On nonexistence of Baras-Goldstein type without positivity assumptions for singular linear and nonlinear parabolic equations |
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Authors: | V A Galaktionov |
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Institution: | (1) Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK |
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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 ⊂ ℝ
N
, N ≥ 3, u
t
= Δ
u
+ in B
1 × (0,T), u|∂B
1 = 0, in the supercritical range c > c
Hardy = does not have a solution for any nontrivial L
1 initial data u
0(x) ≥ 0 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) = ↦ V
n
(x) = min{ , n} (n ≥ 1) diverges; i.e., as n → ∞, u
n
(x,t) → + ∞ in B
1 × (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
= , and , N > 4.
Dedicated to Professor S.I. Pohozaev on the occasion of his 70th birthday |
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Keywords: | |
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