t^{(n−1)/2}e^{−ξ^{2}/2}(1 + |ξ|)^{n} + 1}.

Define ũ(x, t) = −ṽ_{x}(x, t)/ṽ(x, t). Then we have the asymptotic estimate

lim_{t→∞} ∥\frac{∂_{x}^{r}[u(x, t) − ũ(x, t)]}{z_{r+1}(x, t)}∥_{L^{∞}(x)} = 0.

Remarks The extra assumptions on u_{1} are sufficient to cause the integral defining M_{0} in (1) to converge. When n ⩾ 2 is an integer, the definition of z_{r+1} is identical to that made in the case n > 1 not an integer. The definition of ṽ looks the same, but the definition of V (ξ) is different, depending on whether n > 1 is an integer or not.

Our final result gives L^{∞} decay rates for the solution of Burgers’ equation and its x derivatives.

Theorem 1.4. Suppose u is one of the solutions of Burgers’ equation discussed in Theorems 1.1, 1.2, or 1.3, and r ⩾ 0 is an integer. Then we have

∥∂_{x}^{r}u(·, t)∥_{L^{∞}} = \begin{cases}O (\frac{1}{t^{(r+1)/2}} ) & n < 1, t ⩾ 1, \\ O ([\frac{ln(ln t)}{t}]^{(r+1)/2}) & n = 1, t ⩾ e^{2}, \\ O ([\frac{ln t}{t}]^{(r+1)/2}) & n > 1, t ⩾ e. \end{cases}

When |μ| + |n| > 0, and when n ⩾ 1 for r = 0, these decay rates are sharp, since

lim inf_{t→∞} t^{(r+1)/2}∥∂_{x}^{r}u(·, t)∥_{L^{∞}} > 0,  n < 1,
lim inf_{t→∞} [\frac{t}{ln(ln t)}]^{1/2} ∥u(·, t)∥_{L^{∞}} > 0,  n = 1,

lim inf_{t→∞} [\frac{t}{ln t}]^{1/2} ∥u(·, t)∥_{L^{∞}} > 0,  n > 1.

Remarks When n ⩾ 1 the above decay estimates on the derivatives of u are probably sharp, but the final resolution of this issue must wait for study of the higher-order asymptotic theory mentioned in the remark after Theorem 1.2.

Our assumptions on u_{0} imply the asymptotic eigen-expansion

u_{0}(x) = μδ(x) + n p. v. \frac{1}{x} + r(x),

where the first two terms are eigendistributions of ϑ with eigenvalue −1 and e^{s}(e^{sϑ}r) tends to 0 in the sense of tempered distributions as s → ∞. Despite the fact that Burgers’ equation is balanced, the linear pattern persists in the nonlinear case for 12 Daniel B. Dix small initial data in that the large-time behavior is asymptotically self-similar with a profile completely determined by the parameters μ and n. However, this work shows that the linear pattern is broken for large solutions and new essentially nonlinear phenomena appear in the large-time behavior. The phenomenon of the emergence of an asymptotic hybrid between two partially stable modes is also difficult to explain by way of analogy with finite dimensional dynamical systems. It appears to be an essentially infinite dimensional bifurcation in the large-time behavior.

1.6. Plan for the Remainder of the Paper

In section 2 we handle the nonlinear estimates involved in the proofs of Theorems 1.1, 1.2, and 1.3. This can be done in a uniform manner. In sections 3, 4, and 5 we present the linear estimates required to finish the proofs of Theorems 1.1, 1.2, and 1.3 respectively. In section 6 we prove Theorem 1.4 and give an asymptotic description of the solution u in the case n ⩾ 1 in different regions.

2. Nonlinear Estimates

In this section we will show how to prove Theorems 1.1, 1.2, and 1.3, which are all weighted decay estimates of the quantity ∂_{x}^{m}(u−ũ), where u = −v_{x}/v is the solution of Burgers’ equation under study, and ũ = −ṽ_{x}/ṽ is its large-time asymptotic approximation. We will do this assuming the validity of certain weighted asymptotic estimates on the quantities 1/ṽ, ∂_{x}^{γ} ṽ, and ∂_{x}^{γ}(v −ṽ). The exact form of the weighted estimates on these quantities and their proofs differ between the three cases, n < 1, n > 1 not an integer, and n ⩾ 1 an integer. Thus in section 3, 4, and 5 we will prove those estimates separately. But the nonlinear estimates for all three cases can be treated in an unified manner, and so that will be the topic of this section.

We have for every integer r ⩾ 0 that

∂_{x}^{r}(u − ũ) = −∂_{x}^{r+1} ln(v/ṽ) = −∂_{x}^{r+1} ln (1 + \frac{v − ṽ}{ṽ}) .

First we must expand this to see what combinations of powers of v − ṽ and ṽ and their derivatives appear. For this expansion we will employ a multidimensional extension of Faà di Bruno’s formula (see page 823 of [1] and [19]). To give a general statement of this formula we use the standardized multi-index notation of Laurent Schwartz [21], i.e. if m ⩾ 1 is an integer, M = {0, 1, 2, … }, α = (α_{1}, …, α_{m}) ∈ M^{m} is a multi-index, and x = (x_{1}, …, x_{m}) ∈ ℝ^{m}, then ∂^{α} = ∂_{x_{1}}^{α_{1}} ⋯ ∂_{x_{m}}^{α_{m}}, α! = α_{1}! ⋯ α_{m}!, x^{α} = x_{1}^{α_{1}} ⋯ x_{m}^{α_{m}}, and |α| = α_{1} + ⋯ + α_{m}.

Lemma 2.1. Suppose f is a smooth ℝ^{n}-valued function defined on some open subset of ℝ^{m}, and g is a smooth ℝ-valued function defined on an open subset of ℝ^{n}. Let P = M^{m} \ {(0, …, 0)}. Then for every α ∈ M^{m} and every x ∈ ℝ^{m} for which g(f(x)) is defined we have

∂^{α}(g ∘ f)(x) = α! ∑_{β∈M^{n}}(∂^{β}g)(f(x)) ∑_{a} ∏_{γ∈P} \frac{1}{a(γ)!} [\frac{(∂^{γ}f)(x)}{γ!}]^{a(γ)} ,

where the sum in a is over all mappings a P → M^{n} satisfying the conditions

∑_{γ∈P} a(γ) = β,  ∑_{γ∈P} |a(γ)|γ = α.

Remarks As usual, empty sums are understood to have the value zero. Hence the sum in β need only be taken over those β ∈ M^{n} such that |β| ⩽ |α|. Also, the mappings a satisfying the two conditions above will always be supported on the set of those γ such that γ ⩽ α. The values of such mappings will always satisfy a(γ) ⩽ β. Thus the sum in a is always a finite sum. We are also making the convention that 0^{0} = 1. Thus the apparently infinite product over γ is in fact only a finite product.

If we apply this Lemma with m = 1, n = 2, we get

∂_{x}^{r+1}[g(f_{1}(x), f_{2}(x))] = (r + 1)! ∑_{n_{1}⩾0,n_{2}⩾0,integers 1⩽n_{1}+n_{2}⩽r+1}(∂_{y_{1}}^{n_{1}}∂_{y_{2}}^{n_{2}}g)(f_{1}(x), f_{2}(x))·

· ∑_{(a_{1},a_{2})} ∏_{γ=1}^{r+1} \frac{1}{a_{1}(γ)!a_{2}(γ)!} (\frac{f_{1}^{(γ)}(x)}{γ!})^{a_{1}(γ)} (\frac{f_{2}^{(γ)}(x)}{γ!})^{a_{2}(γ)}

where the sum in (a_{1}, a_{2}) is over all pairs of functions defined for integers 1 ⩽ γ ⩽ r + 1, with integer values such that 0 ⩽ a_{1}(γ) ⩽ n_{1}, 0 ⩽ a_{2}(γ) ⩽ n_{2}, and

∑_{γ=1}^{r+1} a_{1}(γ) = n_{1},  ∑_{γ=1}^{r+1} a_{2}(γ) = n_{2},  ∑_{γ=1}^{r+1} γ[a_{1}(γ) + a_{2}(γ)] = r + 1.

We will apply this formula with g(y_{1}, y_{2}) = ln(1 + \frac{y_{1}}{y_{2}} ), f_{1} = v − ṽ, and f_{2} = ṽ. To compute the partial derivatives (∂_{y_{1}}^{n_{1}}∂_{y_{2}}^{n_{2}}g)(y_{1}, y_{2}) in this case we apply Lemma 1.1 again, this time with m = 2, n = 1, g(y) = ln(y), f(x_{1}, x_{2}) = 1+ \frac{x_{1}}{x_{2}} , to obtain (after some manipulation)

∂_{y_{1}}^{n_{1}}∂_{y_{2}}^{n_{2}} ln (1 + \frac{y_{1}}{y_{2}} ) = \frac{1}{y_{2}^{n_{1}+n_{2}}} ∑_{j=max\{1,n_{1}\}}^{n_{1}+n_{2}} \frac{C(j, n_{1}, n_{2})}{(1 + \frac{y_{1}}{y_{2}} )^{j}} (\frac{y_{1}}{y_{2}} )^{j−n_{1}} .

The constant C(j, n_{1}, n_{2}) in the above is defined by

C(j, n_{1}, n_{2}) = (−1)^{n_{2}+j−1}(j − 1)!n_{1}!n_{2}! ∑_{b} [ ∏_{γ_{2}=1}^{n_{2}} \frac{1}{b(0, γ_{2})!}] [ ∏_{γ_{2}=0}^{n_{2}} \frac{1}{b(1, γ_{2})!}] ,

where the sum in b is over all integer valued functions b(γ_{1}, γ_{2}), defined for integer arguments 0 ⩽ γ_{1} ⩽ 1, 0 ⩽ γ_{2} ⩽ n_{2}, γ_{1} + γ_{2} > 0, such that 0 ⩽ b(γ_{1}, γ_{2}) ⩽ j and

∑_{γ_{2}=1}^{n_{2}} b(0, γ_{2})+ ∑_{γ_{2}=0}^{n_{2}} b(1, γ_{2}) = j, ∑_{γ_{2}=0}^{n_{2}} b(1, γ_{2}) = n_{1}, ∑_{γ_{2}=1}^{n_{2}} γ_{2}[b(0, γ_{2})+b(1, γ_{2})] = n_{2}.

