Precise travelling-wave behaviour in problems with doubly nonlinear diffusion

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\geq 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).