# American Institute of Mathematical Sciences

September  2013, 12(5): 2001-2029. doi: 10.3934/cpaa.2013.12.2001

## Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions

 1 Nha Trang Educational College, 01 Nguyen Chanh Str., Nha Trang City, Vietnam 2 Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University Ho Chi Minh City, 227 Nguyen Van Cu Str., Dist.5, Ho Chi Minh City, Vietnam

Received  January 2012 Revised  November 2012 Published  January 2013

The paper is devoted to the study of a nonlinear wave equation with nonlocal boundary conditions of integral forms. First, we establish two local existence theorems by using Faedo-Galerkin method. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions.
Citation: Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001
##### References:
 [1] R. G. C. Almeida and M. L. Santos, Lack of exponential decay of a coupled system of wave equations with memory, NA, Series B: Real World Applications, 40 (2001), 1159-1188. doi: 10.1016/j.nonrwa.2010.08.025.  Google Scholar [2] M. Bergounioux, N. T. Long and A. P. N. Dinh, Mathematical model for a shock problem involving a linear viscoelastic bar, Nonlinear Anal., 43 (2001), 547-561. doi: 10.1016/S0362-546X(99)00218-7.  Google Scholar [3] S. A. Beilin, On a Mixed nonlocal problem for a wave equation, Electronic J. Differential Equations, 2006 (2006), 1-10. doi: http://ejde.math.txstate.edu/Volumes/2006/103/beilin.pdf.  Google Scholar [4] A. Benaissa and S. A. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation, Nonlinear Differ. Equ. Appl., 12 (2005), 391-399. doi: 10.1007/s00030-005-0008-5.  Google Scholar [5] H. R. Clark, Global classical solutions to the Cauchy problem for a nonlinear wave equation, Internat. J. Math. and Math. Sci., 21 (1998), 533-548. doi: 10.1155/S016117129800074X.  Google Scholar [6] Lakshmikantham V and Leela S, "Differential and Integral Inequalities," Vol.1, Academic Press, NewYork, 1969. doi: 10.1016/S0076-5392(08)62290-0.  Google Scholar [7] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [8] N. T. Long and A. P. N. Dinh, On the quasilinear wave equation: $u_{t t}-\Delta u+f(u, u_t)=0$ associated with a mixed nonhomogeneous condition, Nonlinear Anal., 19 (1992), 613-623. doi: 10.1016/0362-546X(92)90097-X.  Google Scholar [9] N. T. Long and T. N. Diem, On the nonlinear wave equation $u_{t t}-u_{x x}=f(x,t,u,u_x,u_t)$ associated with the mixed homogeneous conditions, Nonlinear Anal., 29 (1997), 1217-1230. doi: 10.1016/S0362-546X(97)87360-9.  Google Scholar [10] N. T. Long, A. P. N. Dinh and T. N. Diem, On a shock problem involving a nonlinear viscoelastic bar, J. Boundary Value Prob., Hindawi Publishing Corporation, 2005 (2005), 337-358. doi: 10.1155/BVP.2005.337.  Google Scholar [11] N. T. Long and L. X. Truong, Existence and asymptotic expansion for a viscoelastic problem with a mixed nonhomogeneous condition, NA, TMA, Series A: Theory and Methods, 67 (2007), 842-864. doi: 10.1016/j.na.2006.06.044.  