Therefore we have proved the following:

Lemma 2.2.

∂_{x}^{r}(u − ũ) = − ∑_{n_{1}⩾0,n_{2}⩾0,integers 1⩽n_{1}+n_{2}⩽r+1} \frac{(r + 1)!}{ṽ^{n_{1}+n_{2}}} [ ∑_{j=max\{1,n_{1}\}}^{n_{1}+n_{2}} \frac{C(j, n_{1}, n_{2})}{(1 + \frac{v−ṽ}{ṽ})^{j}} (\frac{v − ṽ}{ṽ})^{j−n_{1}} ] ·

· ∑_{(a_{1},a_{2})} ∏_{γ=1}^{r+1} \frac{1}{a_{1}(γ)!a_{2}(γ)!} (\frac{∂_{x}^{γ}(v − ṽ)}{γ!})^{a_{1}(γ)} (\frac{∂_{x}^{γ} ṽ}{γ!})^{a_{2}(γ)} ,

where the sum in (a_{1}, a_{2}) is as before.

For our weighted estimates we will use the weight function w_{γ}(x, t), which will faithfully reflect the behavior in x and t of ∂_{x}^{γ} ṽ(x, t), and the weight function z_{θ}(x, t) = w_{θ}(x, t)/w_{0}(x, t), which will reflect the behavior of ∂_{x}^{θ} ln ṽ. These functions will have different definitions in the three cases we will discuss in detail later. The main property of these weight functions that we need to assume is the following.

Interpolation Inequality For all nonnegative integers r, a(1), …, a(r+1), such that ∑_{γ=1}^{r+1} γa(γ) = r + 1, there exists a constant C > 0 such that for all x ∈ ℝ and t ⩾ 1 we have

∏_{γ=1}^{r+1} z_{γ}(x, t)^{a(γ)} ⩽ Cz_{r+1}(x, t).

We will assume the following three linear estimates have been established.

Linear Estimates

1.

sup_{t⩾1} ∥\frac{w_{0}(x, t)}{ṽ(x, t)}∥_{L^{∞}(x)} < ∞.

2. For all γ ⩾ 0 an integer we have that

sup_{t⩾1} ∥\frac{∂_{x}^{γ} ṽ(x, t)}{w_{γ}(x, t)}∥_{L^{∞}(x)} < ∞.

3. For all γ ⩾ 0 an integer we have that

lim_{t→∞} ∥\frac{∂_{x}^{γ}[v(x, t) − ṽ(x, t)]}{w_{γ}(x, t)}∥_{L^{∞}(x)} = 0.

Lemma 2.3. Lemma 1.3 Under the assumptions listed above we have for all integers r ⩾ 0

lim_{t→∞} ∥\frac{∂_{x}^{r}[u(x, t) − ũ(x, t)]}{z_{r+1}(x, t)}∥_{L^{∞}(x)} = 0.

Proof By the linear estimates (3) (γ = 0) and (1) we have

0 ⩽ lim sup_{t→∞} sup_{x∈ℝ} |\frac{v − ṽ}{ṽ}| ⩽ sup_{t⩾1} ∥\frac{w_{0}(x, t)}{ṽ(x, t)}∥_{L^{∞}(x)} lim sup_{t→∞} ∥\frac{v(x, t) − ṽ(x, t)}{w_{0}(x, t)}∥_{L^{∞}(x)} = 0.

Thus the quantity

K(t; n_{1}, n_{2}) = sup_{x∈ℝ} | ∑_{j=max\{1,n_{1}\}}^{n_{1}+n_{2}} \frac{C(j, n_{1}, n_{2})}{(1 + \frac{v−ṽ}{ṽ})^{j}} (\frac{v − ṽ}{ṽ})^{j−n_{1}} |

is bounded independently of t for t sufficiently large, and it tends to zero as t → ∞ if n_{1} = 0. By Lemma 1.2, linear estimate (1) and the interpolation estimate we have

|\frac{∂_{x}^{r}[u − ũ]}{z_{r+1}} |

⩽ C ∑_{(n_{1},n_{2})} \frac{w_{0}^{n_{1}+n_{2}}}{|ṽ|^{n_{1}+n_{2}}} K(t; n_{1}, n_{2}) ∑_{(a_{1},a_{2})} \frac{1}{z_{r+1}} ∏_{γ=1}^{r+1} |\frac{∂_{x}^{γ}(v − ṽ)}{w_{0}} |^{a_{1}(γ)} |\frac{∂_{x}^{γ} ṽ}{w_{0}} |^{a_{2}(γ)}

⩽ C′ ∑_{(n_{1},n_{2})} K(t; n_{1}, n_{2}) ∑_{(a_{1},a_{2})} ∏_{γ=1}^{r+1} |\frac{∂_{x}^{γ}(v − ṽ)}{z_{γ}w_{0}} |^{a_{1}(γ)} |\frac{∂_{x}^{γ} ṽ}{z_{γ}w_{0}} |^{a_{2}(γ)}

⩽ C′ ∑_{(n_{1},n_{2})} K(t; n_{1}, n_{2}) ∑_{(a_{1},a_{2})} ∏_{γ=1}^{r+1} ∥\frac{∂_{x}^{γ}(v − ṽ)}{w_{γ}} ∥_{L^{∞}}^{a_{1}(γ)} ∥\frac{∂_{x}^{γ} ṽ}{w_{γ}} ∥_{L^{∞}}^{a_{2}(γ)} .

a_{1}(γ) = 0 for all γ = 1, …, r + 1, if and only if n_{1} = 0. By the linear estimate (2) the sum in (a_{1}, a_{2}) contained in terms where n_{1} = 0 is bounded independently of t, and hence its product with K(t; 0, n_{2}) tends to zero as t → ∞. For terms with n_{1} > 0 we must have a_{1}(γ) > 0 for some 1 ⩽ γ ⩽ r + 1, and so such terms also tend to zero as t → ∞ by linear estimates (2) and (3).

3. Linear Estimates for the Case n < 1

The purpose of this section is first of all to explicitly compute the asymptotic approximation ṽ to v which is appropriate for the case n < 1. The weight functions w_{γ} and z_{γ} will be defined, and the Interpolation Estimate from section 2 verified. Then we will prove the three Linear Estimates from section 2, completing the proof of Theorem 1.1.

The solution ṽ(x, t) of the heat equation with the homogeneous initial data ṽ_{0} is self-similar:

ṽ(x, t) = \frac{1}{\sqrt{2πt}} ∫_{−∞}^{∞} e^{− \frac{(x−y)^{2}}{2t}} ṽ_{0}(y) dy = t^{−n/2}V (xt^{−1/2}),

where

V (ξ) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} e^{−(ξ−y)^{2}/2}ṽ_{0}(y) dy

= \frac{1}{\sqrt{2π}} e^{−ξ^{2}/2}[e^{−μ/2} ∫_{0}^{∞} e^{−y^{2}/2+yξ}y^{−n} dy

+ e^{μ/2} ∫_{0}^{∞} e^{−y^{2}/2−yξ}y^{−n} dy].

16 Daniel B. Dix This is clearly an explicit special function, well-defined, positive and smooth if n < 1, but we would like to express it in terms of Confluent Hypergeometric functions. For this we employ the following expression:

e^{−ξ^{2}/2} ∫_{0}^{∞} e^{−y^{2}/2+yξ}y^{−n} dy
= 2^{− \frac{n+1}{2}} Γ(\frac{1−n}{2})M(\frac{n}{2}, \frac{1}{2}, −\frac{1}{2}ξ^{2}) + 2^{− \frac{n}{2}} Γ(1 − \frac{n}{2})ξM(\frac{1+n}{2}, \frac{3}{2}, −\frac{1}{2}ξ^{2}).

This can be derived using an integral representation of a Parabolic Cylinder function (19.5.3, page 687, [1]), an expression of Parabolic Cylinder functions in terms of Confluent Hypergeometric functions (19.12.3, page 691, [1]), the duplication formula for the Gamma function, and the Kummer transformation M(a, b, z) = e^{z}M(b − a, b, −z). Inserting this into the expression for V yields

V (ξ) = \frac{1}{\sqrt{2π}} \{cosh(\frac{μ}{2})2^{\frac{1−n}{2}} Γ(\frac{1−n}{2})M(\frac{n}{2}, \frac{1}{2}, −\frac{ξ^{2}}{2} )
− sinh(\frac{μ}{2})2^{1− \frac{n}{2}} Γ(1 − \frac{n}{2})ξM(\frac{1+n}{2}, \frac{3}{2}, −\frac{ξ^{2}}{2} )\}.

The right-hand-side makes sense for all n not a positive integer. When n = 0 this reduces to the expression we wrote down in the introduction provided we take V (−∞) = e^{μ/2}, since (see 13.6.19, page 509 of [1])

M(0, \frac{1}{2}, −\frac{ξ^{2}}{2} ) = 1,  and  ξM(\frac{1}{2}, \frac{3}{2}, −\frac{ξ^{2}}{2} ) = ∫_{0}^{ξ} e^{−y^{2}/2} dy.

We also have an expression in terms of Parabolic Cylinder functions (see chapter 19 of [1]), which holds whenever n is not a positive integer:

V (ξ) = e^{−μ/2}\sqrt{\frac{π}{2}} e^{−ξ^{2}/4}V (\frac{1}{2}−n, ξ)+\frac{Γ(1 − n)}{\sqrt{2π}}[e^{μ/2}−cos(πn)e^{−μ/2}]e^{−ξ^{2}/4}U(\frac{1}{2}−n, ξ).

When n ⩽ 0 is an integer, we have the special expression

V (ξ) = e^{−μ/2}He_{−n}^{∗}(ξ) + \frac{[e^{μ/2} − cos(πn)e^{−μ/2}]}{\sqrt{2π}} ∫_{ξ}^{∞}(y − ξ)^{−n}e^{−y^{2}/2} dy,

where He_{−n}^{∗}(ξ) = e^{−ξ^{2}/2}(\frac{d}{dξ})^{−n}e^{ξ^{2}/2} are the Hermite polynomials, with purely imaginary roots. In particular, when n is an even negative integer and μ = 0, we have V (ξ) = He_{−n}^{∗}(ξ), so that U = −V′/V is a rational function (see [22]).

