Entropy solutions for time-fractional porous medium type equations

In this paper we prove existence of entropy solutions to the time-fractional porous medium type equation, $$\partial_t[k\ast(u-u_0)]-\operatorname{div} (A(t,x)\nabla\varphi(u))=f\text{ in }Q_T=(0,T)\times\Omega,$$ with Dirichlet boundary condition, initial condition $u(0,\cdot)=u_0$ in $\Omega$, and...

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Bibliographic Details
Main Authors Schmitz, Kerstin, Wittbold, Petra
Format Journal Article
LanguageEnglish
Published 13.02.2023
Subjects
Mathematics - Analysis of PDEs
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Summary:In this paper we prove existence of entropy solutions to the time-fractional porous medium type equation, $$\partial_t[k\ast(u-u_0)]-\operatorname{div} (A(t,x)\nabla\varphi(u))=f\text{ in }Q_T=(0,T)\times\Omega,$$ with Dirichlet boundary condition, initial condition $u(0,\cdot)=u_0$ in $\Omega$, and $L^1$-data $f\in L^1((0,T)\times\Omega), u_0\in L^1(\Omega)$. To this end we approximate the data by $L^\infty$-functions, use a known existence result of weak solutions for these more regular data, and additionally a known contraction principle for weak solutions, which can be adopted to the entropy solutions.
DOI:10.48550/arxiv.2302.06399