Monotone iterative technique for time-space fractional diffusion equations involving delay

Abstract. This paper considers the initial boundary value problem for the time-space fractional delayed diffusion equation with fractional Laplacian. By using the semigroup theory of operators and the monotone iterative technique, the existence and uniqueness of mild solutions for the abstract time-space evolution equation with delay under some quasimonotone conditions are obtained. Finally, the abstract results are applied to the time-space fractional delayed diffusion equation with fractional Laplacian operator, which improve and generalize the recent results of this issue.

[1]  D. Baleanu,et al.  Lie symmetry analysis, exact solutions and conservation laws for the time fractional modified Zakharov–Kuznetsov equation , 2017 .

[2]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[3]  Yuri Luchko.  On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation , 2017 .

[4]  Ricardo H. Nochetto,et al.  Numerical methods for fractional diffusion , 2017, Comput. Vis. Sci..

[5]  C. Carracedo,et al.  The theory of fractional powers of operators , 2001 .

[6]  Guofei Pang,et al.  What is the fractional Laplacian? A comparative review with new results , 2020, J. Comput. Phys..

[7]  Mark M. Meerschaert,et al.  Space-time fractional diffusion on bounded domains , 2012 .

[8]  A. Hanyga,et al.  Multi–dimensional solutions of space–time–fractional diffusion equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[9]  Joshua L. Padgett,et al.  The quenching of solutions to time-space fractional Kawarada problems , 2018, Comput. Math. Appl..

[10]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[11]  Joseph E. Pasciak,et al.  The approximation of parabolic equations involving fractional powers of elliptic operators , 2016, J. Comput. Appl. Math..

[12]  A. V. Balakrishnan,et al.  Fractional powers of closed operators and the semigroups generated by them. , 1960 .

[13]  R. Nagel,et al.  One-parameter Semigroups of Positive Operators , 1986 .

[14]  Filipe Rocha,et al.  Superdiffusion and epidemiological spreading , 2018, Ecological Complexity.

[15]  Junxiong Jia,et al.  Maximum principles for a time-space fractional diffusion equation , 2016, Appl. Math. Lett..

[16]  M. Kwasnicki.  Ten equivalent definitions of the fractional laplace operator , 2015, 1507.07356.

[17]  L. Toniazzi.  Stochastic classical solutions for space–time fractional evolution equations on a bounded domain , 2019, Journal of Mathematical Analysis and Applications.

[18]  Yongxiang Li,et al.  Existence and uniqueness of periodic solutions for parabolic equation with nonlocal delay , 2016 .

[19]  Yong-sheng Ding,et al.  A generalized Gronwall inequality and its application to a fractional differential equation , 2007 .

[20]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[21]  Yong Zhou,et al.  A class of fractional evolution equations and optimal controls , 2011 .

[22]  Mark M. Meerschaert,et al.  Space–Time Duality for Fractional Diffusion , 2009, Journal of Applied Probability.

[23]  Yongxiang Li,et al.  Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions , 2013, Zeitschrift für angewandte Mathematik und Physik.

[24]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[25]  H. Amann.  Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems , 1976 .

[26]  Herbert Amann,et al.  Periodic Solutions of Semilinear Parabolic Equations , 1978 .

[27]  Yongxiang Li,et al.  Monotone iterative technique for second order delayed periodic problem in Banach spaces , 2015, Appl. Math. Comput..

[28]  YangQuan Chen,et al.  A new collection of real world applications of fractional calculus in science and engineering , 2018, Commun. Nonlinear Sci. Numer. Simul..