Existence of Multiple Periodic Solutions for a Semilinear Wave Equation in an <i>n</i>-Dimensional Ball

Wei, Hui; Ji, Shuguan

doi:10.1515/ans-2018-2036

Cited by 5 publications

(1 citation statement)

References 27 publications

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“…The problem of finding periodic solutions of such equations has been widely considered since the pioneer work of Rabinowitz [26]. For example, see [2,4,6,8,11,27] for one dimensional case and [1,3,5,12,33] for higher dimensional case. Most of these results are based on the spectral properties of wave operator.…”

Section: Introductionmentioning

confidence: 99%

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

Wei¹,

Ji²

2021

Preprint

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We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. The variable coefficient characterizes the inhomogeneity of media and its presence usually leads to the destruction of the compactness of the inverse of linear wave operator with periodic-Dirichlet boundary conditions on its range. In the pioneering work of Barbu and Pavel [7], it gives the existence and regularity of periodic solution for Lipschitz, nonresonant and monotone nonlinearity under the assumption η u > 0 (see Sect. 2 for its definition) on the coefficient u(x) and leaves the case η u = 0 as an open problem. In this paper, by developing the invariant subspace method and using the complete reduction technique and Leray-Schauder theory, we obtain the existence of periodic solutions for such a problem when the nonlinear term satisfies the asymptotic nonresonance conditions. Our result not only does not need any requirements on the coefficient except for the natural positivity assumption (i.e., u(x) > 0), but also does not need the monotonicity assumption on the nonlinearity. In particular, when the nonlinear term is an odd function and satisfies the global nonresonance conditions, there is only one (trivial) solution to this problem on the invariant subspace.

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Section: Introductionmentioning

confidence: 99%

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

Wei¹,

Ji²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

Wei

2022

Sci. China Math.

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Multiple periodic solutions for an asymptotically linear wave equation with x-dependent coefficients

Wei

2021

Journal of Mathematical Physics

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In this paper, we investigate the existence of multiple nontrivial periodic solutions for an asymptotically linear wave equation with x-dependent coefficients. Such a mathematical model can be derived from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By constructing a suitable function space, we characterize the problem as a variational problem, and then we prove that there are at least three nontrivial periodic solutions via variational methods.

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Existence of Multiple Periodic Solutions for a Semilinear Wave Equation in an n-Dimensional Ball

Cited by 5 publications

References 27 publications

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

Multiple periodic solutions for an asymptotically linear wave equation with x-dependent coefficients

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