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

Matano, Hiroshi

Publ. Res. Inst. Math. Sci. 15, 401-454 (1979).

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

Keywords:

asymptotic behavior; semilinear diffusion equations; initial-boundary value problem; second order uniformly elliptic operator; stability of solutions

References:

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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.