Using the known asymptotics of the confluent hypergeometric function (see 13.1.5, page 504 of [1]) we have V (ξ) = (±ξ)^{−n}e^{∓μ/2}[1+o(1)] as ±ξ → ∞, | arg(±ξ)| < π/4. Since V (ξ) is an entire function of ξ, this asymptotic relation also holds (see Theorem 4.2, page 9 of [31]) when differentiated with respect to ξ any number of times, i.e. for every integer γ ⩾ 0 we have

V^{(γ)}(ξ) = \frac{e^{∓μ/2}}{(±ξ)^{n}(−ξ)^{γ}} [\frac{Γ(n + γ)}{Γ(n)} + o(1)] , as ξ → ∞, | arg(±ξ)| ⩽ π/4 − ϵ.

When γ ⩾ 1 this statement gives the leading-order term provided n is not an integer satisfying n ⩽ 0 < n + γ. When n takes on one of these exceptional values, the

Large-Time Behavior for Burgers’ Equation 17

leading-order term we have written vanishes, and it holds that V^{(γ)}(ξ) = o(|ξ|^{−n−γ}) as |ξ| → ∞. If n ⩽ 0 is an integer and also n + γ ⩽ 0 then Γ(n + γ)/Γ(n) = (n + γ − 1) · · · · · (n + 1) · n, as can be obtained by the functional equation Γ(z + 1) = zΓ(z) and analytic continuation in the complex variable n. Therefore we have

U(ξ) = −\frac{V′(ξ)}{V (ξ)} = \frac{1}{ξ} [\frac{n + o(1)}{1 + o(1)}] = \frac{1}{ξ}[n + o(1)], as ξ → ∞, | arg(±ξ)| ⩽ π/4 − ϵ.

Again by Theorem 4.2, page 9 of [31] we have for all γ ⩾ 0

U^{(γ)}(ξ) = \frac{(−1)^{γ}γ!}{ξ^{1+γ}}[n + o(1)], as ξ → ∞, | arg(±ξ)| ⩽ π/4 − ϵ.

This proves an assertion made in the Remark after Theorem 1.1. This argument works for all real values of n except positive integer values, where in the expression for V (ξ) either Γ(\frac{1−n}{2}) or Γ(1 − \frac{n}{2}) does not make sense.

Since ∂_{x}^{γ} ṽ(x, t) = t^{−(n+γ)/2}V^{(γ)}(xt^{−1/2}), we define

w_{γ}(x, t) = \frac{1}{[t^{1/2}(1 + |xt^{−1/2}|)]^{n+γ}} , x ∈ ℝ, t ⩾ 1.

Thus

z_{γ}(x, t) = \frac{w_{γ}(x, t)}{w_{0}(x, t)} = \frac{1}{[t^{1/2}(1 + |xt^{−1/2}|)]^{γ}} , x ∈ ℝ, t ⩾ 1.

The weight function z_{r+1} is a faithful reflection of the behavior in both the x and the t variables of ∂_{x}^{r}ũ(x, t) = t^{−(r+1)/2}U^{(r)}(xt^{−1/2}) except when n = 0.

It is immediately verified that the “Interpolation Inequality” written down in the previous section is in fact an equality with constant C = 1 under these definitions of z_{γ}.

From the integral representation of V we wrote down above it is clear that V (ξ) > 0 for all ξ ∈ ℝ. Hence

\frac{w_{0}(x, t)}{ṽ(x, t)} = \frac{1}{t^{n/2}(1 + |ξ|)^{n}t^{−n/2}V (ξ)} = \frac{1}{(1 + |ξ|)^{n}V (ξ)}

is bounded independently of ξ = xt^{−1/2}. This proves Linear Estimate (1). In a similar manner we have

\frac{∂_{x}^{γ} ṽ(x, t)}{w_{γ}(x, t)} = [t^{1/2}(1 + |ξ|)]^{n+γ}t^{−(n+γ)/2}V^{(γ)}(ξ) = (1 + |ξ|)^{n+γ}V^{(γ)}(ξ),

which is bounded independently of ξ ∈ ℝ. This proves Linear Estimate (2).

It remains then to prove linear estimate (3), which is the heart of the matter. Recall that the Hermite polynomial He_{γ}(x), with real roots, is defined such that

\frac{d^{γ}}{dx^{γ}} e^{−x^{2}/2} = (−1)^{γ}He_{γ}(x)e^{−x^{2}/2}.

Thus we have

\frac{∂_{x}^{γ}[v(x, t) − ṽ(x, t)]}{w_{γ}(x, t)}

= \frac{[t^{1/2}(1 + |ξ|)]^{n+γ}}{\sqrt{2π}t^{(1+γ)/2}} ∫_{−∞}^{∞}(−1)^{γ}He_{γ} (\frac{x−y}{\sqrt{t}} ) e^{− \frac{1}{2} ( \frac{x−y}{\sqrt{t}} )^{2}}[v_{0}(y) − ṽ_{0}(y)] dy

= \frac{(1 + |ξ|)^{n+γ}}{\sqrt{2π}} ∫_{−∞}^{∞}(−1)^{γ}He_{γ}(ξ − y)e^{− \frac{1}{2} (ξ−y)^{2}}t^{n/2}[v_{0}(y\sqrt{t}) − ṽ_{0}(y\sqrt{t})] dy

= I_{1} + I_{2} + I_{3},

where I_{1} is the integral over −1 ⩽ y ⩽ 1, I_{2} is the integral over 1 ⩽ y < ∞, and I_{3} is the integral over −∞ < y ⩽ −1. We will bound each of these separately.

To show that I_{1} → 0 as t → ∞ we use Peetre’s inequality (1 + |ξ|)^{n+γ} ⩽ C(1 + |ξ − y|)^{|n+γ|}(1 + |y|)^{n+γ}, where C > 0 is independent of ξ or y. Also, using the expression for ṽ_{0}(y) given in the introduction we have that there are constants C′ > 0, C′′ > 0, independent of x ∈ ℝ and t ⩾ 1, such that

|I_{1}| ⩽ C′ ∫_{−1}^{1}(1 + |ξ − y|)^{|n+γ|}|He_{γ}(ξ − y)|e^{− \frac{1}{2} (ξ−y)^{2}}t^{n/2}[v_{0}(y\sqrt{t}) − ṽ_{0}(y\sqrt{t})] dy

⩽ C′′ ∫_{−1}^{1}[(t^{−1} + y^{2})^{−n/2} exp(−\frac{1}{2} ∫_{−∞}^{y\sqrt{t}} u_{1}(z) dz + \frac{1}{2} ∫_{y\sqrt{t}}^{∞} u_{1}(z) dz)

− |y|^{−n}e^{− sgn(y)μ/2}] dy

Since n < 1, this quantity tends to zero as t → ∞ by the dominated convergence theorem.

We will show that I_{2} → 0 as t → ∞. The proof for I_{3} is very similar. If we integrate by parts γ times (γ ⩽ r + 1) we get the following:

∫_{1}^{∞}He_{γ}(ξ − y)e^{− \frac{1}{2} (ξ−y)^{2}}t^{n/2}[v_{0}(y\sqrt{t}) − ṽ_{0}(y\sqrt{t})] dy

= ∑_{j=1}^{γ}(−1)^{j−1} \frac{d^{γ−j}}{dy^{γ−j}} [e^{− \frac{1}{2} (ξ−y)^{2}}] t^{(n+j−1)/2}[v_{0}^{(j−1)}(y\sqrt{t}) − ṽ_{0}^{(j−1)}(y\sqrt{t})]|_{y=1}^{y=∞}

+ (−1)^{γ} ∫_{1}^{∞} e^{− \frac{1}{2} (ξ−y)^{2}}t^{(n+γ)/2}[v_{0}^{(γ)}(y\sqrt{t}) − ṽ_{0}^{(γ)}(y\sqrt{t})] dy

Let these terms be denoted by J_{1}, …, J_{γ}, J. Our strategy for dealing with J is to use Peetre’s inequality and Hölder’s inequality

(1 + |ξ|)^{n+γ}|J|

⩽ C ∫_{1}^{∞} e^{− \frac{1}{2} (ξ−y)^{2}}(1 + |ξ − y|)^{|n+γ|}|yt^{1/2}|^{n+γ} |v_{0}^{(γ)}(y\sqrt{t}) − ṽ_{0}^{(γ)}(y\sqrt{t})| dy

⩽ C sup_{ξ∈ℝ} ∫_{1}^{∞} e^{− \frac{1}{2} (ξ−y)^{2}}(1 + |ξ − y|)^{|n+γ|} dy · sup_{y⩾\sqrt{t}} y^{n+γ} |v_{0}^{(γ)}(y) − ṽ_{0}^{(γ)}(y)|

Hence we must show that the supremum over y ⩾ \sqrt{t} tends to zero as t → ∞. As we have seen earlier

v_{0}(y) − ṽ_{0}(y) = (1 + y^{2})^{−n/2} exp(−\frac{1}{2} ∫_{−∞}^{y} u_{1}(z) dz + \frac{1}{2} ∫_{y}^{∞} u_{1}(z) dz)
− |y|^{−n}e^{− sgn(y)μ/2}.

So we have (for 1 ⩽ y < ∞)

v_{0}^{(γ)}(y) − ṽ_{0}^{(γ)}(y)

= \frac{d^{γ}}{dy^{γ}} [(1 + y^{2})^{−n/2}] exp(−\frac{1}{2} ∫_{−∞}^{y} u_{1}(z) dz + \frac{1}{2} ∫_{y}^{∞} u_{1}(z) dz)

− [∏_{h=1}^{γ}(−n − h + 1)]y^{−n−γ}e^{−μ/2}

+ ∑_{l=1}^{γ} \binom{γ}{l}\frac{d^{γ−l}}{dy^{γ−l}} [(1 + y^{2})^{−n/2}] \frac{d^{l}}{dy^{l}} exp(−\frac{1}{2} ∫_{−∞}^{y} u_{1}(z) dz + \frac{1}{2} ∫_{y}^{∞} u_{1}(z) dz).

