Existence of the nontrivial nonnegative periodic solutions for the quasilinear parabolic equation with nonlocal terms (Q813198)

By using the Leray-Schauder degree theory, the authors show the existence of nontrivial nonnegative periodic solutions for a class of quasilinear parabolic equations with nonlocal terms, that is, the existence of a solution for the periodic boundary value problem \[ \begin{alignedat}{2} \frac{\partial u}{\partial t}-D_i(a_{ij}(x,t,u)D_ju)&=(a-\Phi[u])u,\quad &(x,t)&\in Q_{\omega},\\ u(x,t)&=0,&(x,t)&\in \partial\Omega\times [0,\omega],\tag{1}\\ u(x,0)&=u(x,\omega),&x&\in \Omega,\end{alignedat} \] where \(\Omega\subset\mathbb{R}^n\) is a bounded domain with smooth boundary, \(Q_{\omega}=\Omega\times(0,\omega)\), \(\Phi[u]:L^2(\Omega)^+ \to (0,\infty)\), \( L^2(\Omega)^+=\{u\in L^2(\Omega):u\geq 0,\;\text{a.e. in }\Omega\}\); \(a_{ij}\) satisfies some suitable smoothness and structure conditions, the repeated indices denote the summation from \(1\) to \(n\).