×

Existence and multiplicity results of positive doubly periodic solutions for nonlinear telegraph system. (English) Zbl 1159.35011

In this paper, the authors study the following nonlinear telegraph system \[ \left\{ \begin{matrix} u_{tt}-u_{xx}+c_1u_t+a_{11}\left( t,x\right) u+a_{12}\left( t,x\right) v=\lambda b_1\left( t,x\right) f\left( u,v\right) , \\ v_{tt}-v_{xx}+c_2v_t+a_{21}\left( t,x\right) u+a_{22}\left( t,x\right) v=\mu b_2\left( t,x\right) g\left( u,v\right) , \end{matrix} \right. \] with doubly periodic boundary conditions \[ \left\{ \begin{matrix} u\left( t+2\pi ,x\right) =u\left( t,x+2\pi \right) =u\left( t,x\right) ,\left( t,x\right) \in \mathbb{R}^2, \\ v\left( t+2\pi ,x\right) =v\left( t,x+2\pi \right) =v\left( t,x\right) ,\left( t,x\right) \in \mathbb{R}^2, \end{matrix} \right. \] where \(c_i>0\) is constant, \(a_{11},a_{22},b_1,b_2\in C\left( \mathbb{R}^2, \mathbb{R}_{+}\right) ,a_{12},a_{21}\in C\left( \mathbb{R}^2,\mathbb{R} _{-}\right) ,f,g\in C\left( \mathbb{R}_{+}\times \mathbb{R}_{+},\mathbb{R} _{+}\right) ,\) and \(a_{ij},b_i,f,g\) are \(2\pi -\)periodic in \(t\) and \(x.\)
Using upper and lower solution method and the fixed point index theory, the authors build first the maximum principle for telegraph system and then give some sufficient conditions for the existence of one solutions, two solutions or no solution for \(\left( 1\right) +\left( 2\right) \) when \(\lambda ,\mu \) belong to a suitable domains.

MSC:

35B50 Maximum principles in context of PDEs
35L70 Second-order nonlinear hyperbolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] An, Y., Periodic solutions of telegraph-wave coupled system at nonresonance, Nonlinear Anal., 46, 525-533 (2001) · Zbl 0993.35067
[2] Berkovits, J.; Mustonuen, V., On nonresonance for system of semilinear wave equations, Nonlinear Anal., 29, 6, 627-638 (1997) · Zbl 0887.35013
[3] Correa, F. J.S. A.; Souto, M. A.S., On maximum principle for cooperative elliptic systems via fixed point index, Nonlinear Anal., 26, 997-1006 (1996) · Zbl 0846.35020
[4] Dunninger, D. R.; Wang, H., Multiplicity of positive radial solutions for an elliptic systems on an annulus domain, Nonlinear Anal., 42, 803-811 (2000) · Zbl 0959.35051
[5] Dalmasso, R., Existence and uniqueness of positive solutions of semilinear elliptic systems, Nonlinear Anal., 39, 559-568 (2000) · Zbl 0940.35091
[6] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045
[7] Hai, D. D., Uniqueness of positive solutions for a class of semilinear elliptic systems, Nonlinear Anal., 52, 595-603 (2003) · Zbl 1022.35013
[8] Kim, W. S., Doubly-periodic boundary value problem for nonlinear dissipative hyperbolic equations, J. Math. Anal. Appl., 145, 1-6 (1990)
[9] Kim, W. S., Multiple doubly periodic solutions of semilinear dissipative hyperbolic equations, J. Math. Anal. Appl., 197, 735-748 (1996) · Zbl 0870.35069
[10] Lee, Y., Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus, J. Differential Equations, 174, 420-441 (2001) · Zbl 1001.34011
[11] Li, Y., Positive doubly periodic solutions of nonlinear telegraph equations, Nonlinear Anal., 55, 245-254 (2003) · Zbl 1036.35020
[12] Li, Y., Maximum principles and method of upper and lower solutions for time-periodic problems of the telegraph equations, J. Math. Anal. Appl., 327, 997-1009 (2007) · Zbl 1108.35021
[13] Ma, R., Multiple nonnegative solutions of second-order systems of boundary value problems, Nonlinear Anal., 42, 1003-1010 (2000) · Zbl 0973.34014
[14] Mawhin, J.; Ortega, R.; Robles-Perez, A. M., A maximum principle for bounded solutions of the telegraph equations and applications to nonlinear forcings, J. Math. Anal. Appl., 251, 695-709 (2000) · Zbl 0972.35016
[15] Mawhin, J.; Ortega, R.; Robles-Perez, A. M., Maximum principles for bounded solutions of the telegraph equation in space dimensions two or three and applications, J. Differential Equations, 208, 42-63 (2005) · Zbl 1082.35040
[16] Ortega, R.; Robles-Perez, A. M., A maximum principle for periodic solutions of the telegraph equations, J. Math. Anal. Appl., 221, 625-651 (1998) · Zbl 0932.35016
[17] Wang, F.; An, Y., Nonnegative doubly periodic solutions for nonlinear telegraph system, J. Math. Anal. Appl., 338, 91-100 (2008) · Zbl 1145.35010
[18] Yang, X., Existence of positive solutions for \(2m\)-order nonlinear differential systems, Nonlinear Anal., 61, 77-95 (2005) · Zbl 1079.34014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.