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2014, 7(6): 1193-1202. doi: 10.3934/dcdss.2014.7.1193

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

 1 Departmento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia, Colombia 2 Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, United States 3 Department of Mathematics, Universidad Distrital Francisco José de Caldas, Bogotá, Colombia

Received  April 2013 Revised  November 2013 Published  June 2014

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.
Citation: José Caicedo, Alfonso Castro, Rodrigo Duque, Arturo Sanjuán. Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1193-1202. doi: 10.3934/dcdss.2014.7.1193
##### References:
 [1] P. Bates and A. Castro, Existence and uniqueness for a variational hyperbolic system without resonance,, Nonlinear Analysis TMA, 4 (1980), 1151. doi: 10.1016/0362-546X(80)90024-3. [2] M. Berti and L. Biasco, Forced vibrations of wave equations with non-monotone nonlinearities,, Ann. Inst. H. Poincaré Anal. Non Lineaire, 23 (2006), 439. doi: 10.1016/j.anihpc.2005.05.004. [3] 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. [4] R. Brooks and K. Schmitt, The contraction mapping principle and some applications,, Electron. J. Diff. Eqns. Monograph, 90 (2009). [5] J. Caicedo and A. Castro, A semilinear wave equation with derivative of nonlinearity containing multiple eigenvalues of infinite multiplicity,, Contemp. Math., 208 (1997), 111. doi: 10.1090/conm/208/02737. [6] 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. doi: 10.3934/dcds.2009.24.653. [7] 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. doi: 10.1007/s00032-011-0154-7. [8] A. Castro, Semilinear equations with discrete spectrum,, Contemporary Mathematics, 357 (2004), 1. doi: 10.1090/conm/357/06509. [9] A. Castro and B. Preskill, Existence of solutions for a semilinear wave equation with non-monotone nonlinearity,, Discrete and Continuous Dynamical Systems, 28 (2010), 649. doi: 10.3934/dcds.2010.28.649. [10] A. Castro and S. Unsurangsie, A semilinear wave equation with nonmonotone nonlinearity,, Pacific J. Math., 132 (1988), 215. doi: 10.2140/pjm.1988.132.215. [11] D. Gilbarg and N. Trudinger, Eliiptic Partial Differential Equations of Second Order,, Springer Verlag, (1997). [12] H. Hofer, On the range of a wave operator with nonmonotone nonlinearity,, Math. Nachr., 106 (1982), 327. doi: 10.1002/mana.19821060128. [13] J. Mawhin, Periodic solutions of some semilinear wave equations and systems: A survey,, Chaos, 5 (1995), 1651. doi: 10.1016/0960-0779(94)00169-Q. [14] 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. doi: 10.1090/S0002-9939-1985-0766527-X. [15] P. Rabinowitz, Large amplitude time periodic solutions of a semilinear wave equation,, Comm. Pure Appl. Math., 37 (1984), 189. doi: 10.1002/cpa.3160370203. [16] M. Willem, Density of the range of potential operators,, Proc. Amer. Math. Soc., 83 (1981), 341. doi: 10.1090/S0002-9939-1981-0624926-7.

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##### References:
 [1] P. Bates and A. Castro, Existence and uniqueness for a variational hyperbolic system without resonance,, Nonlinear Analysis TMA, 4 (1980), 1151. doi: 10.1016/0362-546X(80)90024-3. [2] M. Berti and L. Biasco, Forced vibrations of wave equations with non-monotone nonlinearities,, Ann. Inst. H. Poincaré Anal. Non Lineaire, 23 (2006), 439. doi: 10.1016/j.anihpc.2005.05.004. [3] 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. [4] R. Brooks and K. Schmitt, The contraction mapping principle and some applications,, Electron. J. Diff. Eqns. Monograph, 90 (2009). [5] J. Caicedo and A. Castro, A semilinear wave equation with derivative of nonlinearity containing multiple eigenvalues of infinite multiplicity,, Contemp. Math., 208 (1997), 111. doi: 10.1090/conm/208/02737. [6] 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. doi: 10.3934/dcds.2009.24.653. [7] 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. doi: 10.1007/s00032-011-0154-7. [8] A. Castro, Semilinear equations with discrete spectrum,, Contemporary Mathematics, 357 (2004), 1. doi: 10.1090/conm/357/06509. [9] A. Castro and B. Preskill, Existence of solutions for a semilinear wave equation with non-monotone nonlinearity,, Discrete and Continuous Dynamical Systems, 28 (2010), 649. doi: 10.3934/dcds.2010.28.649. [10] A. Castro and S. Unsurangsie, A semilinear wave equation with nonmonotone nonlinearity,, Pacific J. Math., 132 (1988), 215. doi: 10.2140/pjm.1988.132.215. [11] D. Gilbarg and N. Trudinger, Eliiptic Partial Differential Equations of Second Order,, Springer Verlag, (1997). [12] H. Hofer, On the range of a wave operator with nonmonotone nonlinearity,, Math. Nachr., 106 (1982), 327. doi: 10.1002/mana.19821060128. [13] J. Mawhin, Periodic solutions of some semilinear wave equations and systems: A survey,, Chaos, 5 (1995), 1651. doi: 10.1016/0960-0779(94)00169-Q. [14] 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. doi: 10.1090/S0002-9939-1985-0766527-X. [15] P. Rabinowitz, Large amplitude time periodic solutions of a semilinear wave equation,, Comm. Pure Appl. Math., 37 (1984), 189. doi: 10.1002/cpa.3160370203. [16] M. Willem, Density of the range of potential operators,, Proc. Amer. Math. Soc., 83 (1981), 341. doi: 10.1090/S0002-9939-1981-0624926-7.
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