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, 40 (2001), 1159. doi: 10.1016/j.nonrwa.2010.08.025.

[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. doi: 10.1016/S0362-546X(99)00218-7.

[3]

S. A. Beilin, On a Mixed nonlocal problem for a wave equation,, Electronic J. Differential Equations, 2006 (2006), 1. doi: http://ejde.math.txstate.edu/Volumes/2006/103/beilin.pdf.

[4]

A. Benaissa and S. A. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation,, Nonlinear Differ. Equ. Appl., 12 (2005), 391. doi: 10.1007/s00030-005-0008-5.

[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. doi: 10.1155/S016117129800074X.

[6]

Lakshmikantham V and Leela S, "Differential and Integral Inequalities," Vol.1,, Academic Press, (1969). doi: 10.1016/S0076-5392(08)62290-0.

[7]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires,, Dunod, (1969).

[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. doi: 10.1016/0362-546X(92)90097-X.

[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. doi: 10.1016/S0362-546X(97)87360-9.

[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., 2005 (2005), 337. doi: 10.1155/BVP.2005.337.

[11]

N. T. Long and L. X. Truong, Existence and asymptotic expansion for a viscoelastic problem with a mixed nonhomogeneous condition,, NA, 67 (2007), 842. doi: 10.1016/j.na.2006.06.044.

[12]

S. A. Messaoudi, Decay of the solution energy for a nonlinearly damped wave equation,, Arab. J. for Science and Engineering, 26 (2001), 63.

[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. doi: 10.1023/A:1012934900316.

[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. doi: 10.1023/A:1013151525487.

[15]

G. P. Menzala, On global classical solutions of a nonlinear wave equation,, Appl. Anal., 10 (1980), 179. doi: 10.1080/00036818008839300.

[16]

S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation,, Math. Nachr., 260 (2003), 58. doi: 10.1002/mana.200310104.

[17]

M. Nakao, Decay of solutions of some nonlinear evolution equations,, J. Math. Anal. Appl., 60 (1977), 542. doi: 10.1016/0022-247X(77)90040-3.

[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. doi: 10.1007\%2FBF02571342.

[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. doi: http://www.math.kobe-u.ac.jp/\symbol{126}fe/xml/mr1374429.xml.

[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, 70 (2009), 3943. doi: 10.1016/j.na.2008.08.004.

[21]

K. Ono, On the global existence and decay of solutions for semilinear telegraph equations,, Int. J. Applied Math., 2 (2000), 1121. doi: 10.1002/(SICI)1099-1476(200004)23:6.

[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. doi: 10.1002/(SICI)1099-1476(20000110)23:1.

[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. doi: http://www.emis.de/journals/EJDE/Volumes/2001/73/santos.pdf.

[24]

M. L. Santos, Decay rates for solutions of a system of wave equations with memory,, Electronic J. Differential Equations, 2002 (2002), 1. doi: 10.1155/S1085337502204133.

[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. doi: 10.1016/S0362-546X(03)00121-4.

[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, 11 (2010), 1289. doi: 10.1016/j.nonrwa.2009.02.018.

show all references

References:
[1]

R. G. C. Almeida and M. L. Santos, Lack of exponential decay of a coupled system of wave equations with memory,, NA, 40 (2001), 1159. doi: 10.1016/j.nonrwa.2010.08.025.

[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. doi: 10.1016/S0362-546X(99)00218-7.

[3]

S. A. Beilin, On a Mixed nonlocal problem for a wave equation,, Electronic J. Differential Equations, 2006 (2006), 1. doi: http://ejde.math.txstate.edu/Volumes/2006/103/beilin.pdf.

[4]

A. Benaissa and S. A. Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation,, Nonlinear Differ. Equ. Appl., 12 (2005), 391. doi: 10.1007/s00030-005-0008-5.

[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. doi: 10.1155/S016117129800074X.

[6]

Lakshmikantham V and Leela S, "Differential and Integral Inequalities," Vol.1,, Academic Press, (1969). doi: 10.1016/S0076-5392(08)62290-0.

[7]

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires,, Dunod, (1969).

[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. doi: 10.1016/0362-546X(92)90097-X.

[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. doi: 10.1016/S0362-546X(97)87360-9.

[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., 2005 (2005), 337. doi: 10.1155/BVP.2005.337.

[11]

N. T. Long and L. X. Truong, Existence and asymptotic expansion for a viscoelastic problem with a mixed nonhomogeneous condition,, NA, 67 (2007), 842. doi: 10.1016/j.na.2006.06.044.

[12]

S. A. Messaoudi, Decay of the solution energy for a nonlinearly damped wave equation,, Arab. J. for Science and Engineering, 26 (2001), 63.

[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. doi: 10.1023/A:1012934900316.

[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. doi: 10.1023/A:1013151525487.

[15]

G. P. Menzala, On global classical solutions of a nonlinear wave equation,, Appl. Anal., 10 (1980), 179. doi: 10.1080/00036818008839300.

[16]

S. A. Messaoudi, Blow up and global existence in a nonlinear viscoelastic wave equation,, Math. Nachr., 260 (2003), 58. doi: 10.1002/mana.200310104.

[17]

M. Nakao, Decay of solutions of some nonlinear evolution equations,, J. Math. Anal. Appl., 60 (1977), 542. doi: 10.1016/0022-247X(77)90040-3.

[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. doi: 10.1007\%2FBF02571342.

[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. doi: http://www.math.kobe-u.ac.jp/\symbol{126}fe/xml/mr1374429.xml.

[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, 70 (2009), 3943. doi: 10.1016/j.na.2008.08.004.

[21]

K. Ono, On the global existence and decay of solutions for semilinear telegraph equations,, Int. J. Applied Math., 2 (2000), 1121. doi: 10.1002/(SICI)1099-1476(200004)23:6.

[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. doi: 10.1002/(SICI)1099-1476(20000110)23:1.

[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. doi: http://www.emis.de/journals/EJDE/Volumes/2001/73/santos.pdf.

[24]

M. L. Santos, Decay rates for solutions of a system of wave equations with memory,, Electronic J. Differential Equations, 2002 (2002), 1. doi: 10.1155/S1085337502204133.

[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. doi: 10.1016/S0362-546X(03)00121-4.

[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, 11 (2010), 1289. doi: 10.1016/j.nonrwa.2009.02.018.

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