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A Degenerate Forward-backward Problem Involving the Spectral Dirichlet Laplacian
Nguyen Ngoc Trong
,
Bui Le Trong Thanh
,
Tan Duc Do
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{equation*}\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}\end{equation*} 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.