Pointwise Bounds and Blow-up for Systems of Semilinear Parabolic Inequalities and Nonlinear Heat Potential Estimates

2015/04/19 by Ghergu, Marius, Taliaferro, Steven D.
#35B09 #35B33 #35B44 #35B45 #35K40 #35R45 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1504.04841

Abstract

We study the behavior for t small and positive of C^2,1 nonnegative solutions u(x,t) and v(x,t) of the system 0≤ u_t-Δu≤ v^λ 0≤ v_t-Δv≤ u^σ in Ω× (0,1), where λ and σ are nonnegative constants and Ω is an open subset of \mathbb Rⁿ, n≥ 1. We provide optimal conditions on λ and σ such that solutions of this system satisfy pointwise bounds in compact subsets of Ω as t→ 0⁺. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials.

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