Existence of weak solutions for fractional porous medium equations with nonlinear term. (November 2016)

• Record Type: Journal Article
• Authors: Zhang, Chang; Zhang, Jin; Zhong, Chengkui
• Is Part Of: Applied mathematics letters. Volume 61(2016:Nov.)
• Page Start: 95
• Page End: 101
• Publication Date: 2016-11
• Subjects: Fractional diffusion -- Porous medium equation -- Weak solution -- Bounded domain; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2016.05.001
• Languages: English
• ISSNs: 0893-9659

Abstract

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.