Advances in Differential Equations

Existence, regularity and representation of solutions of time fractional diffusion equations

Valentin Keyantuo, Carlos Lizama, and Mahamadi Warma

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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 $L^p(\mathbb R ^N)$, $1\le p < \infty$.

Article information

Adv. Differential Equations Volume 21, Number 9/10 (2016), 837-886.

First available in Project Euclid: 14 June 2016

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Mathematical Reviews number (MathSciNet)

Primary: 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 47D62: Integrated semigroups 34A08: Fractional differential equations 35R11: Fractional partial differential equations 45N05: Abstract integral equations, integral equations in abstract spaces


Keyantuo, Valentin; Lizama, Carlos; Warma, Mahamadi. Existence, regularity and representation of solutions of time fractional diffusion equations. Adv. Differential Equations 21 (2016), no. 9/10, 837--886.

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