Existence of weak solutions for fractional porous medium equations with nonlinear term

Abstract

We study the following fractional porous medium equations with nonlinear term { u t + ( − Δ ) σ / 2 ( | u | m − 1 u ) + g ( u ) = h , in Ω × R + , u ( x , t ) = 0 , in ∂ Ω × R + , u ( x , 0 ) = u 0 , in Ω . The authors in de Pablo et al. (2011) and de Pablo et al. (2012) established the existence of weak solutions for the case g ( u ) ≡ 0 . Here, we consider the nonlinear term g is without an upper growth restriction. The nonlinearity of g leads to the invalidity of the Crandall–Liggett theorem, which is the critical method to establish the weak solutions in de Pablo et al. (2011) and de Pablo et al. (2012). In addition, because of g does not have an upper growth restriction, we have to apply the weak compactness theorem in an Orlicz space to prove the existence of weak solutions by using the Implicit Time Discretization method.