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September/October 2016 Existence, regularity and representation of solutions of time fractional diffusion equations
Valentin Keyantuo, Carlos Lizama, Mahamadi Warma
Adv. Differential Equations 21(9/10): 837-886 (September/October 2016).

Abstract

Using regularized resolvent families, we investigate the solvability of the fractional order inhomogeneous Cauchy problem $$ \mathbb{D}_t^\alpha u(t)=Au(t)+f(t), \;t > 0,\;\;0 < \alpha\le 1, $$ where $\mathbb D_t^\alpha$ is the Caputo fractional derivative of order $\alpha$, $A$ a closed linear operator on some Banach space $X$, $f:\;[0,\infty)\to X$ is a given function. We define an operator family associated with this problem and study its regularity properties. When $A$ is the generator of a $\beta$-times integrated semigroup $(T_\beta(t))$ on a Banach space $X$, explicit representations of mild and classical solutions of the above problem in terms of the integrated semigroup are derived. The results are applied to the fractional diffusion equation with non-homogeneous, Dirichlet, Neumann and Robin boundary conditions and to the time fractional order Schrödinger equation $\mathbb{D}_t^\alpha u(t,x)=e^{i\theta}\Delta_pu(t,x)+f(t,x),$ $ t > 0,\; x\in \mathbb R ^N$ where $\pi/2\le \theta < (1-\alpha/2)\pi$ and $\Delta_p$ is a realization of the Laplace operator on $L^p(\mathbb R ^N)$, $1\le p < \infty$.

Citation

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Valentin Keyantuo. Carlos Lizama. Mahamadi Warma. "Existence, regularity and representation of solutions of time fractional diffusion equations." Adv. Differential Equations 21 (9/10) 837 - 886, September/October 2016.

Information

Published: September/October 2016
First available in Project Euclid: 14 June 2016

zbMATH: 1375.47034
MathSciNet: MR3513120

Subjects:
Primary: 34A08, 35R11, 45N05, 47D06, 47D62

Rights: Copyright © 2016 Khayyam Publishing, Inc.

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Vol.21 • No. 9/10 • September/October 2016
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