A Degenerate Forward-backward Problem Involving the Spectral Dirichlet Laplacian

Let \(\varOmega \) $\Omega$ be an open bounded subset of \({\mathbb {R}}\) $\mathbb{R}$ , \(s \in (\frac{1}{2},1)\) $s\in (\frac{1}{2},1)$ and \(\epsilon > 0\) $ϵ>0$ . We investigate the problem \(\begin{aligned} (P_\epsilon ) \quad \left\{ \begin{array}{ll} {\partial }_t u = -(-\Delta )^s \big ( \varphi (u) + \epsilon \, {\partial }_t(\psi (u)) \big ) & \text { in } \varOmega \times (0,T],\\ \varphi (u) + \epsilon \, {\partial }_t(\psi (u)) = 0 & \text { on } {\partial }\varOmega \times (0,T], \\ u = u_0 & \text { in } \varOmega \times \{0\}, \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill (P_{ϵ})\left\{\begin{array}{cc}{\partial }_{t}u=-{(-\mathrm{\Delta })}^s(\phi (u)+ϵ{\partial }_t(\psi (u)))\hfill & \text{in}\Omega \times (0,T],\hfill \\ \phi (u)+ϵ{\partial }_t(\psi (u))=0\hfill & \text{on}\partial \Omega \times (0,T],\hfill \\ u=u_0\hfill & \text{in}\Omega \times \{0\},\hfill \end{array}\right)\end{array}$$ where \(\varphi , \psi \in C^\infty ({\mathbb {R}})\) $\phi ,\psi \in C^{\infty }(\mathbb{R})$ and \(u_0 \in {\mathcal {M}}^+(\varOmega )\) $u_0\in {\mathcal{M}}^{+}(\Omega )$ satisfy certain assumptions. Here \((-\Delta )^s\) ${(-\mathrm{\Delta })}^s$ denotes the spectral Dirichlet Laplacian and \({\mathcal {M}}^+(\varOmega )\) ${\mathcal{M}}^{+}(\Omega )$ is the set of positive Radon measures on \(\varOmega \) $\Omega$ . We show that \((P_\epsilon )\) $(P_{ϵ})$ has a unique weak solution.