Invariant Regions and Global Existence of Uniqueness Weak Solutions for Tridiagonal Reaction-Diffusion Systems
DOI:
https://doi.org/10.24193/subbmath.2024.2.08Keywords:
Semigroups, local weak solution, global weak solution, reaction-diffusion systems, invariant regions, matrice of diffusionAbstract
In this paper we study the existence of uniqueness global weak solutions for m × m reaction-diffusion systems for which two main properties hold: the positivity of the weak 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 we use here in order to prove global existence is in the same spirit of the method developed by Boccardo, Murat, and Puel for a single equation.
Mathematics Subject Classification (2010): 35K57, 35K40, 35K55.
Received 13 November 2021; Accepted 10 April 2022.
References
Abdelmalek, S., Existence of global solutions via invariant regions for a generalized reaction-diffusion system with a tridiagonal Toeplitz matrix of diffusion coefficients, Functional Analysis: Theory, Methods and Applications, 2(2016), 12-27.
Abdelmalek, S., Invariant regions and global solutions for reaction-diffusion systems with a tridiagonal symmetric Toeplitz matrix of diffusion coefficients, Electron. J. Differential Equations., 2014(2014), no. 247, 1-14.
Abdelmalek, S., Invariant regions and global existence of solutions for reaction-diffusion systems with a tridiagonal matrix of diffusion coefficients and nonhomogeneous boundary conditions, Appl. Math., (2007), 1-15.
Abdelmalek, S., Kouachi, S., Proof of existence of global solutions for m-component reaction-diffusion systems with mixed boundary conditions via the Lyapunov functional method, Phys. A., 40(2007), 12335-12350.
Abdelmalek, K., Rebiai, B., Abdelmalek, S., Invariant regions and existence of global solutions to generalized m-component reaction-diffusion system with tridiagonal symmetric Toeplitz diffusion matrix, Adv. Pure Appl. Math., 12(1)(2021), 1-15.
Alaa, N., Solution faibles d’équations paraboliques quasi-linéaires avec données initiales mesures, Ann. Math. Blaise Pascal, 3(1996), 1-15.
Alaa, N., Mesbahi, S., Existence result for triangular Reaction-Diffusion systems with L1 data and critical growth with respect to the gradient, Mediterr. J. Math. Z., 10(2013), 255-275.
Alaa, N., Mounir, I., 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. 9. Andelic, M., Fonseca, C.M., Sufficient conditions for positive definiteness of tridiagonal
matrices revisited, Positivity, 15(2011), no. 1, 155-159.
Baras, P., Hassan, J.C., Veron, L., Compacité de l’opérateur définissant la solution d’une
équation non homogène, C.R. Math. Acad. Sci. Paris, 284(1977), 799-802.
Bensoussan, A., Boccardo, L., Murat, F., On a nonlinear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 5(1989), 347-364.
Boccardo, L., Gallouot, T., Strongly nonlinear elliptic equations having natural growth terms in L1 data, Nonlinear Anal., 19(1992), no. 6, 573-579.
Boccardo, L., Murat, F., Puel, J.P., Existence results for some quasilinear parabolic equations, Nonlinear Anal., 13(1989), 373-392.
Bouarifi, W., Alaa, N., Mesbahi S., Global existence of weak solutions for parabolic triangular reaction-diffusion systems applied to a climate model, An. Univ. Craiova Ser. Mat. Inform., 42(1)(2015), 80-97.
Dall’aglio, A., Orsina, L., Nonlinear parabolic equations with natural growth conditions and L1 data, Nonlinear Anal., 27(1996), 59-73.
Friedman, A., Partial Differential Equations of Parabolic Type, Prentice Pract., New Jersey, 1964.
Henry, D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math, 1984.
Hollis, S.L., Martin, R.H., Pierre, M., Global existence and boundedness in reaction- diffusion systems, SIAM J. Math. Anal., 18(1987), no. 3, 744-761.
Hollis, S.L., Morgan, J., Interior estimates for a class of reaction-diffusion systems from
L1 a priori estimates, J. Differential Equations, 98(1992), no. 2, 260-276.
Hollis, S.L., Morgan, J.J., On the blow-up of solutions to some semilinear and quasilinear reaction-diffusion systems, Rocky Moutain J. Math., 14(1994), 1447-1465.
Kouachi, S., 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.
Kulkarni, D., Schmidh, D., Tsui, S., Eigenvalues of tridiagonal pseudo-Toeplitz matrices, Linear Algebra Appl., 297(1999), 63-80.
Ladyzenskaya, O.A., Solonnikov, V.A., Uralceva, N.N., Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr., 23, AMS, Providence, R.I, 1968.
Landes, R., Mustonen, V., On parabolic initial-boundary value problems with critical growth for the gradient, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 11(1994), 135- 158.
Martin, R.H., Pierre, M., Nonlinear reaction-diffusion systems, in: Nonlinear Equations in the Applied Sciences, Math. Sci. Eng., (1991).
Moumeni, A., Barrouk, N., Existence of global solutions for systems of reaction- diffusion with compact result, International Journal of Pure and Applied Mathematics, 102(2)(2015), 169-186.
Moumeni, A., Barrouk, N., Triangular reaction-diffusion systems with compact result, Global Journal of Pure and Applied Mathematics, 11(6)(2015), 4729-4747.
Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., 44, Springer-Verlag, New York, 1983.
Pierre, M., Schmitt, D., Existence Globale ou Explosion pour les Systèmes de Réaction- Diffusion avec Contrôle de Masse, Thèse de Doctorat, Université Henri Poincaré, Nancy I, 1995.
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