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

Previous Article
Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms
DCDS-S Home
This Issue
Next Article
Alternate steady states for classes of reaction diffusion models on exterior domains

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
Related Papers

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 and 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), 1151-1156. 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), 439-474. 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), 225-236.
[4]	R. Brooks and K. Schmitt, The contraction mapping principle and some applications, Electron. J. Diff. Eqns. Monograph, 90 (2009), 90 pp.
[5]	J. Caicedo and A. Castro, A semilinear wave equation with derivative of nonlinearity containing multiple eigenvalues of infinite multiplicity, Contemp. Math., 208 (1997), 111-132. 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), 653-658. 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), 207-220. doi: 10.1007/s00032-011-0154-7.
[8]	A. Castro, Semilinear equations with discrete spectrum, Contemporary Mathematics, 357 (2004), 1-16. 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, Series A, 28 (2010), 649-658. 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), 215-225. 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), 327-340. doi: 10.1002/mana.19821060128.
[13]	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.
[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), 59-64. 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), 189-206. doi: 10.1002/cpa.3160370203.
[16]	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.

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), 1151-1156. 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), 439-474. 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), 225-236.
[4]	R. Brooks and K. Schmitt, The contraction mapping principle and some applications, Electron. J. Diff. Eqns. Monograph, 90 (2009), 90 pp.
[5]	J. Caicedo and A. Castro, A semilinear wave equation with derivative of nonlinearity containing multiple eigenvalues of infinite multiplicity, Contemp. Math., 208 (1997), 111-132. 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), 653-658. 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), 207-220. doi: 10.1007/s00032-011-0154-7.
[8]	A. Castro, Semilinear equations with discrete spectrum, Contemporary Mathematics, 357 (2004), 1-16. 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, Series A, 28 (2010), 649-658. 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), 215-225. 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), 327-340. doi: 10.1002/mana.19821060128.
[13]	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.
[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), 59-64. 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), 189-206. doi: 10.1002/cpa.3160370203.
[16]	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.

[1]	José F. Caicedo, Alfonso Castro. A semilinear wave equation with smooth data and no resonance having no continuous solution. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 653-658. doi: 10.3934/dcds.2009.24.653
[2]	Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108
[3]	Tong Li, Hailiang Liu. Critical thresholds in a relaxation system with resonance of characteristic speeds. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 511-521. doi: 10.3934/dcds.2009.24.511
[4]	Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055
[5]	Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations and Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039
[6]	Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559
[7]	Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351
[8]	Martin Michálek, Dalibor Pražák, Jakub Slavík. Semilinear damped wave equation in locally uniform spaces. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1673-1695. doi: 10.3934/cpaa.2017080
[9]	Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084
[10]	Zhiguo Wang, Yiqian Wang, Daxiong Piao. A new method for the boundedness of semilinear Duffing equations at resonance. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3961-3991. doi: 10.3934/dcds.2016.36.3961
[11]	Jiabao Su, Zhaoli Liu. A bounded resonance problem for semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 431-445. doi: 10.3934/dcds.2007.19.431
[12]	Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901
[13]	Alfonso Castro, Benjamin Preskill. Existence of solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 649-658. doi: 10.3934/dcds.2010.28.649
[14]	Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307
[15]	Andrzej Nowakowski. Variational approach to stability of semilinear wave equation with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2603-2616. doi: 10.3934/dcdsb.2014.19.2603
[16]	José Caicedo, Alfonso Castro, Arturo Sanjuán. Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1857-1865. doi: 10.3934/dcds.2017078
[17]	Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388
[18]	Nikos I. Kavallaris, Andrew A. Lacey, Christos V. Nikolopoulos, Dimitrios E. Tzanetis. On the quenching behaviour of a semilinear wave equation modelling MEMS technology. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1009-1037. doi: 10.3934/dcds.2015.35.1009
[19]	Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1857-1871. doi: 10.3934/cpaa.2021043
[20]	Elena Braverman, Karel Hasik, Anatoli F. Ivanov, Sergei I. Trofimchuk. A cyclic system with delay and its characteristic equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (1) : 1-29. doi: 10.3934/dcdss.2020001

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
José Caicedo Alfonso Castro Rodrigo Duque Arturo Sanjuán
on Google Scholar
José Caicedo Alfonso Castro Rodrigo Duque Arturo Sanjuán

[Back to Top]