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Blow-Up Rate Estimates and Liouville Type Theorems for a Semilinear Heat Equation with Weighted Source
Authors:Quoc Hung Phan
Institution:1.Institute of Research and Development,Duy Tan University,Da Nang,Vietnam
Abstract:We study the Liouville-type theorem for the semilinear parabolic equation \(u_t-\Delta u =|x|^a u^p\) with \(p>1\) and \(a\in {\mathbb R}\). Relying on the recent result of Quittner (Math Ann, doi: 10.1007/s00208-015-1219-7, 2015), we establish the optimal Liouville-type theorem in dimension \(N=2\), in the class of nonnegative bounded solutions. We also provide a partial result in dimension \(N\ge 3\). As applications of Liouville-type theorems, we derive the blow-up rate estimates for the corresponding Cauchy problem.
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