Entropy solutions for time-fractional porous medium type equations

Abstract

In this paper we prove existence of entropy solutions to the time-fractional porous medium type equation, $${\partial }_t[k\ast (u-u_0)]-\mathrm{div}(A(t,x)\nabla \phi (u))=f\text{ in }{Q}_T=(0,T)\times \Omega ,$$ 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.