On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives

Abstract

Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sense, respectively. By a comparison principle and Perron's method, unique existence for the Cauchy-Dirichlet and Cauchy-Neumann problems are proved.

A Hopf-Lax formula for Hamilton-Jacobi equations with Caputo time-fractional derivative

“…To simplify the notation, since in this argument only the time variable is involved, we omit the dependence of ϕ on x. Because of the continuity of the Caputo derivative of ϕ with respect to t (see [13,Prop. 2.1]), it is sufficient to prove that…”

“…In this section, we briefly review definitions and some results for the continuous problem (1.1) (we refer to [8,13] for more details). For a function f :…”

“…We also recall an existence result for viscosity solutions of (1.1). For a locally bounded function u : Q T → R, u * and u * denote respectively the upper and lower semi-continuous envelope, defined for (t, x) ∈ Q T by Existence and uniqueness results for the problem of (1.1)-(1.2) in a bounded domain with boundary conditions in viscosity sense are discussed in [13].…”

“…Also, the numerical approximation of differential equations with fractional timederivative has been extensively analyzed [3,9,10]. Since in general smooth solutions to Hamilton-Jacobi equations are not expected to exist, for equation (1.1) a theory of weak solutions, in viscosity sense, has been introduced, in [8,13,18]. Most of the results and techniques which hold in the classical case, i.e., for α = 1, have been extended to the fractional case in order to prove the well-posedness of the Hamilton-Jacobi equation (1.1).…”

On a discrete scheme for time fractional fully nonlinear evolution equations

“…It has inspired further research on numerous related topics. We refer to a non-exhaustive list of references [10,14,2,3,7,15,1,13,9,4] and the references therein. Among these results, the authors of [2,1] mainly study regularity of solutions to a space-time nonlocal equation with Caputo's time fractional derivative in the framework of viscosity solutions.…”