A priori \(L^\infty\)-estimate and existence of solutions for some nonlinear parabolic equations
The author studies the following nonlinear parabolic problem \[ \begin{cases} u_t-\text{ div}(a(x,t,u,\nabla u)+\Phi(u)) =b(x,t,u,\nabla u) -\text{div} g,\\ u|_\Sigma =0,\cr u(x,0)=u_0(x)\in L^\infty(\Omega),\end{cases} \] where \(\Omega\) is a bounded domain in \({\mathbb R}^N\), \(\Sigma=\partial \Omega\times(0,T).\) The functions \(a\) and \(b\) are supposed to be Carathéodory functions satisfying suitable coercivity, growth and monotonicity conditions, respectively. It is proved: an \(L^\infty\)-a priori estimate for the solution of the problem and its weak solvability in the space \(L^\infty(Q)\cap L^p(0,T;W^{1,p}_0(\Omega))\cap C([0,T], L^q(\Omega))\) for all \(q\in[1,\infty).\)
coercivity, growth and monotonicity conditions
scientific article; zbMATH DE number 1513301
scientific article