Large time behavior for a quasilinear diffusion equation with weighted source

Abstract

The large time behavior of general solutions to a class of quasilinear diffusion equations with a weighted source term $$ \partial_tu=\Delta u^m+\varrho(x)u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ with $m>1$, $1<p<m$ and suitable functions $\varrho(x)$, is established. More precisely, we consider functions $\varrho\in C(\mathbb{R}^N)$ such that $$ \lim\limits_{|x|\to\infty}(1+|x|)^{-\sigma}\varrho(x)=A\in(0,\infty), $$ with $\sigma\in(\max\{-N,-2\},0)$ such that $L:=\sigma(m-1)+2(p-1)<0$. We show that, for all these choices of $\varrho$, solutions with initial conditions $u_0\in C(\mathbb{R}^N)\cap L^{\infty}(\mathbb{R}^N)\cap L^r(\mathbb{R}^N)$ for some $r\in[1,\infty)$ are global in time and, if $u_0$ is compactly supported, present the asymptotic behavior $$ \lim\limits_{t\to\infty}t^{-\alpha}\|u(t)-V_*(t)\|_{\infty}=0, $$ where $V_*$ is a suitably rescaled version of the unique compactly supported self-similar solution to the equation with the singular weight $\varrho(x)=|x|^{\sigma}$: $$ U_*(x,t)=t^{\alpha}f_*(|x|t^{-\beta}), \qquad \alpha=-\frac{\sigma+2}{L}, \quad \beta=-\frac{m-p}{L}. $$ This behavior is an interesting example of \emph{asymptotic simplification} for the equation with a regular weight $\varrho(x)$ towards the singular one as $t\to\infty$.