A semilinear parabolic problem with singular term at the boundary

Abstract

In this paper, we deal with a class of semilinear parabolic problems related to a Hardy inequality with singular weight at the boundary.

More precisely, we consider the problem

$$\left\{ \begin{array}{l@{\quad}l} u_t-\Delta u=\lambda \frac{u^p}{d^2} &\text{ in }\,\Omega_{T}\equiv\Omega \times (0,T), \\ u>0 &\text{ in }\,{\Omega_T}, \\ u(x,0)=u_0(x)>0 &\text{ in }\,\Omega, \\ u=0 &\text{ on }\partial \Omega \times (0,T), \end{array} \right. $$

(P)

where Ω is a bounded regular domain of ${\mathbbm{R}^N}$, ${d(x)=\text{dist}(x,\partial\Omega)}$, ${p > 0}$, and ${\lambda > 0}$ is a positive constant.

We prove that

• 1. If ${0 < p < 1}$, then (P) has no positive very weak solution.

• 2. If ${p=1}$, then (P) has a positive very weak solution under additional hypotheses on ${\lambda}$ and ${u_0}$.

• 3. If ${p > 1}$, then, for all ${\lambda > 0}$, the problem (P) has a positive very weak solution under suitable hypothesis on ${u_0}$.

Moreover, we consider also the concave–convex-related case.