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Acta Mathematica Vietnamica

A Degenerate Forward-backward Problem Involving the Spectral Dirichlet Laplacian

Nguyen Ngoc Trong , Bui Le Trong Thanh , icon-email Tan Duc Do

Abstract

Let Ω be an open bounded subset of R, s(12,1) and ϵ>0. We investigate the problem (Pϵ){tu=(Δ)s(φ(u)+ϵt(ψ(u))) in Ω×(0,T],φ(u)+ϵt(ψ(u))=0 on Ω×(0,T],u=u0 in Ω×{0}, where φ,ψC(R) and u0M+(Ω) satisfy certain assumptions. Here (Δ)s denotes the spectral Dirichlet Laplacian and M+(Ω) is the set of positive Radon measures on Ω. We show that (Pϵ) has a unique weak solution.