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

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December 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

José Caicedo ^1, , Alfonso Castro ^2, , Rodrigo Duque ^1, and Arturo Sanjuán ^3,

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

Abstract
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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.

Keywords: resonance, Semilinear wave equation, flat on characteristic..

Mathematics Subject Classification: Primary: 35L05, 35L-70; Secondary: 35P3.

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	R. Brooks and K. Schmitt, The contraction mapping principle and some applications,, Electron. J. Diff. Eqns. Monograph, 90 (2009). Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[8]	A. Castro, Semilinear equations with discrete spectrum,, Contemporary Mathematics, 357* (2004), 1. doi: 10.1090/conm/357/06509. Google Scholar*
[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. Google Scholar
[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. Google Scholar*
[11]	D. Gilbarg and N. Trudinger, Eliiptic Partial Differential Equations of Second Order,, Springer Verlag, (1997). Google Scholar
[12]	H. Hofer, On the range of a wave operator with nonmonotone nonlinearity,, Math. Nachr., 106* (1982), 327. doi: 10.1002/mana.19821060128. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	R. Brooks and K. Schmitt, The contraction mapping principle and some applications,, Electron. J. Diff. Eqns. Monograph, 90 (2009). Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[8]	A. Castro, Semilinear equations with discrete spectrum,, Contemporary Mathematics, 357* (2004), 1. doi: 10.1090/conm/357/06509. Google Scholar*
[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. Google Scholar
[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. Google Scholar*
[11]	D. Gilbarg and N. Trudinger, Eliiptic Partial Differential Equations of Second Order,, Springer Verlag, (1997). Google Scholar
[12]	H. Hofer, On the range of a wave operator with nonmonotone nonlinearity,, Math. Nachr., 106* (1982), 327. doi: 10.1002/mana.19821060128. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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