L^∞ estimates and uniqueness results for nonlinear parabolic equations with gradient absorption terms

2012/02/13 by Marie-Françoise Bidaut-Véron, Bidaut-Véron, Marie-Françoise, Nguyen Anh Dao +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1202.2674

arxiv created 2013/03/22 · arxiv updated 2013/03/25

Abstract

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem \% u_t-νΔu+|∇ u|^q=0, u(0)=u₀, . in Q_Ω,T=Ω×(0,T) , where q>1,ν\geqq 0,T∈(0,∞] , and Ω=ℝ^N or Ω is a smooth bounded domain, and u₀∈ L^r(Ω),r\geqq1, or u₀% \inM_b(Ω). We show L^∞ decay estimates, valid for any weak solution, without any conditions as ‖ x‖ →∞, and without uniqueness assumptions. As a consequence we obtain new uniqueness results, when u₀∈ M_b(Ω) and q<(N+2)/(N+1), or u₀∈ L^r(Ω) and q<(N+2r)/(N+r). We also extend some decay properties to quasilinear equations of the model type u_t-Δ_pu+‖ u‖ ^λ-1u|∇ u|^q=0 where p>1,λ\geqq0, and u is a signed solution.

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