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An invariant measure for the equationu tt −u xx +u 3=0

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Abstract

Numerical studies of the initial boundary-value problem of the semilinear wave equationu tt −u xx +u 3=0 subject to periodic boundary conditionsu(t, 0)=u(t, 2π),u t (t, 0)=u t (t, 2π) and initial conditionsu(0,x)=u 0(x),u t(0,x)=v 0(x), whereu 0(x) andv 0(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial stateu 0(x),v 0(x) after undergoing a possibly chaotic evolution. In this paper an appropriate abstract space is considered. In this space a finite measure is constructed. This measure is invariant under the flow generated by the Hamiltonian system which corresponds to the original equation. This enables one to verify the above “returning” property.

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  1. 5 Kremenchugskaia St., Apt. 228, Moscow, USSR

    L. Friedlander

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  1. L. Friedlander
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Communicated by J. Mather

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Friedlander, L. An invariant measure for the equationu tt −u xx +u 3=0. Commun.Math. Phys. 98, 1–16 (1985). https://doi.org/10.1007/BF01211041

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  • DOI: https://doi.org/10.1007/BF01211041

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