## Differential and Integral Equations

- Differential Integral Equations
- Volume 14, Number 1 (2001), 1-18.

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

Daisuke Hirata and Masayoshi Tsutsumi

#### 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 $L^\infty (0,T; L^{\frac{N}{2}} (\Omega))$.

#### Article information

**Source**

Differential Integral Equations, Volume 14, Number 1 (2001), 1-18.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356123371

**Mathematical Reviews number (MathSciNet)**

MR1797928

**Zentralblatt MATH identifier**

1161.35418

**Subjects**

Primary: 35K05: Heat equation

Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35K15: Initial value problems for second-order parabolic equations

#### Citation

Hirata, Daisuke; Tsutsumi, Masayoshi. On the well-posedness of a linear heat equation with a critical singular potential. Differential Integral Equations 14 (2001), no. 1, 1--18. https://projecteuclid.org/euclid.die/1356123371