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**Invariant regions and global existence of solutions for reaction-diffusion systems with a tridiagonal matrix of diffusion coefficients and nonhomogeneous boundary conditions.**
*(English)*
Zbl 1166.35338

Summary: 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. Our techniques are based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of polynomial growth.

### MSC:

35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |

35K55 | Nonlinear parabolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

35K57 | Reaction-diffusion equations |

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\textit{A. Salem}, J. Appl. Math. 2007, Article ID 12375, 15 p. (2007; Zbl 1166.35338)

### References:

[1] | A. Friedman, “Partial Differential Equations of Parabolic Type,” Prentice-Hall, Englewood Cliffs, NJ, USA, 1964. · Zbl 0144.34903 |

[2] | D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1984. |

[3] | A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. · Zbl 0516.47023 |

[4] | 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. · Zbl 0988.35078 |

[5] | J. Smoller, Shock Waves and Reaction-Diffusion Equations, vol. 258 of Fundamental Principles of Mathematical Science, Springer, New York, NY, USA, 1983. · Zbl 0508.35002 |

[6] | 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. · Zbl 1012.35041 |

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