Existence and uniqueness of the weak solution for Keller-Segel model coupled with the heat equation

Ali Slimani; Amar Guesmia

doi:10.5269/bspm.66186

Ali Slimani university of 20 august 1955, Skikda
Amar Guesmia

Résumé

Keller-Segel chemotaxis model is described by a system of nonlinear PDE : a convection diffusion equation for the cell density coupled with a reaction-diffusion equation for chemoattractant concentration. In this work, we study the phenomenon of Keller Segel model coupled with a heat equation, because The heat has an effect the density of the cells as well as the signal of chemical concentration, since the heat is a factor affecting the spread and attraction of cells as well in relation to the signal of chemical concentration, The main objectives of this work is the study of the global existence and uniqueness and boundedness of the weak solution for the problem defined in (2.4) for this we use the technical of Galerkin method.

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Références

Betterton, M. D., and M. P. Brenner, Collapsing bacterial cylinders, Phys. Rev. E, 6 (Year), pp. 1234-1237.

Budrene, E. O., and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature (London), 349(1991), pp. 630-633.

Budrene, E. O., and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature (London), 376 (1995), pp. 49-53.

Brenner, M. P., L. Levitov, and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacteria, Biophys. J., 74 (1995), pp. 1677-1693.

Childress, S., and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), pp. 217-237.

Guesmia, A., and N. Daili, About the existence and uniqueness of solution to fractional Burgers equation, Acta Universitatis Apulensis, 21 (2010), pp. 161-170.

Guesmia, A., and N. Daili, Existence and uniqueness of an entropy solution for Burgers equations, Applied Mathematical Sciences, 2(33) (2008), pp. 1635-1664.

Herrero, M. A., and J. J. L. Velazquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Pisa IV, 35 (1997), pp. 633-683.

Herrero, M. A., E. Medina, and J. J. L. Velazquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), pp. 1739-1754.

Herrero, M. A., E. Medina, and J. J. L. Velazquez, Self-similar blow-up for a reaction-diffusion system, Journal of Computational and Applied Mathematics, 97 (1998), pp. 99-119.

Hillen, T., and A. Potapov, The one-dimensional chemotaxis model: global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), pp. 1783-1801.

Horstmann, D., and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), pp. 52-107.

Jager, W., and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), pp. 819-824.

Keller, E. F., and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), pp. 399-415.

Messikh, C., A. Guesmia, and S. Saadi, Global Existence and Uniqueness of the Weak Solution in Keller-Segel Model, 2249-4626 Print ISSN: 0975-5896.

Murray, J. D., Mathematical Biology, 3rd ed., Vol. I and II, Springer, New York, NY, 2002.

Nagai, T., Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), pp. 581-601.

Osaki, K., and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), pp. 441-469.

Rahai, A., Slimani, A., and Guesmia, A., Existence and uniqueness of solution for a fractional thixotropic model, Mathematical Methods in the Applied Sciences, April 2023, DOI: 10.1002/mma.9283.

Slimani, A., Rahai, A., Guesmia, A., and Bouzettouta, L., Stochastic Chemotaxis model with fractional derivative driven by multiplicative noise, Int. J. Anal. Appl., 19 (2021), pp. 1-32, DOI: 10.28924/2291-8639-19-2021-1.

Slimani, A., Bouzettouta, L., and Guesmia, A., Existence and Uniqueness of the weak solution for Keller-Segel model coupled with Boussinesq equations, Demonstatio Mathematica, June 28, 2021, doi.org/10.1515/dema-2021-0027.

Slimani, A., Sadek, L., and Guesmia, A., Analytical Solution of One-Dimensional Keller-Segel Equations via New Homotopy Perturbation Method, Contemporary Mathematics, March 2024, doi.org/10.37256/cm.5120242604. (1970).