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, $${\mathrm{\partial }}_t[k\ast (u-u_0)]-\mathrm{div}(A(t,x)\mathrm{\nabla }\phi (u))=f\text{ in }{Q}_T=(0,T)\times \mathrm{\Omega },$$ with Dirichlet boundary condition, initial condition $u(0,\cdot )=u_0$ in $\mathrm{\Omega }$, and $L^1$-data $f\in L^1((0,T)\times \mathrm{\Omega }),u_0\in L^1(\mathrm{\Omega })$. To this end we approximate the data by $L^{\mathrm{\infty }}$-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.