On Nonexistence of Baras--Goldstein Type without Positivity Assumptions for Singular Linear and Nonlinear Parabolic Equations

V. A. Galaktionov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and nonlinear partial differential equations, Tome 260 (2008), pp. 130-150

Résumé

The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball $B_1\subset\mathbb R^N$, $N\ge3$, $u_t=\Delta u+\frac c{|x|^2}u$ in $B_1\times(0,T)$, $u|_{\partial B_1}=0$, in the supercritical range $c>c_\mathrm{Hardy}=\bigl(\frac{N-2}2\bigr)^2$ does not have a solution for any nontrivial $L^1$ initial data $u_0(x)\ge0$ in $B_1$ (or for a positive measure $u_0$). More precisely, it was proved that a regular approximation of a possible solution by a sequence $\{u_n(x,t)\}$ of classical solutions corresponding to truncated bounded potentials given by $V(x)=\frac c{|x|^2}\mapsto V_n(x)=\min\bigl \{\frac c{|x|^2},n\bigr\}$ ($n\ge1$) diverges; i.e., as $n\to\infty$, $u_n(x,t)\to+\infty$ in $B_1\times(0,T)$. Similar features of “nonexistence via approximation” for semilinear heat PDEs were inherent in related results by Brezis–Friedman (1983) and Baras–Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data $u_0(x)$ that are assumed to be regular and positive at $x=0$. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., $u_t =\Delta(|u|^{m-1}u)+\frac{|u|^{p-1}u}{|x|^2}$, $m\ge1$, $p>1$, and $u_t=-\Delta^2u+\frac c{|x|^4}u$, $c>c_\mathrm H=\bigl[\frac{N(N-4)}4\bigr]^2$, $N>4$.