Generalized result on the global existence of positive solutions for a parabolic reaction diffusion model with a full diffusion matrix

Nabila Barrouk, Salim Mesbahi

Abstract


In this paper, we study the global existence in time of solutions for a parabolic reaction diffusion model with a full matrix of diffusion coefficients on a bounded domain. The technique used is based on compact semigroup methods and some estimates. Our objective is to show, under appropriate hypotheses, that the proposed model has a global solution with a large choice of nonlinearities.

Keywords


global existence, compact semigroups, reaction diffusion systems.

Full Text:

PDF

References


: N. Alaa, S. Mesbahi, W. Bouarifi, Global existence of weak solutions for parabolic triangular reaction diffusion systems applied to a climate model, An. Univ. Craiova Ser. Mat. Inform., Vol. 42(1) (2015), 80-97.

: N. Alaa, S. Mesbahi, A. Mouida, W. Bouarifi, Existence of solutions for quasilinear elliptic degenerate systems with L¹ data and nonlinearity in the gradient, Electron. J. Diff. Equ., Vol. 2013 (2013), No. 142, 1-13.

: N.D. Alikakos, L^{p}-bounds of solutions of reaction diffusion equations, Comm. Partial Differential Equations 4 (1979), 827-868.

: S. Bonafede, D. Schmitt, Triangular reaction diffusion systems with integrable initial data, Nonlinear analysis 33 (1998), 785-801.

: N.F. Britton, Reaction diffusion equations and their applications to Biology, Academic Press, London, (1986).

: P.C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomath. 28, Springer-Verlag, Berlin, New York, (1979).

: A. Haraux, A. Youkana, On a result of K. Masuda concerning reaction diffusion equations, Tohoku Math. J. 40 (1988), 159-163.

: A. Haraux, M. Kirane, Estimations C¹ pour des problèmes paraboliques semi-lineaires, Ann. Fac. Sci. Toulouse Math., 5 (1983), 265-280.

: D. Henry, Theory of semilinear parabolic equations, Lecture notes in Math, 840, Springer-Verlag, New York (1984).

: S.L. Hollis, R. H. Martin, M. Pierre, Global existence and boundedness in reaction diffusion systems, SIAM J. Math. anal, 18: (1987), 744-761.

: S. Kouachi, Invariant regions and global existence of solutions for reaction diffusion systems with full matrix of diffusion coefficients and nonhomogeneous boundary conditions, Georgian Math. J., 11 (2004), 349-359.

: S. Kouachi, A. Youkana, Global existence for a class of reaction diffusion systems, Bull. Polish. Acad. Sci. Math. 49(3), (2001).

: O.A. Ladyzenskaya, V. A Solonnikov, N. N Uralceva, Linear and quasilinear equations of parabolic type, Trans. Math. Monographs. vol. 23, AMS; Providence, R. I, (1968).

: J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod : Gauthier-Villars, 1969.

: K. Masuda, On the global existence and asymptotic behaviour of solution of reaction diffusion equations, Hokkaido Math, J. 12: (1983), 360-370.

: M. Mebarki, A. Moumeni, Global solution of system reaction diffusion with full matrix, Global Journal of Mathematical Analysis, (2015), 04-25.

: S. Mesbahi, N. Alaa, Mathematical analysis of a reaction diffusion model for image restoration, An. Univ. Craiova Ser. Mat. Inform., Vol. 42 (1) (2015), 70-79.

: S. Mesbahi, N. Alaa, Existence result for triangular reaction diffusion systems with L¹ data and critical growth with respect to the gradient, Mediterr. J. Math., 10 (2013), 255-275.

: A. Moumeni, N. Barrouk, Existence of global solutions for systems of reaction diffusion with compact result, IJPAM. 102(2) (2015), 169-186.

: A. Moumeni, N. Barrouk, Triangular reaction diffusion systems with compact result, GJPAM. 11(6) (2015), 4729-4747.

: J. D. Murray, Mathematical Biology I : An Introduction, volume I, Springer-Verlag, 3rd edition, 2003.

: J. D. Murray, Mathematical Biology II : Spatial Models and Biochemical Applications, volume II, Springer-Verlag, 3rd edition, 2003.

: A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.

: M. Pierre, Global existence in reaction diffusion systems with dissipation of mass : a survey, Milan J. Math., Vol. 78 (2) (2010), 417-455.

: F. Rothe, Global existence of reaction diffusion systems, Lecture Notes in Math, 1072, Springer-Verlag, Berlin, 1984.

: J. Smoller, Shock waves and reaction difussion systems, Springer-Verlag, New York, (1983).




DOI: http://dx.doi.org/10.24193/subbmath.2023.2.11

Refbacks

  • There are currently no refbacks.