Existence and Nonexistence of Global Solutions for Semilinear Heat Equation on Unbounded Domain (Journal of Partial Differential Equations, 2020)
"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."
Keywords: Semilinear heat equation; global existence; critical exponent of Fu-jita' s type