On the initial-boundary value problem for a nonlocal parabolic equation with nonlocal boundary condition

We consider a nonlinear nonlocal parabolic equation $u_{t}=\Delta u + a(x,t)u^{r} \int\limits_{\Omega}u^{p}(y,t)dy - b(x,t)u^{q}$ for $(x,t)\in \Omega\times (0, +\infty)$ with nonlinear nonlocal boundary condition $u(x,t)\rvert_{\partial\Omega\times (0,+\infty)}= \int\limits_{\Omega}k(x,y,t)u^{l}(y,t)dy$ and initial data $u(x,0)=u_{0}(x), x\in \Omega$,where $r,p,q,l$ are positive constants; $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. Nonnegative functions $a(x,t)$ and $b(x,t)$ are defined for $x\in \bar\Omega, t\geq 0$ and local Holder continuous, nonnegative continuous function $k(x,y,t)$ is defined for $x\in \partial\Omega, y\in \bar\Omega, t\geq 0$, nonnegative continuous function $u_{0}(x)$ is defined for $x\in \bar\Omega$ and satisfies the condition $u_{0}(x)=\int\limits_{\Omega}k(x,y,0)u'_{0}(y)dy$ for $x\in \partial\Omega.$ In this paper we study classical solutions. To prove the existence of a local maximal solution, we consider the regularization of the original problem. We establish the existence of a local solution of the regularized problem and the convergence of solutions of this problem to a local maximal solution of the original problem. We introduce definitions of a supersolution and a subsolution. It is shown that a supersolution is not less than a subsolution. We establish the positiveness of solutions of the problem with nontrivial initial data under certain conditions on the data of the problem. As a consequence of the positiveness of solutions and the comparison principle of solutions, we prove the uniqueness theorem