Google Scholar [12] S. A. Messaoudi, Decay of the solution energy for a nonlinearly damped wave equation, Arab. J. for Science and Engineering, 26 (2001), 63-68.  Google Scholar [13] L. A. Medeiros, J. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, Part one, J. Comput. Anal. Appl., 4 (2002), 91-127. doi: 10.1023/A:1012934900316.  Google Scholar [14] L. A. Medeiros, J. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, Part two, J. Comput. Anal. Appl., 4 (2002), 211-263. doi: 10.1023/A:1013151525487.  Google Scholar [15] G. P. Menzala, On global classical solutions of a nonlinear wave equation, Appl. Anal., 10 (1980), 179-195. doi: 10.1080/00036818008839300.  Google Scholar [16] S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58-66. doi: 10.1002/mana.200310104.  Google Scholar [17] M. Nakao, Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl., 60 (1977), 542-549. doi: 10.1016/0022-247X(77)90040-3.  Google Scholar [18] Nakao and Mitsuhiro, Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equations, Math. Z., 206 (1991), 265-276. doi: 10.1007\%2FBF02571342.  Google Scholar [19] M. Nakao and K. Ono, Global existence to the Cauchy problem of the semilinear wave equation with a nonlinear dissipation, Funkcial. Ekvac., 38 (1995), 417-431. doi: http://www.math.kobe-u.ac.jp/\symbol{126}fe/xml/mr1374429.xml.  Google Scholar [20] L. T. P. Ngoc, L. N. K. Hang and N. T. Long, On a nonlinear wave equation associated with the boundary conditions involving convolution, NA, TMA, Series A: Theory and Methods, 70 (2009), 3943-3965. doi: 10.1016/j.na.2008.08.004.  Google Scholar [21] K. Ono, On the global existence and decay of solutions for semilinear telegraph equations, Int. J. Applied Math., 2 (2000), 1121-1136. doi: 10.1002/(SICI)1099-1476(200004)23:6.  Google Scholar [22] J. E. Munoz-Rivera and D. Andrade, Exponential decay of non-linear wave equation with a viscoelastic boundary condition, Math. Methods Appl. Sci., 23 (2000), 41-61. doi: 10.1002/(SICI)1099-1476(20000110)23:1.  Google Scholar [23] M. L. Santos, Asymptotic behavior of solutions to wave equations with a memory condition at the boundary, Electronic J. Differential Equations, 73 (2001), 1-11. doi: http://www.emis.de/journals/EJDE/Volumes/2001/73/santos.pdf.  Google Scholar [24] M. L. Santos, Decay rates for solutions of a system of wave equations with memory, Electronic J. Differential Equations, 2002 (2002), 1-17. doi: 10.1155/S1085337502204133.  Google Scholar [25] M. L. Santos, J. Ferreira, D. C. Pereira and C. A. Raposo, Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary, Nonlinear Anal., 54 (2003), 959-976. doi: 10.1016/S0362-546X(03)00121-4.  Google Scholar [26] L. X. Truong, L. T. P. Ngoc, A. P. N. Dinh and N. T. Long, The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions, NA, Series B: Real World Applications, 11 (2010), 1289-1303. doi: 10.1016/j.nonrwa.2009.02.018.  Google Scholar

