Multiple existence of periodic solutions for a nonlinear parabolic problem with singular nonlinearities
Let \(N\geq 2\) and \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with a smooth boundary \(\partial\Omega\) and \(h: \mathbb{R}\times\overline\Omega\to \mathbb{R}\) be a continuous function such that for each \(x\in\Omega\), \(h(\cdot, x)\) is \(T\)-periodic and continuous, and \(h> 0\) on \(\mathbb{R}\times \Omega\). In this paper, the authors are interested in multiple existence of \(T\)-periodic solutions of the following problem: \[ \begin{alignedat}{2} & u_t-\Delta_x u= g(u)+ h(t,x)\quad &&\text{in }(0,T)\times\Omega,\\ & u=0\quad &&\text{on }(0,T)\times\partial\Omega,\tag{1}\\ & u(0)= u(T)\quad &&\text{in }\overline\Omega,\end{alignedat} \] where \(\Delta_x\) denotes the Laplacian on \(\mathbb{R}^N\). For a proof of solvability of (1) the authors use topological arguments for variational formulation of (1).
scientific article; zbMATH DE number 1945034
scientific article