Approximation of solutions to integro-differential time fractional order parabolic equations in Lp-spaces.

In this paper we study the initial boundary value problem for a class of integro-differential time fractional order parabolic equations with a small positive parameter ε. Using the Laplace transform, Mittag-Leffler operator family, C 0 -semigroup, resolvent operator, and weighted function space, we get the existence of a mild solution. For suitable indices p ∈ [ 1 , + ∞) and s ∈ (1 , + ∞) , we first prove that the mild solution of the approximating problem converges to that of the corresponding limit problem in L p ((0 , T) , L s (Ω)) as ε → 0 + . Then for the linear approximating problem with ε and the corresponding limit problem, we give the continuous dependence of the solutions. Finally, for a class of semilinear approximating problems and the corresponding limit problems with initial data in L s (Ω) , we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data. [ABSTRACT FROM AUTHOR]