Existence of nontrivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms
Using the Leray-Schauder degree theory, the authors show the existence of nontrivial nonnegative periodic solutions for a class of degenerate parabolic equations with nonlocal terms by considering the following periodic boundary value problem \[ \begin{alignedat}{2} \frac{\partial u}{\partial t}-\Delta u^m & =(a-\Phi[u])u,\quad & (x,t) & \in Q_{\omega},\\ u(x,t) & =0, & (x,t) & \in \partial\Omega\times [0,\omega],\tag{1}\\ u(x,0) & =u(x,\omega), & x & \in \Omega,\end{alignedat} \] where \(m>1,\Omega\subset\mathbb{R}^n\) is a bounded domain with smooth boundary, \(Q_{\omega}=\Omega\times(0,\omega),\;\Phi[u]:L^2(\Omega)^+ \to[0,\infty)\), \( L^2(\Omega)^+=\{u\in L^2(\Omega):u\geq 0,\;\text{a. e. in }\Omega\}\). Let \(0<\varepsilon\leq \frac{1}{2}\) be constant, they first showed the existence of nontrivial nonnegative solution \(u_{\varepsilon}\) to the corresponding regularized periodic boundary value problem \[ \begin{alignedat}{2} \frac{\partial u_{\varepsilon}}{\partial t}-\text{div}((mu_{\varepsilon}^{m-1}+\varepsilon)\nabla u_{\varepsilon}) & =(a-\Phi[u_{\varepsilon}])u_{\varepsilon}, \quad &(x,t) & \in Q_{\omega},\\ u_{\varepsilon}(x,t) & =0, &(x,t) & \in \partial\Omega\times [0,\omega],\\ u_{\varepsilon}(x,0) & =u_{\epsilon}(x,\omega), &x & \in \Omega,\end{alignedat} \] and then passing to a limit by letting \(\varepsilon\to 0\) to get the solution \(u\) of (1).
regularized periodic boundary value problem
scientific article; zbMATH DE number 5001891