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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 |

### Keywords:

asymptotic behavior; semilinear diffusion equations; initial-boundary value problem; second order uniformly elliptic operator; stability of solutions
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\textit{H. Matano}, Publ. Res. Inst. Math. Sci. 15, 401--454 (1979; Zbl 0445.35063)

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### References:

[1] | Amann, H., On the existence of positive solutions of nonlinear elliptic boundary value problems, Indiana Univ. Math. J., 21 (1971), 125-146. · Zbl 0219.35037 |

[2] | - , Existence of multiple solutions for nonlinear elliptic boundary value pro- blems, Indiana Univ. Math. J., 21 (1972), 925-935. · Zbl 0222.35023 |

[3] | - , Supersolutions, monotone iterations, and stability, J. Differential Equa- tions, 21 (1976), 363-377. · Zbl 0319.35039 |

[4] | Chafee, N., Asymptotic behavior for solutions of a one-dimensional parabolic equa- tion with homogeneous Neumann boundary conditions, J. Differential Equations, 18 (1975), 111-134, · Zbl 0304.35008 |

[5] | Friedman, A., Partial differential equations oi parabolic type, Prentice -Hall,, 1963. |

[6] | Fujita, H., On the nonlinear equations du + eu = Q and dv/dt=dv + ev, Bull. Amer. Math. Soc., 75 (1969), 132-135. |

[7] | - , On the asymptotic stability of solutions of the equation vt = Av-^-ev9 Proc. Intern. Conference on Functional Analysis and Related Topics, Tokyo 1969, Univ. of Tokyo Press 1970, 252-259. · Zbl 0215.16003 |

[8] | Ito, S., Fundamental solutions of parabolic differential equations and boundary value problems, Japan J. Math., 27 (1957), 55-102. · Zbl 0092.31101 |

[9] | - , A remark on my paper ”A boundary value problem of partial differential equations of parabolic type ” in Duke Mathematical Journal, Proc. Japan Acad., 34 (1958), 463-465. · Zbl 0098.06601 |

[10] | Keller, H. B., Elliptic boundary value problems suggested by nonlinear diffusion processes, Arch. Rat. Mech. Anal., 35 (1969), 363-381. · Zbl 0188.17102 |

[11] | Protter, M. H. and Weinberger, H. F., Maximum principles in differential equations, Prentice-Hall, 1967. · Zbl 0153.13602 |

[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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