Nonlinear degenerate parabolic equations with Neumann boundary condition
The goal of this paper is to give a necessary and sufficient condition for the existence of solutions to the following initial boundary value problem \[ u_t-\Delta v= f(t, x),\;v\in\beta(u)\text{ in }(0,T)\times\Omega,\quad {\partial v\over\partial n}= 0\text{ on }(0,T)\times\partial\Omega,\quad u(0,x)= u_0(x)\text{ in }\Omega,\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\geq 1\). Under some suitable assumptions of \(f\), \(u_0\) and \(\beta\), the authors present a necessary and sufficient condition for existence of solutions for (1). Existence of solutions for (1) based on three steps: approximation, uniform estimate and limiting process.
scientific article; zbMATH DE number 2174935