Advances in Differential Equations

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

Valentin Keyantuo, Carlos Lizama, and Mahamadi Warma

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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.

Export citation