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Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity

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  • For Dirichlet-periodic and double periodic boundary conditions, we prove the existence of solutions to a forced semilinear wave equation with large forcing terms not flat on characteristics. The nonlinearity is assumed to be non-monotone, asymptotically linear, and not resonanant. We prove that the solutions are in $L^{p}$, $(p\geq 2)$, when the forcing term is in $L^{p}$. This is optimal; even in the linear case there are $L^p$ forcing terms for which the solutions are only in $L^p$. Our results extend those in [9] where the forcing term is assumed to be in $L_{\infty}$, and are in contrast with those in [6] where the non-existence of continuous solutions is established for $C^{\infty}$ forcing terms flat on characteristics. 200 words.
    Mathematics Subject Classification: Primary: 35L05, 35L-70; Secondary: 35P30.


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  • [1]

    P. Bates and A. Castro, Existence and uniqueness for a variational hyperbolic system without resonance, Nonlinear Analysis TMA, 4 (1980), 1151-1156.doi: 10.1016/0362-546X(80)90024-3.


    M. Berti and L. Biasco, Forced vibrations of wave equations with non-monotone nonlinearities, Ann. Inst. H. Poincaré Anal. Non Lineaire, 23 (2006), 439-474.doi: 10.1016/j.anihpc.2005.05.004.


    H. Brezis and L. Nirenberg, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Annali della Scuola Norm. Sup. di Pisa, 5 (1978), 225-236.


    R. Brooks and K. Schmitt, The contraction mapping principle and some applications, Electron. J. Diff. Eqns. Monograph, 90 (2009), 90 pp.


    J. Caicedo and A. Castro, A semilinear wave equation with derivative of nonlinearity containing multiple eigenvalues of infinite multiplicity, Contemp. Math., 208 (1997), 111-132.doi: 10.1090/conm/208/02737.


    J. Caicedo and A. Castro, A semilinear wave equation with smooth data and no resonance having no continuous solution, Discrete and Continuous Dynamical Systems, 24 (2009), 653-658.doi: 10.3934/dcds.2009.24.653.


    J. Caicedo, A. Castro and R. Duque, Existence of solutions for a wave equation with non-monotone nonlinearity and a small parameter, Milan Journal of Mathematics, 79 (2011), 207-220.doi: 10.1007/s00032-011-0154-7.


    A. Castro, Semilinear equations with discrete spectrum, Contemporary Mathematics, 357 (2004), 1-16.doi: 10.1090/conm/357/06509.


    A. Castro and B. Preskill, Existence of solutions for a semilinear wave equation with non-monotone nonlinearity, Discrete and Continuous Dynamical Systems, Series A, 28 (2010), 649-658.doi: 10.3934/dcds.2010.28.649.


    A. Castro and S. Unsurangsie, A semilinear wave equation with nonmonotone nonlinearity, Pacific J. Math., 132 (1988), 215-225.doi: 10.2140/pjm.1988.132.215.


    D. Gilbarg and N. Trudinger, Eliiptic Partial Differential Equations of Second Order, Springer Verlag, 1997.


    H. Hofer, On the range of a wave operator with nonmonotone nonlinearity, Math. Nachr., 106 (1982), 327-340.doi: 10.1002/mana.19821060128.


    J. Mawhin, Periodic solutions of some semilinear wave equations and systems: A survey, Chaos, Solitons and Fractals, 5 (1995), 1651-1669.doi: 10.1016/0960-0779(94)00169-Q.


    P. J. McKenna, On solutions of a nonlinear wave equation when the ratio of the period to the length of the interval is irrational, Proc. Amer. Math. Soc., 93 (1985), 59-64.doi: 10.1090/S0002-9939-1985-0766527-X.


    P. Rabinowitz, Large amplitude time periodic solutions of a semilinear wave equation, Comm. Pure Appl. Math., 37 (1984), 189-206.doi: 10.1002/cpa.3160370203.


    M. Willem, Density of the range of potential operators, Proc. Amer. Math. Soc., 83 (1981), 341-344.doi: 10.1090/S0002-9939-1981-0624926-7.

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