Cauchy problem for fractional diffusion-wave equations with variable coefficients

We consider an evolution equation with the Caputo-Dzhrbashyan fractional derivative of order with respect to the time variable, and the second order uniformly elliptic operator with variable coefficients acting in spatial variables. This equation describes the propagation of stress pulses in a viscoelastic medium. Its properties are intermediate between those of parabolic and hyperbolic equations. In this paper, we construct and investigate a fundamental solution of the Cauchy problem, prove existence and uniqueness theorems for such equations.

[1]  Gustaf Gripenberg,et al.  Regularity Properties of Solutions of Fractional Evolution Equations , 2019, Evolution Equations and Their Applications in Physical and Life Sciences.

[2]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[3]  Anatoly N. Kochubei,et al.  Cauchy problem for fractional diffusion equations , 2003 .

[4]  A. Pskhu,et al.  The fundamental solution of a diffusion-wave equation of fractional order , 2009 .

[5]  Global Stability of a Fractional Partial Differential Equation , 2000 .

[6]  A. Kochubei,et al.  Analytic Methods In The Theory Of Differential And Pseudo-Differential Equations Of Parabolic Type , 2004 .

[7]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

[8]  F. Mainardi.  The fundamental solutions for the fractional diffusion-wave equation , 1996 .

[9]  藤田 安啓,et al.  INTEGRODIFFERENTIAL EQUATION WHICH INTERPOLATES THE HEAT EQUATION AND THE WAVE EQUATION II(Martingales and Related Topics) , 1989 .

[10]  Francesco Mainardi,et al.  Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation , 2012, Comput. Math. Appl..

[11]  Francesco Mainardi,et al.  The Wright functions as solutions of the time-fractional diffusion equation , 2003, Appl. Math. Comput..

[12]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[13]  Rico Zacher,et al.  Lyapunov functions and convergence to steady state for differential equations of fractional order , 2008 .

[14]  Andrzej Hanygad,et al.  Multidimensional solutions of time-fractional diffusion-wave equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  J. Prüss.  Evolutionary Integral Equations And Applications , 1993 .

[16]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[17]  A. Kochubei.  Fractional-Parabolic Systems , 2010, 1009.4996.

[18]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[19]  A. Kochubei.  Fractional-hyperbolic systems , 2012, 1209.3983.

[20]  Fractional Abstract Cauchy Problems , 2011 .

[21]  Allaberen Ashyralyev,et al.  Partial Differential Equations of Elliptic Type , 2004 .

[22]  Igor Podlubny,et al.  ADJOINT FRACTIONAL DIFFERENTIAL EXPRESSIONS AND OPERATORS , 2007 .

[23]  G. Simonett,et al.  Quasilinear evolutionary equations and continuous interpolation spaces , 2004 .