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L ∞ stability of an exponentially decreasing solution of the problem Δu+f(x, u)=0 inR n

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Abstract

We consider the Cauchy problemu t=δu+f(x, u), x∈Rn, t>0, and prove that if there exist a strict supersolution\(\bar w\left( x \right)\) and a strict subsolutionw(x) with\(\bar w > w\) then there exists at least one stable equilibrium solution between\(\bar w\) andw provided thatf satifies certain conditions. The stability is with respect to theL ∞ norm. Unlike the case where the spatial domain is bounded, some difficulties occur near |x|=∞ in the present problem. The major part of this paper is devoted to dealing with such difficulties.

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Authors and Affiliations

  1. Department of Mathematics, Hiroshima University, 730, Hiroshima, Japan

    Hiroshi Matano

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  1. Hiroshi Matano
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Additional information

Part of this work was done during the author’s visit to Mathematics Research Center at the University of Wisconsin-Madison; this visit was supported by the U.S. ARO under the Contract number DAAG29-80-C-0041.

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Matano, H. L ∞ stability of an exponentially decreasing solution of the problem Δu+f(x, u)=0 inR n . Japan J. Appl. Math. 2, 85–110 (1985). https://doi.org/10.1007/BF03167040

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  • DOI: https://doi.org/10.1007/BF03167040

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