Dirichlet Parabolic Problems Involving Schrödinger Type Operators with Unbounded Diffusion and Singular Potential Terms in Unbounded Domains

Abstract

We study the well-posedness of autonomous parabolic Dirichlet problems involving Schrodinger type operators of the form $$\begin{aligned} H_{\alpha ,a,b,c}=(1+|x|^\alpha )\Delta +a|x|^\alpha +b|x|^{\alpha -2}+c|x|^{-2}, \end{aligned}$$ with $$\alpha \ge 0$$ , $$a<0$$ and $$b,c\in \mathbb {R}$$ , in regular unbounded domains $$\Omega \subset \mathbb {R}^N$$ containing 0. Under suitable assumptions on $$\alpha $$ , b and c, the solution is governed by a contractive and positivity preserving strongly continuous (analytic) semigroup on the weighted space $$L^p(\Omega , d\mu (x))$$ , $$1<p<\infty $$ , where $$d\mu (x)=(1+|x|^\alpha )^{-1}dx$$ . The proofs are based on some $$L^p$$ -weighted Hardy’s inequality and perturbation techniques.