Stabilization of Solutions to the Cauchy Problem for Fractional Diffusion-Wave Equation
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STABILIZATION OF SOLUTIONS TO THE CAUCHY PROBLEM FOR FRACTIONAL DIFFUSION-WAVE EQUATION A. V. Pskhu
UDC 517.956.8, 517.955
Abstract. Problems on the asymptotic behavior of solutions to the Cauchy problems for the fractional diffusion-wave equation for large values of time are examined. Sufficient conditions of stabilization in the class of rapidly growing functions and necessary and sufficient conditions of stabilization to zero in the case of asymptotically nonnegative initial functions are found. Keywords and phrases: fractional diffusion-wave equation, stabilization, Cauchy problem, fractional derivative, Dzhrbashyan–Nersesyan operator. AMS Subject Classification: 35R11, 35B40
1.
Introduction. We consider the following equation in the upper half-plane {(x, y) : x ∈ R, y > 0}: α ∂2 ∂ − u(x, y) = 0, 0 < α < 2, (1) ∂y α ∂x2
where ∂ α /∂y α is the fractional derivative of order α with the origin point at y = 0. We say that a solution of Eq. (1) stabilizes at a point x ∈ R (respectively, is uniform on a compact set K ⊂ R; is uniform in the whole space) if there exists a finite limit lim u(x, y)
y→∞
at the point x (respectively, this limit is uniform with respect to x ∈ K or uniform with respect to x ∈ R). The first papers on stabilization of solutions of parabolic equations were [16, 27]. Thus, in [27], the problem on stabilization of solutions to the boundary-value problem in a half-strip for the heat equation was examined; this problem appeared due to an applied problem in heat transfer theory. In [16], conditions for the existence of traveling-wave solutions to a quasilinear parabolic equation and the convergence of solutions of the Cauchy problem (as time increases unboundedly) were found; this problem appeared in applications in biology. Subsequently, this direction actively developed and today has an extensive bibliography. We indicate the papers [2–6, 12, 29], which describe the main ideas of the existing approaches to the study of stabilization problems for initial-value and boundary-value problems for parabolic equations and systems and contain reviews of works on this topic. We also mention the papers [9–11, 14, 15, 17, 18, 21–23, 32] devoted to the theory of fractional diffusion and wave equations. We emphasize the paper [15], in which the asymptotic behavior of solutions to the fractional diffusion-wave equation was investigated. Various problems on applications of fractional calculus and the theory of fractional differential equations in physics and modeling are described, for example, in [1, 13, 19, 26, 28]. The main aim of this paper is to find conditions that ensure stabilization of solutions to a fractional diffusion-wave equation. In particular, we find sufficient conditions (Theorem 3) for stabilization of solutions to the Cauchy problem for Eq. (1) and also necessary and sufficient conditions for stabilization (to zero) in the class of asymptotically nonnegative functions (Theorem 4). Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 149,
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