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Systems with Nonlinear Boundary Conditions

In: Nonlinear Parabolic and Elliptic Equations

Author

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  • C. V. Pao

    (North Carolina State University)

Abstract

This chapter gives a treatment for coupled system of equations with nonlinear boundary conditions analogous to that for coupled systems with linear boundary conditions. The system under consideration can be coupled through the boundary conditions or through the differential equations. For systems of two equations coupled through the boundary conditions, a similar classification of the three types of quasimonotone functions is given. The monotone method is used to establish the corresponding existence-comparison theorems for both parabolic and elliptic boundary-value problems. These theorems are then extended to coupled system of an arbitrary finite number of parabolic and elliptic equations coupled through the equations as well as through the boundary conditions. Both quasimonotone and nonquasimonotone functions are considered, and for nonquasimonotone functions generalized upper and lower solutions are introduced. Similar treatment is given to finite coupled parabolic and elliptic systems with nonlocal boundary conditions. Existence-comparison results for positive solutions of parabolic and elliptic systems are also included in the discussion.

Suggested Citation

  • C. V. Pao, 1992. "Systems with Nonlinear Boundary Conditions," Springer Books, in: Nonlinear Parabolic and Elliptic Equations, chapter 0, pages 459-509, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-3034-3_9
    DOI: 10.1007/978-1-4615-3034-3_9
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