This paper is devoted to the study of a wave equation with a boundary condition of many-point type. The existence of weak solutions is proved by using the Galerkin method. Also, the uniqueness and the stability of solutions are established.
References
M. Aassila, M. M. Cavalcanti, and V. N. Domingos Cavalcanti, “Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term,” Calculus of Variations and Partial Differential Equations, vol. 15, no. 2, pp. 155–180, 2002. 1009.35055 MR1930245 10.1007/s005260100096 M. Aassila, M. M. Cavalcanti, and V. N. Domingos Cavalcanti, “Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term,” Calculus of Variations and Partial Differential Equations, vol. 15, no. 2, pp. 155–180, 2002. 1009.35055 MR1930245 10.1007/s005260100096
V. Barbu, I. Lasiecka, and M. A. Rammaha, “On nonlinear wave equations with degenerate damping and source terms,” Transactions of the American Mathematical Society, vol. 357, no. 7, pp. 2571–2611, 2005. 1065.35193 10.1090/S0002-9947-05-03880-8 V. Barbu, I. Lasiecka, and M. A. Rammaha, “On nonlinear wave equations with degenerate damping and source terms,” Transactions of the American Mathematical Society, vol. 357, no. 7, pp. 2571–2611, 2005. 1065.35193 10.1090/S0002-9947-05-03880-8
L. Bociu, P. Radu, and D. Toundykov, “Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping,” Evolution Equations and Control Theory, vol. 2, no. 2, pp. 255–279, 2013. MR3089719 1343.35169 10.3934/eect.2013.2.55 L. Bociu, P. Radu, and D. Toundykov, “Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping,” Evolution Equations and Control Theory, vol. 2, no. 2, pp. 255–279, 2013. MR3089719 1343.35169 10.3934/eect.2013.2.55
A. D. Dang and D. P. Alain, “Mixed problem for some semi-linear wave equation with a nonhomogeneous condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 12, no. 6, pp. 581–592, 1988. 0682.35070 10.1016/0362-546X(88)90016-8 A. D. Dang and D. P. Alain, “Mixed problem for some semi-linear wave equation with a nonhomogeneous condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 12, no. 6, pp. 581–592, 1988. 0682.35070 10.1016/0362-546X(88)90016-8
T. X. Le and G. Giang Vo, “The shock of a rigid body and a nonlinear viscoelastic bar associated with a nonlinear čommentComment on ref. [6?]: Please provide the full name of this journal, if possible.boundary condition,” J. Science Natural Sciences, Uni. Peda. HCM, vol. 8, pp. 70–81, 2006. T. X. Le and G. Giang Vo, “The shock of a rigid body and a nonlinear viscoelastic bar associated with a nonlinear čommentComment on ref. [6?]: Please provide the full name of this journal, if possible.boundary condition,” J. Science Natural Sciences, Uni. Peda. HCM, vol. 8, pp. 70–81, 2006.
L. T. Nguyen and D. T. Bui, “A nonlinear wave equation associated with a nonlinear integral equation involving boundary value,” Electronic Journal of Differential Equations, vol. 2004, pp. 1–21, 2004. MR2108874 1073.35175 L. T. Nguyen and D. T. Bui, “A nonlinear wave equation associated with a nonlinear integral equation involving boundary value,” Electronic Journal of Differential Equations, vol. 2004, pp. 1–21, 2004. MR2108874 1073.35175
L. T. Nguyen, U. V. Le, and T. T. Nguyen, “On a shock problem involving a linear viscoelastic bar,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 2, pp. 198–224, 2005. 1082.35108 10.1016/j.na.2005.05.007 L. T. Nguyen, U. V. Le, and T. T. Nguyen, “On a shock problem involving a linear viscoelastic bar,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 2, pp. 198–224, 2005. 1082.35108 10.1016/j.na.2005.05.007
L. T. Nguyen and G. Giang Vo, “A wave equation associated with mixed nonhomogeneous conditions: global existence and asymptotic expansion of solutions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 7, pp. 1526–1546, 2007. MR2301336 10.1016/j.na.2006.02.007 1114.35117 L. T. Nguyen and G. Giang Vo, “A wave equation associated with mixed nonhomogeneous conditions: global existence and asymptotic expansion of solutions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 7, pp. 1526–1546, 2007. MR2301336 10.1016/j.na.2006.02.007 1114.35117
L. T. Nguyen and G. Giang Vo, “A nonlinear wave equation associated with nonlinear boundary conditions: existence and asymptotic expansion of solutions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 12, pp. 2852–2880, 2007. MR2311642 1115.35086 10.1016/j.na.2006.04.013 L. T. Nguyen and G. Giang Vo, “A nonlinear wave equation associated with nonlinear boundary conditions: existence and asymptotic expansion of solutions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 12, pp. 2852–2880, 2007. MR2311642 1115.35086 10.1016/j.na.2006.04.013
P. Radu, “Weak solutions to the cauchy problem of a semilinear wave equation with damping and source terms,” Advances in Differential Equations, vol. 10, no. 11, pp. 1261–1300, 2005. 1195.35217 euclid.ade/1355867752 P. Radu, “Weak solutions to the cauchy problem of a semilinear wave equation with damping and source terms,” Advances in Differential Equations, vol. 10, no. 11, pp. 1261–1300, 2005. 1195.35217 euclid.ade/1355867752
M. L. Santos, “Asymptotic behavior of solutions to wave equations with a memory condition at the boundary,” Electronic Journal of Differential Equations, vol. 2001, no. 73, pp. 1–11, 2001. MR1872052 0984.35025 M. L. Santos, “Asymptotic behavior of solutions to wave equations with a memory condition at the boundary,” Electronic Journal of Differential Equations, vol. 2001, no. 73, pp. 1–11, 2001. MR1872052 0984.35025
D. Takači and A. Takači, “On the approximate solution of a mathematical model of a viscoelastic bar,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1560–1569, 2007. 1113.74042 10.1016/j.na.2006.07.042 D. Takači and A. Takači, “On the approximate solution of a mathematical model of a viscoelastic bar,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1560–1569, 2007. 1113.74042 10.1016/j.na.2006.07.042
Q. Tiehu, “Global solvability of nonlinear wave equation with a viscoelastic boundary condition,” Chinese Annals of Mathematics Series B, vol. 3, 1993. 0799.35211 Q. Tiehu, “Global solvability of nonlinear wave equation with a viscoelastic boundary condition,” Chinese Annals of Mathematics Series B, vol. 3, 1993. 0799.35211
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires, Dunod, Gauthier-Villars, Paris, France, 1969. 0189.40603 J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires, Dunod, Gauthier-Villars, Paris, France, 1969. 0189.40603
E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955. MR0069338 0064.33002 E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955. MR0069338 0064.33002
J. L. Lions and W. A. Strauss, “Some nonlinear evolution equations,” Bulletin de la Société Mathématique de France, vol. 93, p. 79, 1965. \endinput 0132.10501 10.24033/bsmf.1616 J. L. Lions and W. A. Strauss, “Some nonlinear evolution equations,” Bulletin de la Société Mathématique de France, vol. 93, p. 79, 1965. \endinput 0132.10501 10.24033/bsmf.1616