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 \(t\to\infty\).