Critical Fujita absorption exponent for evolution $p$-Laplacian with inner absorption and boundary flux

Abstract

This paper studies the evolution $p$-Laplacian equation with inner absorption $u_t=(|u_x|^{p-2}u_x)_x-\lambda u^{\kappa}$ in $\mathbb{R}^+\times \mathbb{R}^+$ subject to nonlinear boundary flux $-|u_x|^{p-2}u_x(0,t)=u^q(0,t)$. First, we determine the critical boundary flux exponent $q^*=\max\{\frac{2(p-1)}p$, $\frac{(\kappa+1)(p-1)}{p}\}$ to identify global and nonglobal solutions, relying on or independent of initial data. It is interesting to find that, for the critical case $q=q^*$ (the balanced case between inner absorption and boundary flux) with the absorption coefficient $\lambda > 0$ small, there is the so-called critical Fujita absorption exponent $\kappa_c=2p-1 > \kappa_0=1$ related to a Fujita type conclusion that (i) when $\kappa\le \kappa_0$, all solutions are global; (ii) if $\kappa_0 < \kappa < \kappa_c$, the solutions blow up in finite time under any nontrivial nonnegative initial data; (iii) if $\kappa > \kappa_c$, there are both global and nonglobal solutions, respectively, determined by the sizes of initial data. Obviously, when $\lambda > 0$ large for the balanced case $q=q^*$, the absorption would overcome the boundary flux to yield global solutions. Furthermore, we will show how and in what ways the inner absorption affects the evolution of the support of solutions, where a complete classification for all the nonlinear parameters is given to distinguish localization and non-localization behavior of the global solutions.