Existence and regularity results for solutions to nonlinear parabolic equations (Q2504062)

Existence and regularity is proved for solutions to a class of nonlinear parabolic equations of the form \[ \begin{cases} {{\partial u}\over{\partial t}}-\Delta_p u=f(x,t), & x\in \Omega,\;\;t\in(0,T),\\ u(x,t)=0, & x\in \partial\Omega,\;t\in (0,T),\\ u(x,0)=0, & x\in \Omega, \end{cases} \] with smooth and bounded domain \(\Omega\subset {\mathbb R}^N,\) \(N\geq2,\) \(\Delta_p\) is the \(p\)-Laplace operator, \(p>1,\) and \(f\in L^r(0,T;L^q(\Omega))\) with suitable \(r\geq1\) and \(q\geq 1.\)