Generalized Result on Global Existence of Weak Solutions for Parabolic Reaction-Diffusion Systems

Main Article Content

Abdelhamid Bennoui, Nabila Barrouk, Mounir Redjouh


In this paper, we study global existence of weak solutions for 2 × 2 parabolic reaction-diffusion systems with a full matrix of diffusion coefficients on a bounded domain, such as, we treat the main properties related: the positivity of the solutions and the total mass of the components are preserved with time. Moreover, we suppose that the non-linearities have critical growth with respect to the gradient. 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 non-linearities.

Article Details


  1. N. Alaa, Solutions Faibles d’Équations Paraboliques Quasi-linéaires Avec Données Initiales Mesures, Ann. Math. Blaise Pascal. 3 (1996), 1-15.
  2. N. Alaa, S. Mesbahi, Existence Result for Triangular Reaction Diffusion Systems With L 1 Data and Critical Growth With Respect to the Gradient, Mediterr. J. Math. 10 (2013), 255-275.
  3. N. Alaa, I. Mounir, Global Existence for Reaction-Diffusion Systems with Mass Control and Critical Growth with Respect to the Gradient, J. Math. Anal. Appl. 253 (2001), 532-557.
  4. N.E. Alaa, M. Pierre, Weak Solutions of Some Quasilinear Elliptic Equations with Data Measures, SIAM J. Math. Anal. 24 (1993), 23-35.
  5. N.D. Alikakos, L p Bounds of Solutions of Reaction-Diffusion Equations, Commun. Part. Differ. Equ. 4 (1979), 827-868.
  6. A. Bensoussan, L. Boccardo, F. Murat, On a Non-Linear P.D.E. Having Natural Growth Terms and Unbounded Solutions, Ann. Inst. H. Poincaré, Anal. Non Linéaire. 5 (1988), 347-264.
  7. L. Boccardo, F. Murat, J.P. Puel, Existence Results for Some Quasilinear Parabolic Equations, Nonlinear Anal.: Theory Meth. Appl. 13 (1989), 373-392.
  8. N. Boudiba, Existence globale pour des systèmes de réaction-diffusion paraboliques quasilinéaires, Thèse de troisième cycle, Université des Sciences et de la Technologie Houari Boumediene d’Alger, (1995).
  9. H. Brezis, W. Strauss, Semi-Linear Second Order Elliptic Equations in L 1 , J. Math. Soc. Japan. 25 (1973), 565-590.
  10. N.F. Britton, Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, London, (1986).
  11. A. Dall’aglio, L. Orsina, Nonlinear Parabolic Equations With Natural Growth Conditions and L 1 Data, Nonlinear Anal.: Theory Meth. Appl. 27 (1996), 59-73.
  12. T. Diagana, Some Remarks on Some Strongly Coupled Reaction-Diffusion Equations, (2003).
  13. P.C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer Berlin Heidelberg, Berlin, Heidelberg, 1979.
  14. A. Haraux, A. Youkana, On a Result of K. Masuda Concerning Reaction-Diffusion Equations, Tohoku Math. J. (2). 40 (1988), 159-163.
  15. S.L. Hollis, R.H. Martin, Jr., M. Pierre, Global Existence and Boundedness in Reaction-Diffusion Systems, SIAM J. Math. Anal. 18 (1987), 744-761.
  16. S.L. Hollis, J. Morgan, Interior Estimates for a Class of Reaction-Diffusion Systems from L 1 a Priori Estimates, J. Differ. Equ. 98 (1992), 260–276.
  17. 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.
  18. S. Kouachi, A. Youkana, Global Existence for a Class of Reaction-Diffusion Systems, Bull. Polish. Acad. Sci. Math. 49 (2001), 1-6.
  19. O.A. Ladyzenskaya, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Trans. Math. Monographs. vol. 23, American Mathematical Society, Providence, RI, (1968).
  20. R. Landes, Solvability of Perturbed Elliptic Equations With Critical Growth Exponent for the Gradient, J. Math. Anal. Appl. 139 (1989), 63-77.
  21. R. Landes, V. Mustonen, On Parabolic Initial-Boundary Value Problems With Critical Growth for the Gradient, Ann. Inst. H. Poincaré Anal. Non Linéaire. 11 (1994), 135-158.
  22. R.H. Martin, M. Pierre, Nonlinear Reaction-Diffusion Systems, in: Nonlinear Equations in the Applied Sciences, Math. Sci. Eng. Academic Press, New York (1991).
  23. K. Masuda, On the Global Existence and Asymptotic Behavior of Solutions of Reaction-Diffusion Equations, Hokkaido Math. J. 12 (1983), 360-370.
  24. A. Moumeni, N. Barrouk, Existence of Global Solutions for Systems of Reaction-Diffusion With Compact Result, Int. J. Pure Appl. Math. 102 (2015), 169-186.
  25. A. Moumeni, N. Barrouk, Triangular Reaction-Diffusion Systems With Compact Result, Glob. J. Pure Appl. Math. 11 (2015), 4729-4747.
  26. J.D. Murray, Mathematical Biology, Springer-Verlag, New York, (1993).
  27. M. Pierre, D. Schmitt, Existence Globale ou Explosion Pour les Systèmes De réaction-Diffusion Avec Contrôle de Masse, Thèse de Doctorat, Université Henri Poincaré, Nancy I, (1995).
  28. B. Rebiai, S. Benachour, Global Classical Solutions for Reaction–diffusion Systems With Nonlinearities of Exponential Growth, J. Evol. Equ. 10 (2010), 511-527.
  29. J. Smoller, Shock Waves and Reaction-Difussion Systems, Springer-Verlag, New York, (1983).
  30. A.I. Volpert, V.A. Volpert, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, RI, (1994).