Positive solutions with a singular set for semilinear parabolic equations - Jah, Sidi Hamidou; Riahi, Lotfi (Topological Methods in Nonlinear Analysis, 2025)
We study the existence and large time behavior of positive solutions for the semilinear parabolic equation ut(x,t) =Δ u(x,t)+ V(x)u(x,t) +f(x,u(x,t)) with initial Dirichlet boundary conditions on (DE)× (0,∞), where D is a bounded Lipschitz domain in R^n, n≥ 3, E is a prescribed compact set of D, and V and f are real-valued functions satisfying some general conditions. Our results cover various types of nonlinearities and extend known results proved for the power nonlinearity f(x,u)=W(x)u^p and a singular one-point set E.