A Degenerate Forward-backward Problem Involving the Spectral Dirichlet Laplacian

Abstract

Let \(\varOmega \) be an open bounded subset of \({\mathbb {R}}\), \(s \in (\frac{1}{2},1)\) and \(\epsilon > 0\). We investigate the problem

$$\begin{aligned} (P_\epsilon ) \quad \left\{ \begin{array}{ll} {\partial }_t u = -(-\Delta )^s \big ( \varphi (u) + \epsilon \, {\partial }_t(\psi (u)) \big ) & \text { in } \varOmega \times (0,T],\\ \varphi (u) + \epsilon \, {\partial }_t(\psi (u)) = 0 & \text { on } {\partial }\varOmega \times (0,T], \\ u = u_0 & \text { in } \varOmega \times \{0\}, \end{array}\right. \end{aligned}$$

where \(\varphi , \psi \in C^\infty ({\mathbb {R}})\) and \(u_0 \in {\mathcal {M}}^+(\varOmega )\) satisfy certain assumptions. Here \((-\Delta )^s\) denotes the spectral Dirichlet Laplacian and \({\mathcal {M}}^+(\varOmega )\) is the set of positive Radon measures on \(\varOmega \). We show that \((P_\epsilon )\) has a unique weak solution.