On nonexistence of Baras-Goldstein type without positivity assumptions for singular linear and nonlinear parabolic equations
Виктор Александрович Галактионов · Proceedings of the Steklov Institute of Mathematics · 2008
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 + $$ \tfrac{c} {{|x|^2 }}u $$ in B 1 × (0,T), u|∂B 1 = 0, in the supercritical range c > c Hardy = $$ (\tfrac{{N - 2}} {2})^2 $$ 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) = $$ \tfrac{c} {{|x|^2 }} $$ ↦ V n (x) = min{ $$ \tfrac{c} {{|x|^2 }} $$ , 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 = $$ \Delta (|u|^{m - 1} u) + \tfrac{{|u|^{p - 1} u}} {{|x|^2 }}, m \geqslant , p > 1 $$ , and $$ \Delta ^2 u + \tfrac{c} {{|x|^4 }}u, c > c_H = [\tfrac{{N(N - 4)}} {4}]^2 $$ , N > 4.