Existence and nonexistence of solutions for the heat equation with a superlinear source term

2016/09/21 by Yohei Fujishima, Fujishima, Yohei, Norisuke Ioku +1 · 5 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1609.06394

openalex publication_date 2016/09/21 · openalex created_date 2016/09/30 · openalex updated_date 2026/07/28

Abstract

Classification theory on the existence and non-existence of local in time solutions for initial value problems of nonlinear heat equations are investigated. Without assuming a concrete growth rate on a nonlinear term, we reveal the threshold integrability of initial data which classify existence and nonexistence of solutions via a quasi-scaling and its invariant integral. Typical nonlinear terms, for instance polynomial type, exponential type and its sum, product and composition, can be treated as applications.

Cited by

• The lifespan of positive solutions of heat equation with power-logarithmic nonlinearity on locally finite graph
• Non-uniqueness of positive solutions for supercritical semilinear heat equations without scale invariance
• Asymptotic Blow-up Behavior for the Semilinear Heat Equation with Super-exponential Nonlinearities
• Multiplicity of singular solutions for semilinear elliptic equations with superlinear source terms
• Singular Dirichlet boundary problems for a class of fully nonlinear parabolic equations in one dimension

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