Existence and uniqueness of an attractive nonlinear diffusion system

We establish the existence and uniqueness of an attractive fractional coupled system. Such a system has applications in biological populations of cells. We confirm that the fractional system under consideration admits a global solution in the Sobolev space. The solution is shown to be unique. The technique is founded on analytic method of the fixed point theory and the fractional differential operator is scrutinized from the view of the Riemann-Liouville differential operator. Finally, we illustrate some entropy fractional differential inequalities regarding the solution of the system.

[1]  Thomas Hillen,et al.  Classical solutions and pattern formation for a volume filling chemotaxis model. , 2007, Chaos.

[2]  Youshan Tao,et al.  Competing effects of attraction vs. repulsion in chemotaxis , 2013 .

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

[4]  Vahid Johari Majd,et al.  Solution existence for non-autonomous variable-order fractional differential equations , 2012, Math. Comput. Model..

[5]  Maokang Luo,et al.  Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders , 2014, Commun. Nonlinear Sci. Numer. Simul..

[6]  J. Vázquez,et al.  Quantitative local and global a priori estimates for fractional nonlinear diffusion equations , 2012, 1210.2594.

[7]  Alberto Cabada,et al.  Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations , 2012, Appl. Math. Lett..

[8]  Takashi Suzuki,et al.  Free Energy and Self-Interacting Particles , 2005 .

[9]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[10]  A. Ibrahim,et al.  Fractional differential inclusions with anti-periodic boundary conditions in Banach spaces , 2014 .

[11]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[12]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[13]  F. Mainardi,et al.  Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation , 2013 .

[14]  Michael Winkler,et al.  Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model , 2010 .

[15]  B. Dutta,et al.  On the existence and uniqueness of solutions of a class of initial value problems of fractional order , 2013 .

[16]  H. Srivastava,et al.  Helmholtz and Diffusion Equations Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor-Type Cylindrical Coordinates , 2013 .

[17]  R. Ibrahim,et al.  Boundary fractional differential equation in a complex domain , 2014 .

[18]  R. Ibrahim.  SOLUTIONS TO SYSTEMS OF ARBITRARY-ORDER DIFFERENTIAL EQUATIONS IN COMPLEX DOMAINS , 2014 .

[19]  Shurong Sun,et al.  Existence of solutions for fractional differential equations with integral boundary conditions , 2014 .

[20]  Ahmed Alsaedi,et al.  Maximum principle for certain generalized time and space fractional diffusion equations , 2015 .

[21]  R. Ibrahim.  On Holomorphic Solution for Space- and Time-Fractional Telegraph Equations in Complex Domain , 2012 .

[22]  M. A. Al-Gwaiz,et al.  An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions , 2013 .