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Asymptotic behavior and stability of solutions of semilinear diffusion equations. (English) Zbl 0445.35063

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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[12] Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1972), 979-1000. Note added after submission: After completing this work, the author got acquainted with the following paper: Casten, R. G., and Holland, C. J., Instability results for reac tion diffusion equations with Neumann boundary conditions, J. Differential Equations 27 (1978), 266-273. It contains the same results as in our Theorem 5.1. Further it is shown that any nonconstant solution of (1. 3) is unstable if f(u) is convex or concave. But the problem of finding / and D with which there exists a nonconstant stable solution of (1. 3) is still unsolved there, the answer to which can be found in our Theorem 6. 2 and Corol- lary 6. 3.
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