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##### References:
 [1] R. G. C. Almeida and M. L. Santos, Lack of exponential decay of a coupled system of wave equations with memory, NA, Series B: Real World Applications, 40 (2001), 1159-1188. doi: 10.1016/j.nonrwa.2010.08.025.  Google Scholar [2] M. Bergounioux, N. T. Long and A. P. N. Dinh, Mathematical model for a shock problem involving a linear viscoelastic bar, Nonlinear Anal., 43 (2001), 547-561. doi: 10.1016/S0362-546X(99)00218-7.  Google Scholar [3] S. A. Beilin, On a Mixed nonlocal problem for a wave equation, Electronic J. Differential Equations, 2006 (2006), 1-10. doi: http://ejde.math.txstate.edu/Volumes/2006/103/beilin.pdf.  Google Scholar [4] A. Benaissa and S. A. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation, Nonlinear Differ. Equ. Appl., 12 (2005), 391-399. doi: 10.1007/s00030-005-0008-5.  Google Scholar [5] H. R. Clark, Global classical solutions to the Cauchy problem for a nonlinear wave equation, Internat. J. Math. and Math. Sci., 21 (1998), 533-548. doi: 10.1155/S016117129800074X.  Google Scholar [6] Lakshmikantham V and Leela S, "Differential and Integral Inequalities," Vol.1, Academic Press, NewYork, 1969. doi: 10.1016/S0076-5392(08)62290-0.  Google Scholar [7] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [8] N. T. Long and A. P. N. Dinh, On the quasilinear wave equation: $u_{t t}-\Delta u+f(u, u_t)=0$ associated with a mixed nonhomogeneous condition, Nonlinear Anal., 19 (1992), 613-623. doi: 10.1016/0362-546X(92)90097-X.  Google Scholar [9] N. T. Long and T. N. Diem, On the nonlinear wave equation $u_{t t}-u_{x x}=f(x,t,u,u_x,u_t)$ associated with the mixed homogeneous conditions, Nonlinear Anal., 29 (1997), 1217-1230. doi: 10.1016/S0362-546X(97)87360-9.  Google Scholar [10] N. T. Long, A. P. N. Dinh and T. N. Diem, On a shock problem involving a nonlinear viscoelastic bar, J. Boundary Value Prob., Hindawi Publishing Corporation, 2005 (2005), 337-358. doi: 10.1155/BVP.2005.337.  Google Scholar [11] N. T. Long and L. X. Truong, Existence and asymptotic expansion for a viscoelastic problem with a mixed nonhomogeneous condition, NA, TMA, Series A: Theory and Methods, 67 (2007), 842-864. doi: 10.1016/j.na.2006.06.044.  Google Scholar [12] S. A. Messaoudi, Decay of the solution energy for a nonlinearly damped wave equation, Arab. J. for Science and Engineering, 26 (2001), 63-68.  Google Scholar [13] L. A. Medeiros, J. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, Part one, J. Comput. Anal. Appl., 4 (2002), 91-127. doi: 10.1023/A:1012934900316.  Google Scholar [14] L. A. Medeiros, J. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, Part two, J. Comput. Anal. Appl., 4 (2002), 211-263. doi: 10.1023/A:1013151525487.  Google Scholar [15] G. P. Menzala, On global classical solutions of a nonlinear wave equation, Appl. Anal., 10 (1980), 179-195. doi: 10.1080/00036818008839300.  Google Scholar [16] S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation, Math. Nachr., 260 (2003), 58-66. doi: 10.1002/mana.200310104.  Google Scholar [17] M. Nakao, Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl., 60 (1977), 542-549. doi: 10.1016/0022-247X(77)90040-3.  Google Scholar [18] Nakao and Mitsuhiro, Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipative wave equations, Math. Z., 206 (1991), 265-276. doi: 10.1007\%2FBF02571342.  Google Scholar [19] M. Nakao and K. Ono, Global existence to the Cauchy problem of the semilinear wave equation with a nonlinear dissipation, Funkcial. Ekvac., 38 (1995), 417-431. doi: http://www.math.kobe-u.ac.jp/\symbol{126}fe/xml/mr1374429.xml.  Google Scholar [20] L. T. P. Ngoc, L. N. K. Hang and N. T. Long, On a nonlinear wave equation associated with the boundary conditions involving convolution, NA, TMA, Series A: Theory and Methods, 70 (2009), 3943-3965. doi: 10.1016/j.na.2008.08.004.  Google Scholar [21] K. Ono, On the global existence and decay of solutions for semilinear telegraph equations, Int. J. Applied Math., 2 (2000), 1121-1136. doi: 10.1002/(SICI)1099-1476(200004)23:6.  Google Scholar [22] J. E. Munoz-Rivera and D. Andrade, Exponential decay of non-linear wave equation with a viscoelastic boundary condition, Math. Methods Appl. Sci., 23 (2000), 41-61. doi: 10.1002/(SICI)1099-1476(20000110)23:1.  Google Scholar [23] M. L. Santos, Asymptotic behavior of solutions to wave equations with a memory condition at the boundary, Electronic J. Differential Equations, 73 (2001), 1-11. doi: http://www.emis.de/journals/EJDE/Volumes/2001/73/santos.pdf.  Google Scholar [24] M. L. Santos, Decay rates for solutions of a system of wave equations with memory, Electronic J. Differential Equations, 2002 (2002), 1-17. doi: 10.1155/S1085337502204133.  Google Scholar [25] M. L. Santos, J. Ferreira, D. C. Pereira and C. A. Raposo, Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary, Nonlinear Anal., 54 (2003), 959-976. doi: 10.1016/S0362-546X(03)00121-4.  Google Scholar [26] L. X. Truong, L. T. P. Ngoc, A. P. N. Dinh and N. T. Long, The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions, NA, Series B: Real World Applications, 11 (2010), 1289-1303. doi: 10.1016/j.nonrwa.2009.02.018.  Google Scholar
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