## Journal of Applied Mathematics

### Invariant Regions and Global Existence of Solutions for Reaction-Diffusion Systems with a Tridiagonal Matrix of Diffusion Coefficients and Nonhomogeneous Boundary Conditions

Abdelmalek Salem

#### Abstract

The purpose of this paper is the construction of invariant regions in which we establish the global existence of solutions for reaction-diffusion systems (three equations) with a tridiagonal matrix of diffusion coefficients and with nonhomogeneous boundary conditions after the work of Kouachi (2004) on the system of reaction diffusion with a full 2-square matrix. Our techniques are based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of polynomial growth.

#### Article information

Source
J. Appl. Math., Volume 2007 (2007), Article ID 12375, 15 pages.

Dates
First available in Project Euclid: 27 February 2008

https://projecteuclid.org/euclid.jam/1204126691

Digital Object Identifier
doi:10.1155/2007/12375

Mathematical Reviews number (MathSciNet)
MR2365984

Zentralblatt MATH identifier
1166.35338

#### Citation

Salem, Abdelmalek. Invariant Regions and Global Existence of Solutions for Reaction-Diffusion Systems with a Tridiagonal Matrix of Diffusion Coefficients and Nonhomogeneous Boundary Conditions. J. Appl. Math. 2007 (2007), Article ID 12375, 15 pages. doi:10.1155/2007/12375. https://projecteuclid.org/euclid.jam/1204126691

#### References

• A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964.
• D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1984.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.
• S. Kouachi, Global existence of solutions for reaction-diffusion systems with a full matrix of diffusion coefficients and nonhomogeneous boundary conditions,'' Electronic Journal of Qualitative Theory of Differential Equations, no. 2, pp. 1--10, 2002.
• J. Smoller, Shock Waves and Reaction-Diffusion Equations, vol. 258 of Fundamental Principles of Mathematical Science, Springer, New York, NY, USA, 1983.
• S. Kouachi, Existence of global solutions to reaction-diffusion systems with nonhomogeneous boundary conditions via a Lyapunov functional,'' Electronic Journal of Differential Equations, no. 88, pp. 1--13, 2002.