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Single-point blow-up for a semilinear parabolic system

Ph. Souplet — 2009

Journal of the European Mathematical Society

We consider positive solutions of the system $u_{t} - Δ u = v^{p}$ ; $v_{t} - Δ v = u^{q}$ in a ball or in the whole space, with $p, q > 1$ . Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case $p = q$ . Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for the final...

A remark on the large time behavior of solutions of viscous Hamilton-Jacobi equations

Souplet, Ph. — 2007

Proceedings of Equadiff 11

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