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Stability and Asymptotic Behavior of Solutions

In: Nonlinear Parabolic and Elliptic Equations

Author

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  • C. V. Pao

    (North Carolina State University)

Abstract

The stability analysis for scalar boundary-value problems is extended to coupled system of two equations where the reaction function is quasimonotone. For a given steady-state solution, including the trivial solution, sufficient conditions are obtained to ensure its asymptotic stability for each type of quasimonotone reaction functions. Sufficient conditions are also given for the instability of a steady-state solution. When the reaction function is either quasimonotone nondecreasing or quasimonotone nonincreasing the monotone argument is used to show the convergence of the time-dependent solution to a steady-state solution between upper and lower solutions. This leads to the asymptotic stability of a steady-state solution in a sector when it is unique. This approach is extended to a coupled nonautonomous system where the limit of the reaction function as t → ∞ is either quasimonotone nondecreasing or quasimonotone nonincreasing. In the special case of a homogeneous Neumann boundary condition the asymptotic behavior of the solution is compared with the solution of the corresponding ordinary differential system. Most of the discussions for coupled differential equations are extended to systems that are coupled through the boundary conditions.

Suggested Citation

  • C. V. Pao, 1992. "Stability and Asymptotic Behavior of Solutions," Springer Books, in: Nonlinear Parabolic and Elliptic Equations, chapter 0, pages 511-568, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-3034-3_10
    DOI: 10.1007/978-1-4615-3034-3_10
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