Well-posedness and regularity for fractional damped wave equations
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Well-posedness and regularity for fractional damped wave equations Yong Zhou1,2
· Jia Wei He3
Received: 20 February 2020 / Accepted: 3 November 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper, we study the well-posedness and regularity of mild solutions for a class of time fractional damped wave equations, which the fractional derivatives in time are taken in the sense of Caputo type. A concept of mild solutions is introduced to prove the existence for the linear problem, as well as the regularity of the solution. We also establish a well-posed result for nonlinear problem. By applying finite dimensional approximation method, a compact result of solution operators is presented, following this, an existence criterion shows that the Lipschitz condition or smoothness of nonlinear force functions in some literatures can be removed. As an application, we discuss a case of time fractional telegraph equations. Keywords Damped wave equations · Caputo’s fractional derivative · mild solutions · Well-posedness · Regularity Mathematics Subject Classification 35L05 · 35R11 · 26A33
Communicated by Ansgar Jüngel.
B
Yong Zhou [email protected] Jia Wei He [email protected]
1
Faculty of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China
2
Faculty of Information Technology, Macau University of Science and Technology, Macau 999078, China
3
College of Mathematics and Information Science, Guangxi University, Nanning 530004, China
123
Y. Zhou, J. W. He
1 Introduction Fractional differential equations have gained considerable importance due to their widespread applications in a variety of fields such as physics, chemistry, engineering, biology, geophysics and hydrology. In recent years, partial differential equations with fractional derivatives have been investigated extensively. For details and examples, we refer the reader to a series of papers [1,2,4–10,12,15–18,20–24,26–29,31,32] and the references cited therein. The main purpose of this paper is to investigate the initial/boundary value problems for time fractional damped wave equation β
∂t u + ∂tα u = u + f (u), t > 0,
(1.1)
subject to Dirichlet’s boundary condition u(t, x) = 0, x ∈ ∂, t > 0,
(1.2)
and initial value conditions u(0, x) = φ(x), ∂t u(0, x) = ψ(x), x ∈ ,
(1.3)
where ⊂ Rd (d ≥ 1) is a bounded domain with the sufficiently smooth boundary β ∂, ∂t , ∂tα are standard fractional derivatives in the sense of Caputo type of order β ∈ (1, 2] and α ∈ (0, 1], respectively. f is an appropriate force function which will be special later. Taking the case of β = 2 and α = 1 in (1.1), it becomes the standard damped wave equation, which is an important mathematical model in studying many physic problems. Readers can easily find a large number of related researches that are focused on the well-posedness of some linear or nonlinear Cauchy problems. In addition, various papers have considered to establish the asymptotic behavior and regularity estimates of the solutions, we refer to [3,11,14] a
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