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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Volume 260, Pages 130–150 (Mi tm590)  

This article is cited in 4 scientific papers (total in 4 papers)

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

V. A. Galaktionov

Department of Mathematical Sciences, University of Bath
Full-text PDF (691 kB) Citations (4)
References:
Abstract: 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 $u_t=-\Delta^2u+\frac c{|x|^4}u$, $c>c_\mathrm H=\bigl[\frac{N(N-4)}4\bigr]^2$, $N>4$.
Received in July 2007
English version:
Proceedings of the Steklov Institute of Mathematics, 2008, Volume 260, Pages 123–143
DOI: https://doi.org/10.1134/S0081543808010094
Bibliographic databases:
UDC: 517.9
Language: English
Citation: V. A. Galaktionov, “On Nonexistence of Baras–Goldstein Type without Positivity Assumptions for Singular Linear and Nonlinear Parabolic Equations”, Function theory and nonlinear partial differential equations, Collected papers. Dedicated to Stanislav Ivanovich Pohozaev on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 260, MAIK Nauka/Interperiodica, Moscow, 2008, 130–150; Proc. Steklov Inst. Math., 260 (2008), 123–143
Citation in format AMSBIB
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\paper On Nonexistence of Baras--Goldstein Type without Positivity Assumptions for Singular Linear and Nonlinear Parabolic Equations
\inbook Function theory and nonlinear partial differential equations
\bookinfo Collected papers. Dedicated to Stanislav Ivanovich Pohozaev on the occasion of his 70th birthday
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
     
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