Precise travelling-wave behaviour in problems with doubly nonlinear diffusion

Abstract

We study a family of reaction-diffusion equations of the form ut =Δp​um+h(u) for x∈RN, with a doubly nonlinear diffusion term Δpum involving both the p-Laplacian and the porous medium operators. The reaction term h(u) is also rather general, covering in particular monostable, bistable and combustion type nonlinearities. We consider the so-called slow diffusion regime, which leads to a degenerate behaviour at the level u=0, and so nonnegative solutions with compactly supported initial data have a compact support for any later time, hence generating a free boundary. Equations of this family have a unique (up to translations) travelling wave with a finite front (free boundary). When the initial datum is compactly supported and the solution converges to 1 (which is the case, as we show, for wide classes of such initial data), in the radially symmetric case, we prove that the solution converges to a translation of this unique travelling wave in the radial direction, with a precise logarithmic correction in the position of the free boundary when the dimension N≥2; and in the nonradial case, we obtain the asymptotic location of the free boundary and level sets up to an error term of size O(1). Such precise results have been known in high space dimensions only in the special case p=2 and h(u) a particular monostable nonlinearity from the recent work by Du, Quirós and Zhou (2020). The extension to the much more general situation here relies on several new techniques, including a crucial estimate for the flux, which is new even for the case h(u)≡0 in high space dimensions, and is of independent interest. Most of our results are new also for the special cases (a) p=2 (porous medium diffusion) and (b) m=1 (p-Laplacian diffusion).