Nonlinear parabolic equations with Robin boundary conditions and Hardy-Leray type inequalities

Abstract

We are primarily concerned with the absence of positive solutions of the following problem, ⎧ ∂u ⎪⎨ ∂t=Δ(um ) + V (x)um + λuq in Ω × (0, T ), u(x, 0) = u0 (x) ≥ 0 in Ω, ⎪⎩ ∂u ∂ν=β(x)u on ∂Ω × (0, T ), where 0 < m < 1, V ∈ L1loc(Ω),β∈ L1loc(∂Ω), λ ∈ R, q > 0, Ω ⊂ RN is a bounded open subset of RN with smooth boundary ∂Ω, and∂u is the ∂ν outer normal derivative of u on ∂Ω. Moreover, we also present some new sharp Hardy and Leray type inequalities with remainder terms that provide us concrete potentials to use in the partial differential equation of our interest.

Publication Title

Contemporary Mathematics

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