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 - \u0394u = u^p with Neumann boundary value \\frac{\u2202u}{\u2202\u03bd} = 0 on some unbounded domains, where p > 1, \u03bd is the outward normal vector on boundary \u2202\u03a9. We prove that there exists a critical exponent p_c = p_c(\u03a9) > 1 such that if p \u2208 (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>"