Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms

Abstract

We want to analyse both regularizing effect and long, short time decay concerning parabolic Cauchy-Dirichlet problems of the type \begin{equation*} \begin{cases} \begin{array}{ll} u_t-\text{div} (A(t,x)|\nabla u|^{p-2}\nabla u)=γ|\nabla u|^q & \text{in}\,\,Q_T,\\ u=0 &\text{on}\,\,(0,T)\times\partialΩ,\\ u(0,x)=u_0(x) &\text{in}\,\, Ω. \end{array} \end{cases} \end{equation*} We assume that $A(t,x)$ is a coercive, bounded and measurable matrix, the growth rate $q$ of the gradient term is superlinear but still subnatural, $\gamma >0$, the initial datum $u_0$ is an unbounded function belonging to a well precise Lebesgue space $L^{\sigma }(\Omega )$ for $\sigma =\sigma (q,p,N)$.

First correction: added reference [10], corrected typos, added Definition 4.1, corrected Definitions 3.1 and 5.2, results unchanged. Second correction: corrected typos, results unchanged

References in corpus (3)

• Diffuse measures and nonlinear parabolic equations
• Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term
• Local and global time decay for parabolic equations with super linear first order terms