On the existence of positive solutions of a class of parabolic reaction diffusion systems

Authors

  • Mounir REDJOUH Laboratory LAMAHIS, Department of Mathematics, Faculty of Sciences, 20 August 1955 University, Skikda, 21000, Algeria, e-mail: redjouhmounir@gmail.com
  • Salim MESBAHI Fundamental and Numerical Mathematics Laboratory, Department of Mathematics, Faculty of Sciences, Ferhat Abbas University, Setif, 19000, Algeria, e-mail: salimbra@gmail.com https://orcid.org/0000-0002-2455-3991

DOI:

https://doi.org/10.24193/subbmath.2022.3.04

Keywords:

Reaction diffusion systems, parabolic systems, positive solutions, Green function.

Abstract

In this paper, we show the existence of continuous positive solutions of a class of nonlinear parabolic reaction diffusion systems with initial conditions using techniques of functional analysis and potential analysis.

Mathematics Subject Classification (2010): 34B27, 35B09, 35K41, 35K57.

Received 18 October 2019; Accepted 17 January 2020.

References

Alaa, N., Mesbahi, S., Bouarifi, W., Global existence of weak solutions for parabolic triangular reaction diffusion systems applied to a climate model, An. Univ. Craiova Ser. Mat. Inform., 42(2015), no. 1, 80-97.

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

Arakelian, N., Gauthier, P.M., Approximation, Complex Analysis and Potential Theory, NATO Sci. Ser. II, Vol. 37, 2001.

Armitage, D.H., Gardiner, S.J., Classical Potential Theory, Springer, London, 2001.

Bachar, I., Mˆaagli, H., Zeddini, N., Estimates on the green function and existence of positive solutions of nonlinear singular elliptic equations, Commun. Contemp. Math., 5(2003), no. 3, 401-434.

Ghanmi, A., Mˆaagli, H., Radulescu, V., Zeddini, N., Large and bounded solutions for a class of nonlinear Schr¨odinger stationary systems, Anal. Appl. (Singap.), 7(2009), no. 4, 391-404.

Ghanmi, A., Mˆaagli, H., Turki, S., Zeddini, N., Existence of positive bounded solutions for some nonlinear elliptic systems, J. Math. Anal. Appl., 352(2009), 440-448.

Ghergu, M., Radulescu, V., Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Ser. Math. Appl., vol. 37, Oxford University Press, 2008.

Gontara, S., Turki, S., Existence and asymptotic behavior of positive continuous solutions for some nonlinear parabolic systems, Nonlinear Analysis, 72(2010), no. 3, 1514-1521.

Lions, J.L., Quelques M´ethodes de R´esolution des Probl`emes aux Limites Non Lin´eaires, Dunod-Gauthier Villars, 1969.

Ma, R., Existence of positive radial solutions for elliptic systems, J. Math. Anal. Appl., 201(1996), 375-386.

Mˆaagli, H., Masmoudi, S., Zribi, M., Asymptotic behaviour of positive solutions for heat equation with nonlinear term, J. Math. Anal. Appl., 333(2007), 1272-1289.

Mˆaagli, H., Masmoudi, S., Zribi, M., On a parabolic problem with nonlinear term in a half space and global behavior of solutions, J. Differential Equations, 246(2009), 3417-3447.

Mˆaagli, H., Zribi, M., On a new Kato class and singular solutions of a nonlinear elliptic equation in bounded domains, Positivity, 9(2005), 667-686.

Mesbahi, S., Alaa, N., 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.

Mesbahi, S., Alaa, N., Mathematical analysis of a reaction diffusion model for image restoration, An. Univ. Craiova Ser. Mat. Inform., 42(2015), no. 1, 70-79.

Pierre, M., Global existence in reaction-diffusion systems with dissipation of mass: a survey, Milan J. Math., 78(2010), no. 2, 417-455.

Port, S.C., Stone, C.J., Brownian Motion and Classical Potential Theory, Academic Press, 1978.

Zhang, Q.S., On a parabolic equation with a singular lower order term, Trans. Am. Math. Soc., 348(1996), no. 7, 2811-2844.

Zhang, Q.S., Global existence and local continuity of solutions for semilinear parabolic equations, Comm. Partial Differential Equations, 22(1997), 1529-1557.

Zhao, Z., On the existence of positive solutions of nonlinear elliptic equations A probabilistic potential theory approach, Duke Math. J., 69(1993), 247-258.

Downloads

Published

2022-09-20

How to Cite

REDJOUH, M., & MESBAHI, S. (2022). On the existence of positive solutions of a class of parabolic reaction diffusion systems. Studia Universitatis Babeș-Bolyai Mathematica, 67(3), 501–511. https://doi.org/10.24193/subbmath.2022.3.04

Issue

Section

Articles