Precise travelling-wave behaviour in problems with doubly nonlinear diffusion

Abstract

We study a family of reaction-diffusion equations of the form $u_t={\Delta }_p{u}^m+h(u)$ for $x\in {\mathbb{R}}^N$, with a doubly nonlinear diffusion term ${\Delta }_p{u}^m$ 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 $N\ge 2$; 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 $h(u)\equiv 0$ 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).