Electron. J. Differential Equations, Vol. 2017 (2017), No. 222, pp. 1-42.

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

Valentin Keyantuo, Carlos Lizama, Mahamadi Warma

Abstract:
We study the solvability of the fractional order inhomogeneous Cauchy problem
$$ \mathbb{D}_t^\alpha u(t)=Au(t)+f(t), \quad t>0,\;1<\alpha\le 2, $$
where A is a closed linear operator in some Banach space X and $f:[0,\infty)\to X$ a given function. Operator families associated with this problem are defined and their regularity properties are investigated. In the case where A is a generator of a $\beta$-times integrated cosine family $(C_\beta(t))$, we derive explicit representations of mild and classical solutions of the above problem in terms of the integrated cosine family. We include applications to elliptic operators with Dirichlet, Neumann or Robin type boundary conditions on $L^p$-spaces and on the space of continuous functions.

Submitted October 26, 2016. Published September 18, 2017.
Math Subject Classifications: 47D06, 35K20, 35L20, 45N05.
Key Words: Fractional derivative; subordination principle; elliptic operator; integrated cosine family; Dirichlet, Neumann and Robin boundary conditions.

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  Valentin Keyantuo
University of Puerto Rico, Department of Mathematics
Faculty of Natural Sciences
Rio Piedras Campus, P.O. Box 70377
San Juan, PR 00936-8377, USA
email: valentin.keyantuo1@upr.edu
Carlos Lizama
Universidad de Santiago de Chile
Departamento de Matemática
Facultad de Ciencias
Casilla 307-Correo 2, Santiago, Chile
email: carlos.lizama@usach.cl
Mahamadi Warma
University of Puerto Rico
Department of Mathematics
Faculty of Natural Sciences
Rio Piedras Campus, P.O. Box 70377
San Juan, PR 00936-8377, USA
email: mjwarma@gmail.com, mahamadi.warma1@upr.edu

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