Supersolutions for a class of semilinear heat equations

2011/11/01 by Robinson, James C., Sierzega, Mikolaj · 3 citations
#35K08 #35K58 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1111.0258

Abstract

A semilinear heat equation u_t=Δu+f(u) with nonnegative initial data in a subset of L¹(Ω) is considered under the assumption that f is nonnegative and nondecreasing and Ω⊆ \Rⁿ. A simple technique for proving existence and regularity based on the existence of supersolutions is presented, then a method of construction of local and global supersolutions is proposed. This approach is applied to the model case f(s)=s^p, ϕ∈ L^q(Ω): new sufficient conditions for the existence of local and global classical solutions are derived in the critical and subcritical range of parameters. Some possible generalisations of the method to a broader class of equations are discussed.

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