It is not difficult to show that

sup_{y⩾1} y^{n+γ} |\frac{d^{γ}}{dy^{γ}} [(1 + y^{2})^{−n/2}]| < ∞

and that we have

lim_{t→∞} sup_{y⩾\sqrt{t}} y^{n+γ} |\frac{d^{γ}}{dy^{γ}} [(1 + y^{2})^{−n/2}] exp(−\frac{1}{2} ∫_{−∞}^{y} u_{1}(z) dz + \frac{1}{2} ∫_{y}^{∞} u_{1}(z) dz)

− [∏_{h=1}^{γ}(−n − h + 1)]y^{−n−γ}e^{−μ/2}| = 0.

Also, by Faà di Bruno’s formula we have for all l ⩾ 1 that

\frac{d^{l}}{dy^{l}} exp(−\frac{1}{2} ∫_{−∞}^{y} u_{1}(z) dz + \frac{1}{2} ∫_{y}^{∞} u_{1}(z) dz)

= exp(−\frac{1}{2} ∫_{−∞}^{y} u_{1}(z) dz + \frac{1}{2} ∫_{y}^{∞} u_{1}(z) dz)l! ∑_{(a_{1},...,a_{l})} ∏_{δ=1}^{l} \frac{1}{a_{δ}!} [\frac{−u_{1}^{(δ−1)}(y)}{δ!}]^{a_{δ}} ,

where the sum in (a_{1}, …, a_{l}) is over all l-tuples of nonnegative integers such that

∑_{δ=1}^{l} δa_{δ} = l.

The assumption of Theorem 0.1 is that |y|^{δ}u_{1}^{(δ−1)}(y) → 0^{+} as |y| → ∞, for δ = 1, …, r + 1. Thus for all 1 ⩽ l ⩽ r + 1 we have

lim_{t→∞} sup_{y⩾\sqrt{t}} y^{l} |\frac{d^{l}}{dy^{l}} exp(−\frac{1}{2} ∫_{−∞}^{y} u_{1}(z) dz + \frac{1}{2} ∫_{y}^{∞} u_{1}(z) dz)| = 0.

Thus finally we have shown that

lim_{t→∞} sup_{y⩾\sqrt{t}} y^{n+γ} |v_{0}^{(γ)}(y) − ṽ_{0}^{(γ)}(y)| = 0.

Using this result we have shown that lim_{t→∞} sup_{ξ∈ℝ}(1 + |ξ|)^{n+γ}|J| = 0.

For the boundary term J_{j}, we note that the limit y → ∞ makes no contribution. Since (1+|ξ|)^{n+γ}|He_{γ−j}(ξ −1)| exp[−\frac{1}{2}(ξ −1)^{2}] is bounded independently of ξ ∈ ℝ, we have by the last displayed result (where γ is replaced by j − 1) again that lim_{t→∞} sup_{ξ∈ℝ}(1 + |ξ|)^{n+γ}|J_{j}| = 0, j = 1, …, γ. Thus we have shown that I_{2} → 0 as t → ∞, uniformly in ξ.

4. Linear Estimates for the Case n > 1, n ∉ Z

Now suppose n > 1 and n ∉ Z. As in section 3, our goal in this section is to derive the form of ṽ, define the weight functions, and to verify the interpolation inequality and the three linear estimates from section 2.

As was discussed in section 1.4 we wish to find the solution ṽ_{1}(x, t) of the heat equation which has as initial data the distribution

∑_{j=0}^{N−1} \frac{(−1)^{j}M_{j}}{j!} δ^{(j)} + ṽ_{0}, M_{j} = ∫_{−∞}^{∞} x^{j}v_{0}(x) dx,

where the distribution ṽ_{0} associated to the homogeneous function

ṽ_{0}(y) = |y|^{−n} cosh(μ/2) − |y|^{−n} sgn(y) sinh(μ/2).

is defined by the rule

⟨ṽ_{0}, ϕ⟩ = ∫_{−∞}^{∞} ṽ_{0}(y) [ϕ(y) − ∑_{s=0}^{N−1} \frac{ϕ^{(s)}(0)}{s!}y^{s}] dy

whenever ϕ is a test function, and N = [[n]] is the greatest integer less than or equal to n. The solution ṽ_{1} has the form

ṽ_{1}(x, t) = ∑_{s=0}^{N−1} \frac{(−1)^{s}M_{s}}{s!t^{(1+s)/2}} G^{(s)}(xt^{−1/2}) + t^{−n/2}V (xt^{−1/2}),

where G(ξ) = e^{−ξ^{2}/2}/\sqrt{2π} and

V (ξ) = (G ∗ ṽ_{0})(ξ) = ∫_{−∞}^{∞} [G(ξ − y) − ∑_{s=0}^{N−1} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}] ṽ_{0}(y) dy.

In order to express this V (ξ) in terms of special functions we use the Fourier integral theorem. Using formulae (12) and (13) on page 173 of [18] we can compute the Fourier transform of the distribution ṽ_{0} to be

(Fṽ_{0})(k) = \frac{π|k|^{n−1}}{Γ(n)} [\frac{cosh(\frac{μ}{2})}{cos(\frac{π}{2} n)} + i sgn(k)\frac{sinh(\frac{μ}{2})}{sin(\frac{π}{2} n)}] .

Thus by the Fourier integral theorem we have

V (ξ) = \frac{1}{2Γ(n)} ∫_{−∞}^{∞} e^{ikξ−k^{2}/2}|k|^{n−1} [\frac{cosh(\frac{μ}{2})}{cos(\frac{π}{2} n)} + i sgn(k)\frac{sinh(\frac{μ}{2})}{sin(\frac{π}{2} n)}] dk.

Thus the function V (ξ) is a linear combination of |D|^{n−1}G and |D|^{n−1}HG. These can be expressed in terms of Confluent Hypergeometric functions through the formula

∫_{0}^{∞} e^{ikz−k^{2}/2}k^{n−1} dk = \sqrt{π}Γ(n) [\frac{M(\frac{n}{2}, \frac{1}{2}, −\frac{z^{2}}{2} )}{2^{n/2}Γ(\frac{n+1}{2})} + i\frac{zM(\frac{n+1}{2}, \frac{3}{2}, −\frac{z^{2}}{2} )}{2^{(n−1)/2}Γ(\frac{n}{2})}]

which follows from 19.5.3, page 687, and 19.12.3, page 691, of [1]. Therefore

[|D|^{n−1}G](ξ) = \frac{Γ(n)}{\sqrt{π}} \frac{M(\frac{n}{2}, \frac{1}{2}, −\frac{ξ^{2}}{2} )}{2^{n/2}Γ(\frac{n+1}{2})} = \frac{2^{n/2}}{2π} Γ(\frac{n}{2})M(\frac{n}{2}, \frac{1}{2}, −\frac{ξ^{2}}{2} ),

[|D|^{n−1}HG](ξ) = \frac{Γ(n)}{\sqrt{π}} \frac{ξM(\frac{n+1}{2}, \frac{3}{2}, −\frac{ξ^{2}}{2} )}{2^{(n−1)/2}Γ(\frac{n}{2})} = \frac{2^{(n+1)/2}}{2π} Γ(\frac{n+1}{2})ξM(\frac{n+1}{2}, \frac{3}{2}, −\frac{ξ^{2}}{2} ).

So we can express V (ξ) in the form

V (ξ) = \frac{π}{Γ(n)} [\frac{cosh(\frac{μ}{2})}{cos(\frac{π}{2} n)}[|D|^{n−1}G](ξ) − \frac{sinh(\frac{μ}{2})}{sin(\frac{π}{2} n)}[|D|^{n−1}HG](ξ)]

= \sqrt{π} [\frac{cosh(\frac{μ}{2})}{cos(\frac{π}{2} n)} \frac{M(\frac{n}{2}, \frac{1}{2}, −\frac{ξ^{2}}{2} )}{2^{n/2}Γ(\frac{n+1}{2})} − \frac{sinh(\frac{μ}{2})}{sin(\frac{π}{2} n)} \frac{ξM(\frac{n+1}{2}, \frac{3}{2}, −\frac{ξ^{2}}{2} )}{2^{(n−1)/2}Γ(\frac{n}{2})}] .

This formula agrees with our previous expression for V (ξ) derived when n < 1, as one can see by applying the reflection formula for the Gamma function. However, this formula has a pole at every positive integral value of n.

When n > 1 we have that V (ξ) = 0 has a real solution ξ. To see that this assertion holds, note that the large |ξ| asymptotics of V (ξ) derived in the previous section apply to the case n > 1 (not an integer) as well. Furthermore, since the Fourier transform of V is integrable, V is continuous. Hence V is an integrable function which is positive for |ξ| sufficiently large. Since n > 1 we have that ∫_{−∞}^{∞} V (ξ) dξ = V̂ (0) = 0, and hence V (ξ) must become negative at some value of ξ. Thus V (ξ) = 0 has roots.

Each term of the expression for ṽ_{1} gives rise to a term in the large-time asymptotic expansion of v(x, t). However, the asymptotic ordering of these terms is the same as the order in which they are listed only when ξ = xt^{−1/2} is a constant. The slow decay of V (ξ) and the rapid decay of G^{(s)}(ξ) as |ξ| → ∞ means that the last term becomes the most important if |ξ| tends to ∞ rapidly enough as t → ∞. The main balance is between the first and last terms of ṽ_{1}. Thus when n > 1 and n ∉ Z we define the following special solution of the heat equation:

ṽ(x, t) = t^{−1/2}G(xt^{−1/2}) ∫_{−∞}^{∞} v_{0}(y) dy + t^{−n/2}V (xt^{−1/2}).

22 Daniel B. Dix This solution will be our large-time asymptotic approximation to the solution v(x, t) arising from the initial data v_{0}. Better estimates of the difference ∂_{x}^{γ}[v(x, t)−ṽ_{1}(x, t)] can be obtained than for the difference ∂_{x}^{γ}[v(x, t) − ṽ(x, t)]. But using ṽ_{1} instead ṽ (defining ũ_{1} = −∂_{x} ln ṽ_{1}) still does not lead to estimates of ∂_{x}^{r}(u−ũ_{1}) with the same weight function as in the case n < 1. This is because ∂_{x}^{r}(u−ũ_{1}) depends not only on ∂_{x}^{γ}(v −ṽ_{1}), but also on ∂_{x}^{γ} ṽ_{1}, and and we have not seen how to fully compensate the slower decay of this latter quantity by the better estimates obtainable on ∂_{x}^{γ}(v−ṽ_{1}).

