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Stability Analysis

In: Nonlinear Parabolic and Elliptic Equations

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  • C. V. Pao

    (North Carolina State University)

Abstract

The existence of upper and lower solutions for the elliptic boundary-value problem ensures the existence of at least one steady-state solution. By constructing a suitable pair of upper and lower solutions for the parabolic boundary-value problem it is sometimes possible to determine whether the steady-state solution is stable or unstable. This construction leads to various sufficient conditions for the stability or instability of a given steady-state solution. When the initial function in the parabolic problem is an upper solution or a lower solution of the corresponding elliptic problem the time-dependent solution is monotone in t and converges to a steady-state solution. This monotone convergence property gives a close relationship between the stability and the uniqueness property of a steady-state solution and is a basic tool for determining the stability property of the maximal and the minimal solutions when there are multiple steady-state solutions. These results are applied to a number of specific models that involve either linear or nonlinear boundary conditions.

Suggested Citation

  • C. V. Pao, 1992. "Stability Analysis," Springer Books, in: Nonlinear Parabolic and Elliptic Equations, chapter 0, pages 183-227, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-3034-3_5
    DOI: 10.1007/978-1-4615-3034-3_5
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