On directional blow-up for a semilinear heat equation with space-dependent reaction

We consider nonnegative solutions u of the Cauchy problem for a semilinear heat equation with space-dependent reaction: u(t) = Delta u + mu(x)u(p), u( x, 0) = u(0)(x), where mu(x) >= 0 satisfies some condition and the initial data u(0)(x) ((sic) 0) satisfies parallel to(mu) over tildeu(0)parallel to(L infinity(RN)) < infinity with <(mu)over tilde> = mu(1/(p-1)). We study weighted solutions (mu) over tilde which blow up at minimal blow-up time. Such a weighted solution blows up at space infinity in some direction (directional blow-up). We call this direction a blow-up direction of (mu) over tildeu. We give a sufficient and necessary condition on u(0) for a weighted solution to blow up at minimal blow-up time. Moreover, we completely characterize blow-up directions of (mu) over tildeu by the profile of the initial data. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.