Global Existence for Nonlinear Parabolic Problems With Measure Data— Applications to Non-uniqueness for Parabolic Problems With Critical Gradient terms

摘要

In the present article we study global existence for a nonlinear parabolic equation having a reaction term and a Radon measure datum:(Ψ(v))_t - Δ_pv = f(x, t)( 1 + Ψ(v)) + μin Ω x (0, +∞) ,v(x,= 0on lΩ× (0, +∞) ,v(x, 0) = v_0(x)in Ω,where I < p < N, Ω is a bounded open subset of R~N (N≥ 2), Δ_pu= div(|Δ_u|~(p-2)Δ_u) is the so called p-Laplacian operator,Ψ(s)=[(1+|s|/(p-1))~(p-1)-1] sign s., Ψ(v_0) ∈ L~1 (Ω), μ is a finite Radon measure and f ∈ × (0, T)) for every T > 0. Then we apply this existence result to show wild nonuniqueness for a connected nonlinear parabolic problem having a gradient term with natural growth.