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

Author:

Zhao Yongqiang,Tang Yanbin

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.

Publisher

Springer Science and Business Media LLC

Subject

Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis

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