Asymptotics of the Fast-Diffusion Equation with Critical Exponent
Victor A. Galaktionov, Lambertus A. Peletier, Juan L. Vazquez · SIAM Journal on Mathematical Analysis · 2000
We study the large-time behavior of the solutions of the initial-value problem for the nonlinear diffusion equation $$ u_t= abla\cdot (u^{-\s} abla u) \quad \mbox{in ${\bf R}^n \times {\bf R}_+$} \leqno{{\rm (ND)}} $$ in dimensions $ n \geq 3$ with nonnegative initial data $u(x,0)\in L^1({\bf R}^n)$ when the exponent takes on the \textit{critical value} $\sigma =2/n$. This represents a borderline case in the study of the problem and offers marked qualitative and technical differences with the neighboring cases $\sigma \approx 2/n$, $\sigma eq 2/n$. In particular, it marks the transition between two completely different asymptotic behavior types. It is known that solutions exist globally in time and conserve the L 1 -norm for this problem. We prove that they decay exponentially in time with a complicated law: $$ \log \|u(\cdot,t)\|_\infty \sim - \kappa M^{-2/(n-2)} t^{n/(n-2)} \quad \mbox{as}\quad t \to \infty, $$ where $M=\int u(x,0)dx$ is the conserved mass and the constant $\kappa > 0$ depends only on the dimension n. This strongly differs from the comparatively simple self-similar asymptotics of the case $\sigma <2/n$. The description is split into an inner and an \textit{outer} region, conveniently matched at a transition layer. The analysis of the outer region can be done independently and the behavior is governed by a first-order conservation law which acts as the reduced asymptotic equation. The uniqueness theory for first-order conservation laws is one of the great contributions of S. N. Kruzhkov to mathematics. The behavior in the inner parabolic region is then studied by means of a semiconvexity argument which makes it possible to translate into this region the precise behavior from the outer region.