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Hans Lindblad and I have just uploaded to the arXiv our joint paper “Asymptotic decay for a one-dimensional nonlinear wave equation“, submitted to Analysis & PDE.  This paper, to our knowledge, is the first paper to analyse the asymptotic behaviour of the one-dimensional defocusing nonlinear wave equation

${}-u_{tt}+u_{xx} = |u|^{p-1} u$ (1)

where $u: {\bf R} \times {\bf R} \to {\bf R}$ is the solution and $p>1$ is a fixed exponent.  Nowadays, this type of equation is considered a very simple example of a non-linear wave equation (there is only one spatial dimension, the equation is semilinear, the conserved energy is positive definite and coercive, and there are no derivatives in the nonlinear term), and indeed it is not difficult to show that any solution whose conserved energy

$E[u] := \int_{{\bf R}} \frac{1}{2} |u_t|^2 + \frac{1}{2} |u_x|^2 + \frac{1}{p+1} |u|^{p+1}\ dx$

is finite, will exist globally for all time (and remain finite energy, of course).  In particular, from the one-dimensional Gagliardo-Nirenberg inequality (a variant of the Sobolev embedding theorem), such solutions will remain uniformly bounded in $L^\infty_x({\bf R})$ for all time.

However, this leaves open the question of the asymptotic behaviour of such solutions in the limit as $t \to \infty$.  In higher dimensions, there are a variety of scattering and asymptotic completeness results which show that solutions to nonlinear wave equations such as (1) decay asymptotically in various senses, at least if one is in the perturbative regime in which the solution is assumed small in some sense (e.g. small energy).  For instance, a typical result might be that spatial norms such as $\|u(t)\|_{L^q({\bf R})}$ might go to zero (in an average sense, at least).   In general, such results for nonlinear wave equations are ultimately based on the fact that the linear wave equation in higher dimensions also enjoys an analogous decay as $t \to +\infty$, as linear waves in higher dimensions spread out and disperse over time.  (This can be formalised by decay estimates on the fundamental solution of the linear wave equation, or by basic estimates such as the (long-time) Strichartz estimates and their relatives.)  The idea is then to view the nonlinear wave equation as a perturbation of the linear one.

On the other hand, the solution to the linear one-dimensional wave equation

$-u_{tt} + u_{xx} = 0$ (2)

does not exhibit any decay in time; as one learns in an undergraduate PDE class, the general (finite energy) solution to such an equation is given by the superposition of two travelling waves,

$u(t,x) = f(x+t) + g(x-t)$ (3)

where $f$ and $g$ also have finite energy, so in particular norms such as $\|u(t)\|_{L^\infty_x({\bf R})}$ cannot decay to zero as $t \to \infty$ unless the solution is completely trivial.

Nevertheless, we were able to establish a nonlinear decay effect for equation (1), caused more by the nonlinear right-hand side of (1) than by the linear left-hand side, to obtain $L^\infty_x({\bf R})$ decay on the average:

Theorem 1. (Average $L^\infty_x$ decay) If $u$ is a finite energy solution to (1), then $\frac{1}{2T} \int_{-T}^T \|u(t)\|_{L^\infty_x({\bf R})}$ tends to zero as $T \to \infty$.

Actually we prove a slightly stronger statement than Theorem 1, in that the decay is uniform among all solutions with a given energy bound, but I will stick to the above formulation of the main result for simplicity.

Informally, the reason for the nonlinear decay is as follows.  The linear evolution tries to force waves to move at constant velocity (indeed, from (3) we see that linear waves move at the speed of light $c=1$).  But the defocusing nature of the nonlinearity will spread out any wave that is propagating along a constant velocity worldline.  This intuition can be formalised by a Morawetz-type energy estimate that shows that the nonlinear potential energy must decay along any rectangular slab of spacetime (that represents the neighbourhood of a constant velocity worldline).

Now, just because the linear wave equation propagates along constant velocity worldlines, this does not mean that the nonlinear wave equation does too; one could imagine that a wave packet could propagate along a more complicated trajectory $t \mapsto x(t)$ in which the velocity $x'(t)$ is not constant.  However, energy methods still force the solution of the nonlinear wave equation to obey finite speed of propagation, which in the wave packet context means (roughly speaking) that the nonlinear trajectory $t \mapsto x(t)$ is a Lipschitz continuous function (with Lipschitz constant at most $1$).

And now we deploy a trick which appears to be new to the field of nonlinear wave equations: we invoke the Rademacher differentiation theorem (or Lebesgue differentiation theorem), which asserts that Lipschitz continuous functions are almost everywhere differentiable.  (By coincidence, I am teaching this theorem in my current course, both in one dimension (which is the case of interest here) and in higher dimensions.)  A compactness argument allows one to extract a quantitative estimate from this theorem (cf. this earlier blog post of mine) which, roughly speaking, tells us that there are large portions of the trajectory $t \mapsto x(t)$ which behave approximately linearly at an appropriate scale.  This turns out to be a good enough control on the trajectory that one can apply the Morawetz inequality and rule out the existence of persistent wave packets over long periods of time, which is what leads to Theorem 1.

There is still scope for further work to be done on the asymptotics.  In particular, we still do not have a good understanding of what the asymptotic profile of the solution should be, even in the perturbative regime; standard nonlinear geometric optics methods do not appear to work very well due to the extremely weak decay.