In keeping with this new (non-self-similar) expression for ṽ we will introduce new weight functions (as usual ξ = xt^{−1/2})

w_{γ}(x, t) = \frac{(1 + |ξ|)^{γ}e^{−ξ^{2}/2}}{t^{(1+γ)/2}} + \frac{1}{[t^{1/2}(1 + |ξ|)]^{n+γ}} = \frac{t^{(n−1)/2}e^{−ξ^{2}/2}(1 + |ξ|)^{n+2γ} + 1}{[t^{1/2}(1 + |ξ|)]^{n+γ}}.

This function is larger than the weight function by the same name in section 3. The other weight function is

z_{γ}(x, t) = \frac{w_{γ}(x, t)}{w_{0}(x, t)} = \frac{1}{[t^{1/2}(1 + |ξ|)]^{γ}} \frac{t^{(n−1)/2}e^{−ξ^{2}/2}(1 + |ξ|)^{n+2γ} + 1}{t^{(n−1)/2}e^{−ξ^{2}/2}(1 + |ξ|)^{n} + 1}.

The first factor is the weight function which was used in section 3. The second factor is not bounded; in fact it grows logarithmically in t in some regions. Despite the complicated nature of these weight functions, we will now prove that the Interpolation Estimate of section 2 is satisfied.

Lemma 4.1. Suppose n > 1, and for all integers γ ⩾ 0 the weight functions z_{γ} are defined as in the previous paragraph. Then for all integers r ⩾ 0 we have

∏_{γ=1}^{r+1} z_{γ}(x, t)^{a_{γ}} ⩽ z_{r+1}(x, t), x ∈ ℝ, t ⩾ 1,

where a_{1}, …, a_{r+1} are nonnegative integers such that ∑_{γ=1}^{r+1} γa_{γ} = r + 1.

Proof Define a = t^{(n−1)/2}e^{−ξ^{2}/2}(1 + |ξ|)^{n} > 0 and σ = (1 + |ξ|)^{2} ⩾ 1. Then

z_{γ} = \frac{1}{[t^{1/2}(1 + |ξ|)]^{γ}} \frac{1 + aσ^{γ}}{1 + a} .

Set θ = \frac{a}{1+a} ∈ (0, 1). After some manipulation it is found that we must show that

∏_{γ=1}^{r+1}[1 − θ + θσ^{γ}]^{a_{γ}} ⩽ 1 − θ + θσ^{r+1},

for all θ ∈ (0, 1) and σ > 1. Let f(θ) denote the left-hand-side of this inequality. Since f(0) = 1 and f(1) = σ^{r+1} we see that the above inequality will follow if we show that f′′(θ) ⩾ 0 for all θ ∈ [0, 1]. But this fact follows from direct computation, since σ > 1.

Now we will establish the Linear estimates in a series of lemmata.

Lemma 4.2. Suppose n > 1 and n ∉ Z. Then there exists T ⩾ 1 such that

sup_{t⩾T} ∥\frac{w_{0}(x, t)}{ṽ(x, t)}∥_{L^{∞}(x)} < ∞.

Proof Define M_{0} = ∫_{−∞}^{∞} v_{0}(y) dy. Choose M > 0 sufficiently large so that V (ξ) ⩾ \frac{1}{2} min\{e^{−μ/2}, e^{μ/2}\}|ξ|^{−n} for all |ξ| ⩾ M. So if |ξ| ⩾ M we have

t^{n/2}ṽ(x, t) = M_{0}t^{\frac{n−1}{2}} G(ξ) + V (ξ) ⩾ \frac{M_{0}}{\sqrt{2π}} t^{\frac{n−1}{2}} e^{−ξ^{2}/2} + \frac{1}{2} min\{e^{−μ/2}, e^{μ/2}\}\frac{1}{|ξ|^{n}}

⩾ min \{\frac{M_{0}}{\sqrt{2π}} , \frac{e^{−μ/2}}{2} , \frac{e^{μ/2}}{2} \} [t^{\frac{n−1}{2}} e^{−ξ^{2}/2} + \frac{1}{(1 + |ξ|)^{n}} ] .

Now choose T > 0 sufficiently large so that for all t ⩾ T we have

\frac{M_{0}}{2\sqrt{2π}} ⩽ \frac{M_{0}}{2\sqrt{2π}} t^{\frac{n−1}{2}} e^{−M^{2}/2} + min_{|ξ|⩽M} V (ξ).

So if |ξ| ⩽ M and t ⩾ T we have

t^{n/2}ṽ(x, t) = M_{0}t^{\frac{n−1}{2}} G(ξ) + V (ξ)

⩾ \frac{M_{0}}{2\sqrt{2π}} t^{\frac{n−1}{2}} e^{−ξ^{2}/2} + \frac{M_{0}}{2\sqrt{2π}} t^{\frac{n−1}{2}} e^{−M^{2}/2} + min_{|ξ|⩽M} V (ξ)

⩾ \frac{M_{0}}{2\sqrt{2π}} t^{\frac{n−1}{2}} e^{−ξ^{2}/2} + \frac{M_{0}}{2\sqrt{2π}} ⩾ \frac{M_{0}}{2\sqrt{2π}} [t^{\frac{n−1}{2}} e^{−ξ^{2}/2} + \frac{1}{(1 + |ξ|)^{n}} ] .

Therefore we have

sup_{t⩾T} ∥\frac{w_{0}(x, t)}{ṽ(x, t)}∥_{L^{∞}(x)} ⩽ 2 min \{\frac{M_{0}}{\sqrt{2π}} , e^{−μ/2}, e^{μ/2}\}^{−1} < ∞.

Lemma 4.3. Suppose n > 1 and n ∉ Z. Then for all integers γ ⩾ 0 we have that

sup_{t⩾1} ∥\frac{∂_{x}^{γ} ṽ(x, t)}{w_{γ}(x, t)}∥_{L^{∞}(x)} < ∞.

Proof Immediate.

The first step in the proof of Linear Estimate (3) is the following.

Lemma 4.4. Suppose n > 1 and n ∉ Z. Suppose ṽ(x, t), ṽ_{1}(x, t), and w_{γ}(x, t) are defined as above. Then for all integers γ ⩾ 0 we have

lim_{t→∞} ∥\frac{∂_{x}^{γ}[ṽ_{1}(x, t) − ṽ(x, t)]}{w_{γ}(x, t)}∥_{L^{∞}(x)} = 0.

Proof There is nothing to prove unless N ⩾ 2. In that case

∂_{x}^{γ}[ṽ_{1}(x, t) − ṽ(x, t)] = ∑_{s=1}^{N−1} \frac{(−1)^{s}}{s!t^{(1+s+γ)/2}} G^{(s+γ)}(ξ) ∫_{−∞}^{∞} y^{s}v_{0}(y) dy.

We will prove the same weighted estimate on each term of this sum. After some manipulation we get

|\frac{G^{(s+γ)}(ξ)}{t^{(1+s+γ)/2}w_{γ}(x, t)}| ⩽ C \frac{t^{(n−1−s)/2}(1 + |ξ|)^{n+2γ+s}e^{−ξ^{2}/2}}{t^{(n−1)/2}(1 + |ξ|)^{n+2γ}e^{−ξ^{2}/2} + 1}

= C [\frac{1 + |ξ|}{t^{1/2}}]^{s} · \frac{t^{(n−1)/2}(1 + |ξ|)^{n+2γ}e^{−ξ^{2}/2}}{t^{(n−1)/2}(1 + |ξ|)^{n+2γ}e^{−ξ^{2}/2} + 1}.

Let ϵ > 0 be given. Clearly, if 1 + |ξ| ⩽ ϵt^{1/2} then this quotient is bounded independently of ξ and t ⩾ 1 by Cϵ^{s}. On the other hand if 1 + |ξ| ⩾ ϵt^{1/2} then the quotient is bounded by

Ct^{(n−1−s)/2}(1 + |ξ|)^{n+2γ+s}e^{−ξ^{2}/2}.

Since (1 + |ξ|)^{a}e^{−ξ^{2}/2} is an exponentially decreasing function of |ξ| for sufficiently large |ξ|, the above quantity must be less than ϵ if t is sufficiently large.

Since

\frac{1}{w_{γ}(x, t)} ⩽ [t^{1/2}(1 + |ξ|)]^{n+γ},

in order to complete the proof of Linear Estimate (3) it suffices to prove the following.

Lemma 4.5. Suppose n > 1 and n ∉ Z. For every integer 0 ⩽ γ ⩽ r we have

lim_{t→∞} ∥[t^{1/2}(1 + |ξ|)]^{n+γ}∂_{x}^{γ}[v(x, t) − ṽ_{1}(x, t)]∥_{L^{∞}(x)} = 0.

Proof We have the following formulae:

v(x, t) = ∫_{−∞}^{∞} \frac{1}{\sqrt{t}}G (\frac{x − y}{\sqrt{t}} ) v_{0}(y) dy = ∫_{−∞}^{∞} G(ξ − y)v_{0}(y\sqrt{t}) dy,

ṽ_{1}(x, t) = ∑_{s=0}^{N−1} \frac{(−1)^{s}G^{(s)}(ξ)}{s!} ∫_{−∞}^{∞} y^{s}v_{0}(y\sqrt{t}) dy

+ ∫_{−∞}^{∞} [G(ξ − y) − ∑_{s=0}^{N−1} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}] ṽ_{0}(y\sqrt{t}) dy.

Subtracting these two we get

v(x, t) − ṽ_{1}(x, t) = ∫_{−∞}^{∞} [G(ξ − y) − ∑_{s=0}^{N−1} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}] [v_{0}(y\sqrt{t}) − ṽ_{0}(y\sqrt{t})] dy.

Therefore

∂_{x}^{γ}[v(x, t) − ṽ_{1}(x, t)]

= \frac{1}{t^{γ/2}} ∫_{−∞}^{∞} [G^{(γ)}(ξ − y) − ∑_{s=0}^{N−1} \frac{(−1)^{s}G^{(s+γ)}(ξ)}{s!}y^{s}] [v_{0}(y\sqrt{t}) − ṽ_{0}(y\sqrt{t})] dy.

