Existence and Nonexistence of Global Solutions for Semilinear Heat Equation on Unbounded Domain

Abstract

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In this paper, we consider the existence and nonexistence of global solutions to the semilinear heat equation u_t - Δu = u^p with Neumann boundary value \\frac{∂u}{∂ν} = 0 on some unbounded domains, where p > 1, ν is the outward normal vector on boundary ∂Ω. We prove that there exists a critical exponent p_c = p_c(Ω) > 1 such that if p ∈ (1, p_c], for nonnegative and nontrivial initial data, all positive solutions blow up in finite time; if p > p_c, for suitably small nonnegative initial data, there exists a global positive solution.<\/p>"