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 Schrödinger 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.