On the well-posedness of a linear heat equation with a critical singular potential

Abstract

In this paper, our main concern is the well-posedness of the initial boundary value problem for a linear heat equation with a time-dependent, strongly singular potential $V \in C([0,T]; L^{\frac{N}{2}} (\Omega))$: $$ \begin{cases} u_t -\Delta u = V u & \text{in~} (0,T) \times \Omega, \\ u = 0 & \text{on~} (0,T) \times \partial \Omega, \\ u(0,x) =u_0(x) & \text{in~} \Omega, \end{cases} $$ where $u_0$ is initial data in $L^p(\Omega)$, $p \geq 1$. We show that the problem is well-posed on $L^p(\Omega)$, $p > 1$ within some appropriate class of solutions, and in turn the well-posedness breaks down on $L^1(\Omega)$. Furthermore, we also present some nonuniqueness results for the time-bounded potential class $L^\infty (0,T; L^{\frac{N}{2}} (\Omega))$.