We consider positive solutions of a semilinear parabolic system coupled by power nonlinearities, in a ball or in the whole space. Relatively little is known on the blow-up set for parabolic systems and, up to now, no result was available for this basic system except for the very special case of equal powers. 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 blow-up profiles.
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Philippe Souplet, Single-point blow-up for a semilinear parabolic system. J. Eur. Math. Soc. 11 (2009), no. 1, pp. 169–188DOI 10.4171/JEMS/145