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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.

show all references

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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