Travelling-wave behaviour in doubly nonlinear reaction-diffusion equations

Abstract

We study a family of reaction-diffusion equations that present a doubly nonlinear character given by a combination of the p-Laplacian and the porous medium operators. We consider the so-called slow diffusion regime, corresponding to a degenerate behaviour at the level 0, in which nonnegative solutions with compactly supported initial data have a compact support for any later time. For some results we will also require p ≥ 2 to avoid the possibility of a singular behaviour away from 0.Problems in this family have a unique (up to translations) travelling wave with a finite front. When the initial datum is bounded, radially symmetric and compactly supported, we will prove that solutions converging to 1 (which exist, as we show, for all the reaction terms under consideration for wide classes of initial data) do so by approaching a translation of this unique traveling wave in the radial direction, but with a logarithmic correction in the position of the front when the dimension is bigger than one. As a corollary we obtain the asymptotic location of the free boundary and level sets in the non-radial case up to an error term of size O(1). In dimension one we extend our results to cover the case of non-symmetric initial data, as well as the case of bounded initial data with supporting sets unbounded in one direction of the real line. A main technical tool of independent interest is an estimate for the flux.Most of our results are new even for the special cases of the porous medium equation and the p-Laplacian evolution equation.

Convergence in relative error for the Porous Medium equation in a tube

“…TW fronts arise frequently in the study of reaction-diffusion equations (see for instance [17,22,1,8] and [10,9,7,20,4,5,3,16,18,15] for nonlinear diffusion models) and are not expected in purely diffusive models. The reason why TW solutions appear in our context is the geometry of the domain: when Ω is a tube, solutions spread along the longitudinal direction, with a loss of mass through the fixed boundary ∂Ω which, in turn, is compensated by a linear reaction term appearing in the rescaled problem.…”