Large-Time Behavior for Burgers’ Equation 25

Now we intend to integrate by parts γ times, first over the intervals [−A, −ϵ] and [ϵ, A], and then let ϵ → 0^{+} and A → ∞. Each time another derivative is applied to v_{0}(y\sqrt{t}) − ṽ_{0}(y\sqrt{t}) the singularity at y = 0 gets worse, so we must choose the antiderivative of the other part so that at each stage we have an absolutely convergent integral. The contributions of the endpoints vanish in the limit ϵ → 0^{+} and A → ∞. This process yields finally

∂_{x}^{γ}[v(x, t) − ṽ_{1}(x, t)] =

∫_{−∞}^{∞} [G(ξ − y) − ∑_{s=0}^{N−1+γ} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}] [v_{0}^{(γ)}(y\sqrt{t}) − ṽ_{0}^{(γ)}(y\sqrt{t})] dy.

ṽ_{0}^{(γ)}(y) is the γth derivative of the ordinary function ṽ_{0}(y) defined for y ≠ 0, not a distributional derivative. Therefore

[t^{1/2}(1 + |ξ|)]^{n+γ}∂_{x}^{γ}[v(x, t) − ṽ_{1}(x, t)]

= ∫_{−∞}^{∞} [G(ξ − y) − ∑_{s=0}^{N−1+γ} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}] (\frac{1 + |ξ|}{|y|})^{n+γ} ·

· |y\sqrt{t}|^{n+γ}[v_{0}^{(γ)}(y\sqrt{t}) − ṽ_{0}^{(γ)}(y\sqrt{t})] dy

= I_{1} + I_{2} + I_{3},

where I_{1} is the integral over |y| ⩽ 1, I_{2} is the integral over y ⩾ 1, and I_{3} is the integral over y ⩽ −1. We will estimate these three separately.

To deal with I_{1} we use the integral form of the Taylor remainder

G(ξ−y)− ∑_{s=0}^{N−1+γ} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s} = \frac{(−y)^{N+γ}}{(N − 1 + γ)!} ∫_{0}^{1}(1−z)^{N−1+γ}G^{(N+γ)}(ξ−yz) dz.

By Peetre’s inequality, (1 + |ξ|)^{n+γ} ⩽ C(1 + |ξ − yz|)^{n+γ}(1 + |y|z)^{n+γ}, we have that

|G(ξ − y) − ∑_{s=0}^{N−1+γ} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}| (\frac{1 + |ξ|}{|y|})^{n+γ} ⩽ C|y|^{N−n},

where C > 0 is independent of ξ and |y| ⩽ 1. From section 3 we have the following expression

|y|^{n+γ}[v_{0}^{(γ)}(y) − ṽ_{0}^{(γ)}(y)]

= |y|^{n+γ} \frac{d^{γ}}{dy^{γ}} [(1 + y^{2})^{−n/2}] exp(−\frac{1}{2} ∫_{−∞}^{y} u_{1}(z) dz + \frac{1}{2} ∫_{y}^{∞} u_{1}(z) dz)

− [∏_{h=1}^{γ}(−n − h + 1)]e^{− sgn(y)μ/2}

+ ∑_{l=1}^{γ} \binom{γ}{l}|y|^{n+γ−l} \frac{d^{γ−l}}{dy^{γ−l}} [(1 + y^{2})^{−n/2}] ·

· |y|^{l} \frac{d^{l}}{dy^{l}} exp(−\frac{1}{2} ∫_{−∞}^{y} u_{1}(z) dz + \frac{1}{2} ∫_{y}^{∞} u_{1}(z) dz).

As was shown in section 3, this quantity tends to zero as |y| → ∞, and hence it is bounded for all sufficiently large y. It also can be seen to be bounded for y in any compact interval. Thus by the dominated convergence theorem, |I_{1}| → 0 as t → ∞, uniformly for ξ ∈ ℝ.

The arguments required to deal with I_{2} and I_{3} are essentially the same; we will explain the one for I_{2}. We apply Hölder’s inequality, and exactly as in section 3 we show that

lim_{t→∞} sup_{y⩾\sqrt{t}} y^{n+γ} |v_{0}^{(γ)}(y) − ṽ_{0}^{(γ)}(y)| = 0.

It remains then to bound

∫_{1}^{∞} |G(ξ − y) − ∑_{s=0}^{N−1+γ} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}| (\frac{1 + |ξ|}{|y|})^{n+γ} dy

independently of ξ. Since G^{(s)}(ξ)(1 + |ξ|)^{n+γ} is bounded independently of ξ, we have

∫_{1}^{∞} | ∑_{s=0}^{N−1+γ} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}| (\frac{1 + |ξ|}{|y|})^{n+γ} dy ⩽ C ∫_{1}^{∞} y^{N−1−n} dy < ∞,

where C > 0 is independent of ξ. On the other hand, by Peetre’s inequality we have

(\frac{1 + |ξ|}{|y|})^{n+γ} ⩽ C′(1 + |ξ − y|)^{n+γ}

for y ⩾ 1. Furthermore

∫_{1}^{∞} |G(ξ − y)|(1 + |ξ − y|)^{n+γ} dy

is bounded independently of ξ. Combining these two bounds yields the desired result, and finishes the proof.

5. Linear Estimates for the Case n ⩾ 1, n ∈ Z

Now suppose n ⩾ 1 is an integer. As in the previous two sections our task here is to define ṽ and the corresponding weight functions, prove the interpolation estimate and the linear estimates from section 2. This will complete the proof of Theorem 1.3.

As was discussed in section 1.4 we wish to find the solution ṽ_{1}(x, t) of the heat equation which has as initial data the distribution ∑_{j=0}^{n−1} \frac{(−1)^{j}M_{j}}{j!} δ^{(j)} + ṽ_{0} where

M_{j} = ∫_{−∞}^{∞} x^{j}v_{0}(x) dx,  j = 1, …, n − 2,

M_{n−1} = ∫_{−∞}^{∞} x^{n−1}[v_{0}(x) − ṽ_{0}(x)H(x^{2} − 1)] dx,

and where the distribution ṽ_{0} associated to the homogeneous function

ṽ_{0}(y) = |y|^{−n} cosh(μ/2) − |y|^{−n} sgn(y) sinh(μ/2).

is defined by the rule

⟨ṽ_{0}, ϕ⟩ = ∫_{−∞}^{∞} ṽ_{0}(y) [ϕ(y) − ∑_{j=0}^{n−2} \frac{ϕ^{(j)}(0)}{j!}y^{j} − \frac{ϕ^{(n−1)}(0)}{(n − 1)!}y^{n−1}H(1 − y^{2})] dy,

whenever ϕ is a test function. The absolute convergence of the integral defining M_{n−1} is equivalent to

∫_{1}^{∞} \frac{1}{x} |∫_{x}^{∞} u_{1}(±z) dz| dx < ∞,

as can be seen by substituting for v_{0}(x) its expression in terms of u_{1}(x) and using the fact that |e^{y}−1| ⩽ C|y| for |y| ⩽ 1. We will assume both of these conditions (one with the + sign and the other with the − sign), in addition to the usual assumption that u_{1} ∈ L^{1}.

Using formulae (20) and (21), page 174, formula (38), page 180, and formula (43), page 181, of [18] we have that

(Fṽ_{0})(k) = \frac{(−ik)^{n−1}}{(n − 1)!} \{L_{n} [ln |k| − ψ(n)] − i sgn(k)\frac{π}{2} N_{n}\} .

where γ = lim_{N→∞} ∑_{s=1}^{N} \frac{1}{s} − ln N is the Euler constant, ψ(z) = Γ′(z)/Γ(z) is the Digamma function, and ψ(n) = −γ + ∑_{s=1}^{n−1} \frac{1}{s} for all integers n ⩾ 1, and

L_{n} = (−1)^{n}e^{\frac{μ}{2}} − e^{− \frac{μ}{2}} = \begin{cases}2 sinh(\frac{μ}{2}) & n even \\ −2 cosh(\frac{μ}{2}) & n odd \end{cases} ,

N_{n} = (−1)^{n}e^{\frac{μ}{2}} + e^{− \frac{μ}{2}} = \begin{cases}2 cosh(\frac{μ}{2}) & n even \\ −2 sinh(\frac{μ}{2}) & n odd \end{cases} .

The part of ṽ_{1} arising from the term involving ln |k| is

\frac{(−1)^{n−1}}{(n − 1)!} \frac{L_{n}}{2π} ∫_{−∞}^{∞} e^{ikx− \frac{1}{2} k^{2}t}(ik)^{n−1} ln |k| dk

= \frac{(−1)^{n−1}L_{n}}{(n − 1)!t^{n/2}} [(ln |D| G^{(n−1)})(ξ) − \frac{1}{2} ln t G^{(n−1)}(ξ)] .

where ξ = xt^{−1/2}. Thus we have

ṽ_{1}(x, t) = ∑_{s=0}^{n−2} \frac{(−1)^{s}M_{s}}{s!t^{(1+s)/2}} G^{(s)}(ξ)

+ \frac{(−1)^{n−1}}{(n − 1)!t^{n/2}} (M_{n−1} − L_{n}ψ(n) − \frac{1}{2}L_{n} ln t)G^{(n−1)}(ξ) + t^{−n/2}V (ξ),

V (ξ) = \frac{(−1)^{n−1}}{(n − 1)!} [L_{n}(ln |D| G^{(n−1)})(ξ) + \frac{π}{2} N_{n}(HG^{(n−1)})(ξ)].

This expression for V (ξ) only applies when n ⩾ 1 is an integer.

Using the expressions we derived in section 4 for |D|^{n−1}G and |D|^{n−1}HG in terms of confluent hypergeometric functions, as well as the known large ξ asymptotics for confluent hypergeometric functions (13.1.5, page 504 of [1]) we obtain the asymptotics

(|D|^{n−1}G)(ξ) ∼ \frac{Γ(n)}{π} cos(n\frac{π}{2} )|ξ|^{−n}  as |ξ| → ∞,
(|D|^{n−1}HG)(ξ) ∼ \frac{Γ(n)}{π} sin(n\frac{π}{2} ) sgn(ξ)|ξ|^{−n}  as |ξ| → ∞.

