Renormalized solutions for nonlinear parabolic equations with general measure data

Abstract

We prove the existence of parabolic initial boundary value problems of the type $$\displaylines{ u_t-\text{div}(a_{\epsilon}(t,x,u_{\epsilon},\nabla u_{\epsilon})) =\mu_{\epsilon}\quad\text{in }Q:=(0,T)\times\Omega,\cr u_{\epsilon}=0\quad\text{on }(0,T)\times\partial \Omega,\quad u_{\epsilon}(0)=u_{0,\epsilon}\quad\text{in }\Omega, }$$ with respect to suitable convergence of the nonlinear operators $a_{\epsilon}$ and of the measure data $\mu_{\epsilon}$. As a consequence, we obtain the existence of a renormalized solution for a general class of nonlinear parabolic equations with right-hand side measure.

Keywords

• Nonlinear parabolic problems
• p-capacity
• renormalized solution
• stability
• general measure