Affiliations: [a] Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, Brazil. E-mail: [email protected] | [b] Instituto de Matemática e Computação, Universidade Federal de Itajubá, Itajubá, MG, Brazil. E-mails: [email protected], [email protected]
Correspondence:
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Corresponding author: Jacson Simsen, Instituto de Matemática e Computação, Universidade Federal de Itajubá, 37500-903 Itajubá, MG, Brazil. E-mail: [email protected].
Abstract: We study the asymptotic behavior of parabolic p(x)-Laplacian problems of the form
∂uλ∂t−div(Dλ|∇uλ|p(x)−2∇uλ)+a|uλ|p(x)−2uλ=B(uλ)
in L2(Rn), where n⩾1, p∈L∞(Rn) such that 2<p−:=ess infp(x)⩽p(x)⩽p+:=ess supp(x), Dλ∈L∞(Rn), ∞>M⩾Dλ(x)⩾σ>0 a.e. in Rn, λ∈[0,∞), B:L2(Rn)→L2(Rn) is a globally Lipschitz map and a:Rn→R is a non-negative continuous function such that there exists R1>0 with {x∈Rn;a(x)=0}⊂BR1(0), infx∈Rn∖BR1(0)a(x)>0, and
∫Rn∖BR1(0)1a(x)2/(p(x)−2)dx<+∞.
We also study the sensitivity of the problem according to the variation of the diffusion coefficients.
