On Existence and Uniqueness of Solutions for Semilinear Fractional Wave Equations

Abstract

Abstract. Let Ω be a C 2 -bounded domain of R d , d = 2, 3, and fix Q = (0, T ) × Ω with T ∈ (0, +∞].In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation ∂ α t u + Au = f b (u) in Q where 1 < α < 2, ∂ α t corresponds to the Caputo fractional derivative of order α, A is an elliptic operator and the nonlinearity f b ∈ C 1 (R) satisfiesWe first provide a definition of local weak solutions of this problem by applying some properties of the associated linear equation ∂ α t u + Au = f (t, x) in Q. Then, we prove existence of local solutions of the semilinear fractional wave equation for some suitable values of b > 1. Moreover, we obtain an explicit dependence of the time of existence of solutions with respect to the initial data that allows longer time of existence for small initial data.

Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations

“…However, Theorem 1.4 can be applied to more general boundary conditions and it can also be applied to the stable recovery of a coefficient depending on both time and space variables (see Corollary1.5).Applying Theorem 1.4, we prove in Corollary 1.5 the stable recovery of the coefficient of order zero q provided ∂ x d q = 0. It seems that this result is the first result of stable recovery of a coefficient In Theorem 1.4we have restricted our analysis to α ∈ (0, 1] in order to simplify the statement of this theorem and its proof.The proof of Proposition 1.3 is based on properties of solutions of fraction diffusion and propertiesof Mittag-Leffler functions considered in several works like[7,20,26,30]. 1.8.…”

On existence and regularity of a terminal value problem for the time fractional diffusion equation

“…One of the most important things, when we consider the well-posedness of a PDE, is the boundedness of solution operators. Corresponding to the initial value problem (1.1), (1.2), (1.4), the solution operators are usually bounded in L 2 (D); see e.g., [53,55,58,41,40]. Unfortunately, some solution operators of FVP (1.1)-(1.3) are not bounded on L 2 (D) at t = 0.…”

Well-posedness results for a class of semi-linear super-diffusive equations

“…In (1.1), we consider the Dirichlet operator 3 which is also the case investigated in detail by [27] (assuming only that d ≤ 3, and Ω is of class C 2 and a ij ∈ C 1 (Ω)). Let A be the realization in L 2 (Ω) of A with the Dirichlet boundary condition u = 0 in ∂Ω.…”

“…Here we are interested in existence, uniqueness and regularity results for the semi-linear equation (1.1) when 1 < α < 2, under appropriate conditions on the data. In contrast to the works of [27], [28], which only consider the Laplacian for the operator and a general notion of integral solutions, the main novelties of the present paper are as follows:…”

“…We refer to [36,41,40,42] and their references for more information on the operator (−∆) s . • We employ a notion of energy (weak) solution that is much stronger than the notion of integral solution devised by [27,28]. Besides, in some cases we prove the existence of strong energy solutions that have the additional property that they also satisfy the fractional wave equation pointwise (see also Section 5.1, for further comments).…”