Large time behavior for a quasilinear diffusion equation with weighted source
Razvan Gabriel Iagar, Marta Latorre, Ariel Sánchez
TL;DR
The work analyzes the large-time behavior of solutions to $u_t=\Delta u^m+\varrho(x)u^p$ in $\mathbb{R}^N$ with $m>1$, $1<p<m$ and spatially inhomogeneous sources, under weights that decay like $\varrho(x)\sim A|x|^{\sigma}$ with $\sigma\in(\max\{-N,-2\},0)$ and $L:=\sigma(m-1)+2(p-1)<0$. It shows global existence and finite speed of propagation for broad data, and that for compactly supported initial data the solution converges, after an appropriate self-similar rescaling, to a rescaled self-similar solution $U_*(x,t)=t^{\alpha}f_*(|x|t^{-\beta})$ of the singular-weight equation, with $V_*(x,t)=A^{1/(m-p)}U_*(x,A^{(m-1)/(m-p)}t)$ and $\alpha=-(\sigma+2)/L$, $\beta=-(m-p)/L$. The core method blends sub/supersolution construction, self-similar changes of variables, and the S-theorem of Galaktionov–Vázquez to demonstrate an asymptotic simplification: the regular-weight problem asymptotically behaves like the singular-weight one. The general weights case follows by a limiting argument in the self-similar variables, showing that the same rescaled singular-profile governs the large-time dynamics with $V_*(x,t)$ reflecting the constant $A$ from $\varrho(x)$. Overall, the paper links the long-time dynamics to a Hardy–Henon-type self-similar profile and establishes robust convergence under broad weight assumptions.
Abstract
The large time behavior of general solutions to a class of quasilinear diffusion equations with a weighted source term $$ \partial_tu=Δu^m+\varrho(x)u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ 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 $$ \lim\limits_{|x|\to\infty}(1+|x|)^{-σ}\varrho(x)=A\in(0,\infty), $$ with $σ\in(\max\{-N,-2\},0)$ such that $L:=σ(m-1)+2(p-1)<0$. 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 $$ \lim\limits_{t\to\infty}t^{-α}\|u(t)-V_*(t)\|_{\infty}=0, $$ where $V_*$ is a suitably rescaled version of the unique compactly supported self-similar solution to the equation with the singular weight $\varrho(x)=|x|^σ$: $$ U_*(x,t)=t^αf_*(|x|t^{-β}), \qquad α=-\frac{σ+2}{L}, \quad β=-\frac{m-p}{L}. $$ This behavior is an interesting example of \emph{asymptotic simplification} for the equation with a regular weight $\varrho(x)$ towards the singular one as $t\to\infty$.