Quasi-Linear (Stochastic) Partial Differential Equations with Time-Fractional Derivatives

Abstract

In this paper we develop a method to solve (stochastic) evolution equations on Gelfand triples with time-fractional derivative based on monotonicity techniques. Applications include deterministic and stochastic quasi-linear partial differential equations with time-fractional derivatives, including time-fractional (stochastic) porous media equations (including the case where the Laplace operator is also fractional) and p-Laplace equations as special cases.

Small time asymptotics for SPDEs with locally monotone coefficients

“…On the other hand, some very interesting quasilinear SPDEs have been studied a lot recently, such as stochastic porous media equation and stochastic p-Laplace equation, see e.g. [9,26,27,28,29,38,39,42,43,46,49,58]) and references therein. We would like to investigate whether small time asymptotics (LDP) results also hold for those SPDE models or not?…”

“…It was first investigated in the seminal works of Pardoux [48] and Krylov and Rozovskii [34], where they adapted the monotonicity tricks to prove the existence and uniqueness of solutions for a class of SPDE. Recently, this framework has been substantially extended by the second named author and Röckner in [40,41,42,43] for more general class of SPDE with coefficients satisfying the generalized coercivity and local monotonicity conditions. In recent years, various properties for SPDEs with monotone or locally monotone coefficients has been intensively investigated in the literature, such as small noise LDP [39,45,49,59], random attractors [26,27,28,29], Harnack inequality and applications [38], Wong-Zakai approximation and support theorem [46], ultra-exponential convergence [58], and existence of optimal controls [16].…”

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

“…Of importance for physics are also fractional kinetic equations with application to statistical mechanics and fractional stochastic PDEs. For these developments we refer to [37], [45] and [51], [55], [79], [80] and references therein.…”

Classical and generalized solutions of fractional stochastic differential equations

“…∂ β t f (t) = f (0+) = lim t→0,t>0 f (t), seems to achieve the right balance between mathematical utility and physical relevance [8] and has been recently used in the study of large classes of stochastic partial differential equations [4,11].…”