These hold uniformly for all complex values of n, ℜn ⩾ ϵ > 0. Using these we have the following large |ξ| asymptotics of HG^{(n−1)}:

(HG^{(n−1)})(ξ) ∼ \frac{(−1)^{n−1}(n − 1)!}{π}ξ^{−n} as |ξ| → ∞,

for every positive integer n ⩾ 1. By Theorem 9.4 of [34] we can differentiate the asymptotic expansions of |D|^{n−1}G and |D|^{n−1}HG with respect to n. Thus we obtain

(ln |D| G^{(n−1)})(ξ) ∼ \frac{(−1)^{n}(n − 1)!}{2} sgn(ξ)ξ^{−n} as |ξ| → ∞,

for every positive integer n ⩾ 1. Thus we have that

V (ξ) ∼ [cosh(\frac{μ}{2}) − sinh(\frac{μ}{2}) sgn(ξ)]|ξ|^{−n},

as |ξ| → ∞. Because this term is slowly decaying in ξ, it will become dominant as t → ∞, |ξ| → ∞, despite the fact that it is not so important when t → ∞ and ξ is a fixed constant.

So when n ⩾ 2 is an integer, we can use the same weight functions w_{γ}(x, t) and z_{γ}(x, t) as in section 4. Hence the interpolation estimate for n ⩾ 2 has already been proved in Lemma 4.1. But when n = 1 we define

w_{γ}(x, t) = \frac{ln(t)(1 + |ξ|)^{γ}e^{− \frac{1}{2} ξ^{2}}}{t^{(1+γ)/2}} + \frac{1}{[t^{1/2}(1 + |ξ|)]^{1+γ}} = \frac{ln(t)(1 + |ξ|)^{1+2γ}e^{− \frac{1}{2} ξ^{2}} + 1}{[t^{1/2}(1 + |ξ|)]^{1+γ}},

z_{γ}(x, t) = \frac{w_{γ}(x, t)}{w_{0}(x, t)} = \frac{1}{[t^{1/2}(1 + |ξ|)]^{γ}} \frac{ln(t)(1 + |ξ|)^{1+2γ}e^{− \frac{1}{2} ξ^{2}} + 1}{ln(t)(1 + |ξ|)e^{− \frac{1}{2} ξ^{2}} + 1}.

The proof of the interpolation estimate for the case n = 1 is extremely similar to the proof of Lemma 4.1. (We let a = ln(t)(1+|ξ|)e^{−ξ^{2}/2}, which is positive for t > 1, and then the rest of the proof applies without change.)

When n ⩾ 2 is an integer we define

ṽ(x, t) = \frac{M_{0}}{t^{1/2}} G(ξ) + \frac{V (ξ)}{t^{n/2}}.

Large-Time Behavior for Burgers’ Equation 29

This is not a solution of the heat equation, but adding the extra term to make it a solution is not necessary, since the extra term is negligible in comparison with the weight function w_{0}(x, t) defined in section 4. ṽ(x, t) is positive valued for all sufficiently large t. When n = 1 we define

ṽ(x, t) = \frac{1}{t^{1/2}} \{[M_{0} + cosh(\frac{μ}{2})(−2γ + ln t)]G(ξ) + V (ξ)\}.

This is a solution of the heat equation. It is positive valued for all sufficiently large t.

Linear Estimate (1) can now be proved as in Lemma 4.2. That Lemma already applies when n ⩾ 2 is an integer. When n = 1 the definition of ṽ(x, t) and w_{0}(x, t) are different, but the proof of Lemma 4.2 can be modified to apply to this case quite easily: t^{\frac{n−1}{2}} should be replaced by ln t, and certain constants should be adjusted slightly.

Linear Estimate (2) for the case n ⩾ 1 an integer follows as trivially as it did in Lemma 4.3. The proof of Linear Estimate (3) will take place in two steps, analogous to Lemmata 4.4 and 4.5. The statement of Lemma 4.4 now holds for all integers n ⩾ 1, where ṽ, ṽ_{1}, and w_{γ} have the appropriate definitions given above. The proof is exactly along the lines of the proof of Lemma 4.4

The following is the analog of Lemma 4.5.

Lemma 5.1. Suppose n ⩾ 1 and 0 ⩽ γ ⩽ r+1 are integers. Suppose the conditions from Theorem 1.3 are imposed on u_{1}. Then

lim_{t→∞} ∥[t^{1/2}(1 + |ξ|)]^{n+γ}∂_{x}^{γ}[v(x, t) − ṽ_{1}(x, t)]∥_{L^{∞}(x)} = 0.

Proof For each x and t > 0 define the test function ϕ(y) = \frac{1}{\sqrt{t}}G (\frac{x−y}{\sqrt{t}} ). Then

v(x, t) − ṽ_{1}(x, t) = ⟨v_{0} − ∑_{s=0}^{n−1} \frac{(−1)^{s}}{s!} M_{s}δ^{(s)} − ṽ_{0}, ϕ⟩

= ∫_{−∞}^{∞}\{v_{0}(y)ϕ(y) − v_{0}(y) ∑_{s=0}^{n−2} \frac{ϕ^{(s)}(0)}{s!}y^{s} − [v_{0}(y) − ṽ_{0}(y)H(y^{2} − 1)]\frac{ϕ^{(n−1)}(0)}{(n − 1)!}y^{n−1}

− ṽ_{0}(y)[ϕ(y) − ∑_{s=0}^{n−2} \frac{ϕ^{(s)}(0)}{s!}y^{s} − \frac{ϕ^{(n−1)}(0)}{(n − 1)!}y^{n−1}H(1 − y^{2})]\} dy

= ∫_{−∞}^{∞}[v_{0}(y) − ṽ_{0}(y)][ϕ(y) − ∑_{s=0}^{n−1} \frac{ϕ^{(s)}(0)}{s!}y^{s}] dy

= ∫_{−∞}^{∞} [G(ξ − y) − ∑_{s=0}^{n−1} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}] [v_{0}(y\sqrt{t}) − ṽ_{0}(y\sqrt{t})] dy.

The proof proceeds exactly along the lines of Lemma 4.5, with N = n. We differentiate the above formula γ times with respect to x and then integrate by parts γ times, γ ⩽ r + 1. The resulting weighted error is split into three integrals I_{1}, I_{2}, I_{3} as before. I_{1} is dealt with as in the proof of Lemma 4.5. The estimate of I_{2} requires a slightly modified argument. We must show that

I_{2} = ∫_{1}^{∞} [G(ξ − y) − ∑_{s=0}^{n−1+γ} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}] (\frac{1 + |ξ|}{|y|})^{n+γ} ·
· |y\sqrt{t}|^{n+γ}[v_{0}^{(γ)}(y\sqrt{t}) − ṽ_{0}^{(γ)}(y\sqrt{t})] dy

tends to zero as t → ∞, uniformly for ξ ∈ ℝ. We write this as the difference of two integrals. The first

∫_{1}^{∞} G(ξ − y) (\frac{1 + |ξ|}{|y|})^{n+γ} |y\sqrt{t}|^{n+γ}[v_{0}^{(γ)}(y\sqrt{t}) − ṽ_{0}^{(γ)}(y\sqrt{t})] dy

is estimated using Hölder’s inequality and Peetre’s inequality as in the proof of Lemma 4.5. In the second we can multiply and divide by |y|, and take out the supremum of the first factor:

∫_{1}^{∞} | ∑_{s=0}^{n−1+γ} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}| (\frac{1 + |ξ|}{|y|})^{n+γ} |y|·

· |y\sqrt{t}|^{n+γ} |v_{0}^{(γ)}(y\sqrt{t}) − ṽ_{0}^{(γ)}(y\sqrt{t})| \frac{dy}{|y|}

⩽ sup_{ξ∈ℝ} sup_{y⩾1} \{| ∑_{s=0}^{n−1+γ} \frac{(−1)^{s}G^{(s)}(ξ)}{s!}y^{s}| (\frac{1 + |ξ|}{|y|})^{n+γ} |y|\}

· ∫_{√\overline{t}}^{∞} |y|^{n+γ} |v_{0}^{(γ)}(y) − ṽ_{0}^{(γ)}(y)| \frac{dy}{|y|}.

The supremum is obviously finite. It is not difficult, using expressions for v_{0}^{(γ)}(y) − ṽ_{0}^{(γ)}(y) derived in section 3, to show that our assumptions on u_{1} imply that this last integrand is integrable. Hence this quantity tends to zero as t → ∞ independently of ξ.

6. L^{∞} decay estimates

In this section we investigate the consequences of Theorems 1.2 and 1.3 by giving a more understandable description of the large-time asymptotics of u(x, t) in the case n ⩾ 1. We analyze ũ(x, t) in various regions, and compute sharp L^{∞} decay estimates of the solution u(x, t).

First we will examine the leading-order asymptotic approximation ũ(x, t) in the case n > 1. Recall that

ṽ(x, t) = t^{−1/2}G(ξ)M_{0} + t^{−n/2}V (ξ)
ṽ_{x}(x, t) = t^{−1}G′(ξ)M_{0} + t^{−(n+1)/2}V′(ξ),

Depending on how |ξ| → ∞ as t → ∞ one or the other of the two addends dominate. If θ ∈ ℝ then define the ratios

R_{θ} = \frac{e^{ξ^{2}/2}}{t^{(n−1)/2}(1 + |ξ|)^{n+θ}} , R_{θ}^{−1} = t^{(n−1)/2}(1 + |ξ|)^{n+θ}e^{−ξ^{2}/2}.

The ratios R_{0} and R_{2} allow us to assess which of the two addends is the dominant one, since for some constants c_{0}, c_{2} we have as t → ∞, |ξ| → ∞

\frac{t^{−n/2}V (ξ)}{t^{−1/2}G(ξ)} ∼ c_{0}R_{0}, \frac{t^{−(n+1)/2}V′(ξ)}{t^{−1}G′(ξ)} ∼ c_{2}R_{2}.

There are five disjoint asymptotic regions where ũ(x, t) simplifies asymptotically. These are the following.

I: R_{0} → 0, t → ∞. In this region we have

ũ(x, t) = − \frac{\frac{1}{t}G′(ξ)M_{0} + \frac{1}{t}(1 + |ξ|)e^{−ξ^{2}/2}O(R_{2})}{t^{−1/2}G(ξ)M_{0}[1 + O(R_{0})]} = \frac{x}{t} + o (\frac{1 + |ξ|}{t^{1/2}}) .

