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

We study the nonnegative solutions of the viscous Hamilton–Jacobi problem {ut−νΔu+|∇u|q=0,u(0)=u0, in QΩ,T=Ω×(0,T)QΩ,T=Ω×(0,T), where q>1,ν≧0,T∈(0,∞]q>1,ν≧0,T∈(0,∞], and Ω=RNΩ=RN or ΩΩ is a smooth bounded domain, and u0∈Lr(Ω),r≧1u0∈Lr(Ω),r≧1, or u0∈Mb(Ω)u0∈Mb(Ω). We show L∞L∞ decay estimates, valid for any weak solution, without any conditions a  s |x|→∞|x|→∞, and without uniqueness assumptions  . As a consequence we obtain new uniqueness results, when u0∈Mb(Ω)u0∈Mb(Ω) and q<(N+2)/(N+1)q<(N+2)/(N+1), or u0∈Lr(Ω)u0∈Lr(Ω) and q<(N+2r)/(N+r)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,ut−Δpu+|u|λ−1u|∇u|q=0, where p>1,λ≧0p>1,λ≧0, and uu is a signed solution.