From fast to very fast diffusion in the nonlinear heat equation

Abstract:

We study the asymptotic behavior of the sign-changing solution of the equation $\displaystyle u_t=\nabla \cdot (\vert u\vert ^{{-\alpha }} \nabla u)+f ,$ when the diffusion becomes very fast, i.e. as $\displaystyle \alpha \uparrow 1.$ We prove that a solution $u_\alpha (t)$ converges in $\displaystyle L^1(\Omega ),$ uniformly for $t$ in subsets with compact support in $(0,T),$ to a solution of $\displaystyle u_t=\nabla \cdot (\vert u\vert ^{-1} \nabla u)+f .$ In contrast with the case of $\alpha <1,$ we prove that the singularity $0$ created in the limiting problem, i.e. $\alpha =1,$ is an obstruction to the existence of sign-changing solutions. More precisely, we prove that, for each $t\geq 0,$ the limiting solutions are either positive or negative or identically equal to $0$ in all $\Omega .$ This causes the limit to be singular, in the sense that a boundary layer appears at $t=0,$ when one lets $\alpha \uparrow 1.$