Classification of Global and Blow-up Sign-changing Solutions of a Semilinear Heat Equation in the Subcritical Fujita Range: Second-order Diffusion

摘要

It is well known from the seminal paper by Fujita [22] for 1 < p < p_0, and Hayakawa [36] for the critical case p = p_0, that all the solutions u ≥ 0 of the semilinear heat equation u_t = Δu + |u|~(p-1)u in ?~N × ?_+, in the range 1 < p ≤ p_0 = 1 + 2/N, (0.1) with arbitrary initial data u_0(x) ≥ 0, 0, blow-up in finite time, while for p > p_0 there exists a class of sufficiently "small" global in time solutions. This fundamental result from the 1960-70s (see also [39] for related contributions), was a cornerstone of further active blow-up research. Nowadays, similar Fujita-type critical exponents p_0, as important characteristics of stability, unstability, and blow-up of solutions, have been calculated for various nonlinear PDEs. The above blow-up conclusion does not include solutions of changing sign, so some of them may remain global even for p ≤ p_0. Our goal is a thorough description of blow-up and global in time oscillatory solutions in the subcritical range in (0.1) on the basis of various analytic methods including nonlinear capacity, variational, category, fibering, and invariant manifold techniques. Two countable sets of global solutions of changing sign are shown to exist. Most of them are not radially symmetric in any dimension N ≥ 2 (previously, only radial such solutions in ?~N or in the unit ball B_1 c ?~N were mostly studied). A countable sequence of critical exponents, at which the whole set of global solutions changes its structure, is detected: p_l = 1 + 2/N+l, l = 0, 1, 2,..... See [47, 48] for earlier interesting contributions on sign changing solutions.