Existence of Solution of Space–Time Fractional Diffusion-Wave Equation in Weighted Sobolev Space

In this paper, we consider Cauchy problem of space-time fractional diffusion-wave equation. Applying Laplace transform and Fourier transform, we establish the existence of solution in terms of Mittag-Leffler function and prove its uniqueness in weighted Sobolev space by use of Mikhlin multiplier theorem. e estimate of solution also shows the connections between the loss of regularity and the order of fractional derivatives in space or in time.

Fractional derivatives describe the property of memory and heredity of many materials, which is the major advantage compared with integer order derivatives. Fractional diffusion-wave equations are obtained from the classic diffusion equation and wave equation by replacing the integral order derivative terms by fractional derivatives of order 훼 ∈ (0, 1) ∪ (1, 2). It has attracted considerable attention recently for various reasons, which include modeling of anomalous diffusive and subdiffusive systems, description of fractional random walk, wave propagation phenomenon, multiphase fluid flow problems, and electromagnetic theory. Nigmatullin [1,2] pointed out that many of the universal electromagnetic, acoustic, and mechanical responses can be modeled accurately using the fractional diffusion-wave equations. Schneider and Wyss [3] presented the diffusion and wave equations in terms of integro-differential equations, and obtained the associated Greens functions in closed form in terms of the Foxs functions. Mbodje and Montseny [4] investigated the existence, uniqueness, and asymptotic decay of the wave equation with fractional derivative feedback, and showed that the method developed can easily be adapted to a wide class of problems involving fractional derivative or integral operators of the time variable. In fact, more numerical algorithms present an efficient method in solving the related problem [5][6][7][8]. e development of analytical methods is delayed since there are no analytic solutions in many cases [9][10][11][12]. Additional background, survey, and more applications of this field in science, engineering, and mathematics can be found in [13][14][15][16][17][18][19][20] and the references therein. e fractional wave equation has been researched in all probability for the first time in [21] with the same order in space and in time, i.e., 1 = 2 , where an explicit formula for the fundamental solution of this equation was established.
en this feature was shown to be a decisive factor for inheriting some crucial characteristics of the wave equation like a constant propagation velocity of both the maximum of its fundamental solution and its gravity and mass centers in [22]. Moreover, the first, the second, and the Smith centrovelocities of the damped waves described by the fractional wave equation are constant and depend just on the equation order.
While the fractional wave equation contains fractional derivatives of the same order in space and in time, we establish existence of solution of Cauchy problem to fractional wave equation (1) with different order in space and in time in weighted Sobolev spaces. e powers of the weighted show the connections between the loss of the regularity and the is paper is organized as follows: In Section 2, the related fractional calculus definition and Laplace transform are introduced, the explicit solution of fractional differential equation is given by use of Mittag-Leffler functions. In Section 3, based on the main result given in Section 2, we show the existence and uniqueness of solution of space-time fractional diffusion-wave equation.

Laplace Transform and Fractional Calculus
In this section, we recall some necessary definitions and properties of fractional calculus, then use Laplace transform to consider initial value problem of the related fractional differential equation.
en we have the following estimate where denotes a positive constant.

Theorem 2. Consider the problem (13), then there is a explicit solution which is given in the integral form
Proof. According to Definition 1-3, taking Laplace transform with respect to in both sides of Eq. (13), we obtain e inverse Laplace transform using Lemma 3 yields en substitute (15)(16)(17)(18) into (13) which yields eorem 2. ☐

Fourier Transform and the Main Result
In this section, based on the results of eorem 2, Mikhlin multiplier theorem, Mattag-Leffler function and Fourier transf orm, we establish the existence and uniqueness of solution of Cauchy problem of space-time fractional diffusion-wave equation in weighted Sobolev space.