Approximation of solutions to integro-differential time fractional order parabolic equations in L p $L^{p}$ -spaces

Abstract

Abstract 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 $C_{0}$ -semigroup, resolvent operator, and weighted function space, we get the existence of a mild solution. For suitable indices p ∈ [ 1 , + ∞ ) $p\in [1,+\infty )$ and s ∈ ( 1 , + ∞ ) $s\in (1,+\infty )$ , 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 ) ) $L^{p}((0,T), L^{s}(\Omega ))$ as ε → 0 + $\varepsilon \rightarrow 0^{+}$ . 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 ) $L^{s}(\Omega )$ , we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data.

Keywords

• Integro-differential equation
• Time fractional order parabolic equation
• Limit problem
• Analytic semigroup
• C 0 $C_{0}$ -semigroup
• Mittag-Leffler function