Local and global estimates of solutions of Hamilton-Jacobi parabolic equation with absorption

2014/07/08 by Marie-Françoise Bidaut-Véron, Bidaut-Véron, Marie-Françoise
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1407.1969

arxiv created 2014/07/11 · arxiv updated 2014/07/14

Abstract

We obtain new a priori estimates for the nonnegative solutions of the equation u_t-Δu+|∇ u|^q=0 in Q_Ω,T=Ω×( 0,T) , T\leqq∞, where q>0, and Ω=ℝ^N, or Ω is a smooth bounded domain of ℝ^N and u=0 on ∂Ω×( 0,T) . In case Ω=ℝ^N, we show that any solution u∈ C^2,1(Q_ℝ^N,T) of equation (1.1) in Q_{ℝ^N ,T} (in particular any weak solution if q\leqq2), without condition as \vert x\vert →∞, satisfies the universal estimate \vert ∇ u(.,t)\vert ^q\leqq(1)/(q-1)\fracu(.,t)% t,\qquadin Q_ℝ^N,T. Moreover we prove that the growth of u is limited by C(t+t^-1/(q-1% )(1+\vert x\vert ^q^′), where C depends on u. We also give existence properties of solutions in Q_Ω,T, for initial data locally integrable or even unbounded Radon measures. We give a nonuniqueness result in case q>2. Finally we show that besides the local regularizing effect of the heat equation, u satisfies a second effect of type L_loc^R% -L_loc^∞, due to the gradient term.

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