Existence and uniqueness of periodic solutions for a nonlinear reaction-diffusion problem (Q1580326)

This paper deals with the existence and uniqueness of periodic solutions for the following nonlinear degenerate reaction-diffusion problem \[ u_t=\text{div}(\bigtriangledown \varphi(u))+c(x,t,u)\quad\text{in }Q:=\Omega\times {\mathbb R}, \] \[ -\frac{\partial\varphi(u)}{\partial \nu}=g(\varphi(u)),\quad\text{on }\partial\Omega\times{\mathbb R},\tag{1} \] \[ u(x,t+\omega)=u(x,t)\text{ and }u\geq 0\quad \text{in }Q, \] where \(\Omega\) is a bounded domain in \({\mathbb R}^N\) with smooth boundary \(\partial\Omega\) and \(\nu\) denotes the outward unit normal to \(\partial\Omega\). Equation (1) models the filtration of a fluid through porous medium. Under some appropriate hypothesis, the author proves existence and uniqueness of the nonnegative weak periodic solution. To establish the result, he uses the Schauder fixed point theorem and some regularizing arguments.