Weak Periodic Solution for Semilinear Parabolic Problem with Singular Nonlinearities and $$L^{1}$$ Data

Abstract

We consider a periodic parabolic problem involving singular nonlinearity and homogeneous Dirichlet boundary condition modeled by $$\begin{aligned} \dfrac{\partial u}{\partial t}-\Delta u =\dfrac{f}{u^{\gamma }} \text { in }Q_{T}, \end{aligned}$$ where $$T>0$$ is a period, $$\Omega $$ is an open regular bounded subset of $$\mathbb {R}^{N}$$ , $$Q_{T}=]0,T[\times \Omega $$ , $$\gamma \in \mathbb {R}$$ and f is a nonnegative integrable function periodic in time with period T. Under a suitable assumptions on f, we establish the existence of a weak T-periodic solution for all ranges of value of $$\gamma $$ .