Stochastic Chemotaxis Model with Fractional Derivative Driven by Multiplicative Noise

Main Article Content

Ali Slimani
Amira Rahai
Amar Guesmia
Lamine Bouzettouta

Abstract

We introduce stochastic model of chemotaxis by fractional Derivative generalizing the deterministic Keller Segel model. These models include fluctuations which are important in systems with small particle numbers or close to a critical point. In this work, we study of nonlinear stochastic chemotaxis model with Dirichlet boundary conditions, fractional Derivative and disturbed by multiplicative noise. The required results prove the existence and uniqueness of mild solution to time and space-fractional, for this we use analysis techniques and fractional calculus and semigroup theory, also studying the regularity properties of mild solution for this model.

Article Details

References

  1. L. Debbi, Well-Posedness of the Multidimensional Fractional Stochastic Navier–Stokes Equations on the Torus and on Bounded Domains, J. Math. Fluid Mech. 18 (2016) 25–69.
  2. P.M. de Carvalho-Neto, G. Planas, Mild solutions to the time fractional Navier–Stokes equations in RN , J. Differ. Equ. 259 (2015), 2948–2980.
  3. S. Zitouni, K. Zennir, L. Bouzettouta, Uniform decay for a viscoelastic wave equation with density and time-varying delay in Rn, Filomat. 33 (2019), 961–970.
  4. L. Bouzettouta, F. Hebhoub, K. Ghennam, S. Benferdi, Exponential Stability for a Nonlinear Timoshenko System with Distributed Delay, Int. J. Anal. Appl. 19 (2021), 77-90.
  5. A. Guesmia, N. Daili, Existence and uniqueness of an entropy solution for Burgers equations, Appl. Math. Sci. 2 (2008), 1635-1664, .
  6. A. Guesmia, N. Daili, About the existence and uniqueness of solution to fractional burgers equation, Acta Univ. Apul. 21(2010), 161-170.
  7. E.F. Keller, L.A. Segel, Model for chemotaxis, J. Theor. Biol. 30 (1971), 225–234.
  8. R. Kruse, Strong and weak approximation of semilinear stochastic evolution equations, Springer, New York, 2014.
  9. F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett. 9 (1996), 23–28.
  10. K.A. Khelil, F. Bouchelaghem, L. Bouzettouta, Exponential stability of linear Levin-Nohel integro-dynamic equations on time scales. Int. J. Appl. Math. Stat. 56 (2017), 138-149.
  11. G. Zou, B. Wang, Stochastic Burgers’ equation with fractional derivative driven by multiplicative noise, Computers Math. Appl. 74 (2017), 3195–3208.
  12. N. Dib, A. Guesmia, N. Daili, On the solution of stochastic generalized burgers equation, Commun. Math. Appl. 9 (2018), 521-528.
  13. S. Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos Solitons Fractals. 28 (2006), 930–937.
  14. C. Mesikh, A. Guesmia, S.Saadi, Global existence and uniqueness of the weak solution in Keller segel model, Glob. J. Sci. Front. Res. F, 14 (2014), 46-55.
  15. T. Nagai, T. Senba, T. Suzuki, Chemotactic collapse in a parabolic system of mathematical biology. Hiroshima Math. J. 30 (2000), 463-497.
  16. A. Rahai, A. Guesmia, Global Existence and Uniqueness of the Weak Solution in Thixotropic Model, Int. J. Anal. Appl. 19 (2021), 193-204.
  17. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
  18. D. Yang, m-Dissipativity for Kolmogorov Operator of a Fractional Burgers Equation with Space-time White Noise, Potential Anal. 44 (2016), 215–227.
  19. X.-J. Yang, H.M. Srivastava, T. Machado, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Therm. Sci. 20 (2016), 753–756.
  20. Y. Zhou, F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Computers Math. Appl. 59 (2010), 1063–1077.