Existence, regularity and representation of solutions of time fractional diffusion equations.(English)Zbl 1375.47034

Summary: Using regularized resolvent families, we investigate the solvability of the fractional order inhomogeneous Cauchy problem $\mathbb{D}_t^\alpha u(t)=Au(t)+f(t), \;t > 0,\;\;0 < \alpha\leq 1,$ where $$\mathbb D_t^\alpha$$ is the Caputo fractional derivative of order $$\alpha$$, $$A$$ a closed linear operator on some Banach space $$X$$, $$f:\;[0,\infty)\rightarrow X$$ 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 $$\pi/2\leq \theta < (1-\alpha/2)\pi$$ and $$\Delta_p$$ is a realization of the Laplace operator on $$L^p(\mathbb R ^N)$$, $$1\leq p < \infty$$.

MSC:

 47D06 One-parameter semigroups and linear evolution equations 47D62 Integrated semigroups 35R11 Fractional partial differential equations 45N05 Abstract integral equations, integral equations in abstract spaces
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