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

Abstract

AbstractIn 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 $p\in [1,+\infty )$ p ∈ [ 1 , + ∞ ) and $s\in (1,+\infty )$ s ∈ ( 1 , + ∞ ) , we first prove that the mild solution of the approximating problem converges to that of the corresponding limit problem in $L^{p}((0,T), L^{s}(\Omega ))$ L p ( ( 0 , T ) , L s ( Ω ) ) as $\varepsilon \rightarrow 0^{+}$ 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 $L^{s}(\Omega )$ L s ( Ω ) , we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data.