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

الأصل · EN

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem {array [c]c% uₜ-νΔu+|∇ u|q=0, u(0)=u₀, array. in QΩ,ₜ=Ω×(0,T), where q>1,ν 0,T∈(0,∞], and Ω=Rⁿ or Ω is a smooth bounded domain, and u₀∈ Lʳ(Ω),r1, or u₀% (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∈ Mb(Ω) and q<(N+2)/(N+1), or u₀∈ Lʳ(Ω) and q<(N+2r)/(N+r). We also extend some decay properties to quasilinear equations of the model type uₜ-Δₚu+ u λ⁻¹u|∇ u|q=0 where p>1,λ0, and u is a signed solution.