II_{±}: R_{0} → ∞, R_{2} → 0, ±ξ → ∞, t → ∞. In this region we have

ũ(x, t) = − \frac{\frac{1}{t}G′(ξ)M_{0} + \frac{1}{t}(1 + |ξ|)e^{−ξ^{2}/2}O(R_{2})}{t^{−n/2}V (ξ)[1 + O(R_{0}^{−1})]} = ±\frac{e^{±μ/2}}{\sqrt{2πt}} M_{0}R_{1}^{−1}[1 + o(1)].

III_{±}: R_{2} → ∞, ±ξ → ∞, t → ∞. In this region we have

ũ(x, t) = −\frac{t^{−(n+1)/2}V′(ξ)[1 + O(R_{2}^{−1})]}{t^{−n/2}V (ξ)[1 + O(R_{0}^{−1})]} = \frac{n}{x}[1 + o(1)].

The region II_{±} is quite narrow. In fact, using the asymptotic technique described on page 13 of [31], we have that the value of ξ where the ratio R_{θ} has a positive constant value is given asymptotically by

ξ = \sqrt{2 ln(R_{θ}t^{\frac{n−1}{2}} )} \{1 + (n + θ)\frac{ln [1 + \sqrt{2 ln(R_{θ}t^{\frac{n−1}{2}} )}]}{2 ln(R_{θ}t^{\frac{n−1}{2}} )} + o((ln t^{(n−1)/2})^{−1})\} ,

as t → ∞. Nevertheless, the region II_{±} is nonempty, since it includes the values of ξ where R_{θ} is constant, 0 < θ < 2. In this region ũ(x, t) ∼ c_{1}^{±}t^{−1/2}R_{1}^{−1}, and the ratio R_{1} ranges throughout the interval (0, ∞). This allows the solution to accomplish a rapid transition from the “rarefaction” in region I to the self-similar solution in regions III_{±}.

This analysis extends to the case n = 1 essentially by replacing t^{(n−1)/2} by ln t in the definition of R_{θ}. We omit the details.

The proof of the sharp decay estimate in Theorem 1.4 in the case n < 1 is essentially given in the remark after Theorem 1.1. Now we will prove the decay estimates stated in Theorem 1.4 for the cases n ⩾ 1. We begin with the observation that

sup_{t⩾e^{2}} ∥\frac{∂_{x}^{r}ũ(x, t)}{z_{r+1}(x, t)}∥_{L^{∞}(x)} < ∞.

This is proved along the similar lines as Lemma 2.3 using Linear Estimate (2) and Faà di Bruno’s formula. Next, we note that

∥z_{r+1}(·, t)∥_{L^{∞}} = \begin{cases}O ([\frac{ln(ln t)}{t}]^{(r+1)/2}) & n = 1, t ⩾ e^{2}, \\ O ([\frac{ln t}{t}]^{(r+1)/2}) & n > 1, t ⩾ e. \end{cases}

This follows from the expressions

z_{r+1}(x, t) = \frac{1}{[t^{1/2}(1 + |ξ|)]^{r+1}} \frac{R_{2r+2}^{−1} + 1}{R_{0}^{−1} + 1} = (\frac{1 + |ξ|}{t^{1/2}})^{r+1} \frac{1 + R_{2r+2}}{1 + R_{0}} .

When R_{2r+2} ⩽ 1 we have that

|ξ| = \begin{cases}O([ln(ln t)]^{1/2}) & n = 1, t ⩾ e^{2}, \\ O([ln(t^{(n−1)/2})]^{1/2}) & n > 1, t ⩾ e, \end{cases}

and hence the estimate follows from the second expression. On the other hand if R_{2r+2}^{−1} ⩽ 1 then by the first expression we have the stronger estimate: |z_{r+1}(x, t)| = O(t^{−(r+1)/2}). Finally, the estimate stated in Theorem 1.4 follows, since

∥∂_{x}^{r}u∥_{L^{∞}} ⩽ ∥\frac{∂_{x}^{r}(u − ũ) + ∂_{x}^{r}ũ}{z_{r+1}} ∥_{L^{∞}} ∥z_{r+1}∥_{L^{∞}}

⩽ (∥\frac{∂_{x}^{r}[u − ũ]}{z_{r+1}} ∥_{L^{∞}} + ∥\frac{∂_{x}^{r}ũ}{z_{r+1}} ∥_{L^{∞}}) ∥z_{r+1}∥_{L^{∞}}

= O(∥z_{r+1}∥_{L^{∞}})

by Theorems 1.2 and 1.3.

Now we will show that when n ⩾ 1 and r = 0 then the above decays estimates are sharp, i.e.

lim inf_{t→∞} [\frac{t}{ln(ln t)}]^{1/2} ∥u(·, t)∥_{L^{∞}} > 0,  n = 1,

lim inf_{t→∞} [\frac{t}{ln t}]^{1/2} ∥u(·, t)∥_{L^{∞}} > 0,  n > 1.

When t is sufficiently large, we will find below a value x(t) > 0 such that ũ_{x}(x(t), t) = 0 at x = x(t) = ξ(t)t^{1/2}. Then

∥u(·, t)∥_{L^{∞}} ⩾ |u(x(t), t)| ⩾ (|\frac{ũ(x(t), t)}{z_{1}(x(t), t)}| − |\frac{u(x(t), t) − ũ(x(t), t)}{z_{1}(x(t), t)}|) · |z_{1}(x(t), t)|

Since by Theorems 1.2 and 1.3 we have

lim_{t→∞} |\frac{u(x(t), t) − ũ(x(t), t)}{z_{1}(x(t), t)}| = 0

it suffices to show that the limits

lim_{t→∞} [\frac{t}{ln(ln t)}]^{1/2} |z_{1}(x(t), t)|  n = 1,

lim_{t→∞} [\frac{t}{ln t}]^{1/2} |z_{1}(x(t), t)|  n > 1,

exist and are positive, and that

lim inf_{t→∞} |\frac{ũ(x(t), t)}{z_{1}(x(t), t)}| = 1.

Define

ϕ(t) = \begin{cases}\frac{M_{0}}{\sqrt{2π}} t^{(n−1)/2} & n > 1, \\ \frac{1}{\sqrt{2π}} [M_{0} + cosh(\frac{μ}{2})(−2γ + ln t)] & n = 1. \end{cases}

We suppose t is large enough so that this function assumes positive values. Then ṽ(x, t) = t^{−n/2}[ϕ(t)e^{−ξ^{2}/2} + V (ξ)], where the function V (ξ) is defined either in section 4 or section 5, depending on the value of n ⩾ 1. ũ_{x} = 0 if and only if ṽ_{x}^{2} − ṽṽ_{xx} = 0, i.e.

[−ϕ(t)ξe^{−ξ^{2}/2} + V′(ξ)]^{2} − [ϕ(t)e^{−ξ^{2}/2} + V (ξ)] [ϕ(t)(ξ^{2} − 1)e^{−ξ^{2}/2} + V′′(ξ)] = 0.

For all sufficiently large t we want to find a positive solution ξ = ξ(t) of this equation. After some manipulation we find that we must solve the equation

ϕ(t)ξ^{n−2}e^{−ξ^{2}/2} = ψ(ξ),

where ψ(ξ) is the (unique) positive solution of the quadratic equation

ψ^{2} − ψ {[ξ^{n}V (ξ)](1 − ξ^{−2}) + 2[ξ^{n+1}V′(ξ)]ξ^{−2} − [ξ^{n+2}V′′(ξ)]ξ^{−4}}
− {ξ^{2n+2}[V (ξ)V′′(ξ) − V′(ξ)^{2}]} ξ^{−6} = 0.

Regardless of whether V (ξ) is defined in section 4 or section 5 we found that it has the following asymptotics:

ξ^{n}V (ξ) ∼ e^{−μ/2},
ξ^{n+1}V′(ξ) ∼ −ne^{−μ/2},
ξ^{n+2}V′′(ξ) ∼ n(n + 1)e^{−μ/2},

as ξ → ∞ (n ⩾ 1). Thus ξ^{2n+2}[V (ξ)V′′(ξ) − V′(ξ)^{2}] ∼ ne^{−μ} as ξ → ∞, implying the uniqueness of the positive root when ξ is sufficiently large. Clearly we have ψ(ξ) ∼ e^{−μ/2} as ξ → ∞. Using the technique of page 13 [31] we can show that the positive solution ξ = ξ(t) of the equation e^{μ/2}ϕ(t)ξ^{n−2}e^{−ξ^{2}/2} = e^{μ/2}ψ(ξ) satisfies

ξ(t) ∼ \sqrt{2 ln[e^{μ/2}ϕ(t)]}, as t → ∞.

34 Daniel B. Dix It follows from this that R_{θ}(ξ(t), t) → 0 as t → ∞ for all θ ⩾ 0, and hence

z_{1}(x(t), t) = (\frac{1 + ξ(t)}{t^{1/2}}) \frac{1 + R_{2}}{1 + R_{0}} ∼ (\frac{2 ln[e^{μ/2}ϕ(t)]}{t})^{1/2}

as t → ∞, as claimed. In a similar way we can compute the asymptotic behavior of ũ(x, t) evaluated at x = x(t):

ũ(x(t), t) = \frac{1}{t^{1/2}} \frac{[ϕ(t)ξe^{−ξ^{2}/2} − V′(ξ)]}{[ϕ(t)e^{−ξ^{2}/2} + V (ξ)]} = \frac{ξ}{t^{1/2}} \frac{[1 − \frac{ξ^{n+1}V′(ξ)}{ϕ(t)ξ^{n+2}e^{−ξ^{2}/2}} ]}{[1 + \frac{ξ^{n}V (ξ)}{ϕ(t)ξ^{n}e^{−ξ^{2}/2}} ]}

= \frac{ξ}{t^{1/2}} \frac{[1 − \frac{ξ^{n+1}V′(ξ)}{ψ(ξ)ξ^{4}}]}{[1 + \frac{ξ^{n}V (ξ)}{ψ(ξ)ξ^{2}}]} ∼ \frac{ξ(t)}{t^{1/2}} \frac{[1 − \frac{−ne^{−μ/2}}{e^{−μ/2}ξ(t)^{4}} ]}{[1 + \frac{e^{−μ/2}}{e^{−μ/2}ξ(t)^{2}} ]}

∼ (\frac{2 ln[e^{μ/2}ϕ(t)]}{t})^{1/2}

as t → ∞. Thus lim_{t→∞} ũ(x(t), t)/z_{1}(x(t), t) = 1 as claimed. This completes the proof of Theorem 1.4.

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