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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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Full Text: Euclid