Global Existence, Uniqueness of Weak Solutions and Determining Functionals for Nonlinear Wave Equations

ABSTRACT

We consider the initial-boundary value problem for a nonlinear wave equation with strong structural damping and nonlinear source terms in IR. We prove the global existence and uniqueness of weak solutions of the problem and then we will study the determining modes on the phase space by using energy methods and the concept of the completeness defect.

Cite this paper

Ü. Dinlemez, "Global Existence, Uniqueness of Weak Solutions and Determining Functionals for Nonlinear Wave Equations,"*Advances in Pure Mathematics*, Vol. 3 No. 5, 2013, pp. 451-457. doi: 10.4236/apm.2013.35064.

Ü. Dinlemez, "Global Existence, Uniqueness of Weak Solutions and Determining Functionals for Nonlinear Wave Equations,"

References

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[2] N. I. Karachalios and N. M. Staurakakis, “Global Existence and Blow up Results for Some Nonlinear Wave Equations on ,” Advances in Differential Equations, Vol. 6, No. 2, 2001, pp. 155-174.

[3] A. O. Celebi and D. Ugurlu, “Determining Functionals for the Strongly Damped Nonlinear Wave Equation,” Journal of Dynamical Systems and Geometric Theories, Vol. 5, No. 2, 2007, pp. 105-116. doi:10.1080/1726037X.2007.10698530

[4] I. D. Chueshov, “Theory of Functionals That Uniquely Determine Long-Time Dynamics of Infinitive Dimensional Dissipative Systems,” Russian Mathematical Surveys, Vol. 53, No. 4, 1998, pp. 1-58. doi:10.1070/RM1998v053n04ABEH000057

[5] I. D. Chueshov and V. K. Kalantarov, “Determining Functionals for Nonlinear Damped Wave Equations,” Matematicheskaya Fizika, Analiz, Geometriya, Kharkovskii Matematicheskii Zhurnal, Vol. 8, No. 2, 2001, pp. 215-227.

[6] B. Cockburn, D. A. Jones and E. S. Titi, “Determining Degrees of Freedom for Nonlinear Dissipative Systems,” Comptes Rendus de I’Académie des Sciences Paris Série I Mathématique, Vol. 321, 1995, pp. 563-568.

[7] J. Duan, E. S. Titi and P. Holmes, “Regularity, Approximation and Asymptotic Dynamics for a Generalized Ginzburg-Landou Equation,” Nonlinearty, Vol. 6, No. 6, 1993, pp. 915-933. doi:10.1088/0951-7715/6/6/005

[8] D. Henry, “Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics,” Springer-Verlag, New York, 1981.

[9] P. Massatt, “Limiting Behavior for Strongly Damped Nonlinear Wave Equations,” Journal of Differential Equations, Vol. 48, No. 3, 1983, pp. 334-349. doi:10.1016/0022-0396(83)90098-0

[10] H. Takeda and S. Yoshikawa, “On the Initial Value Problem of the Semilinear Beam Equation with Weak Damping II: Asymptotic Profiles,” Journal of Differential Equations, Vol. 253, No. 11, 2012, pp. 3061-3080.

[1] F. Chen, B. Guo and P. Wang, “Long Time Behavior of Strongly Damped Nonlinear Wave Equations,” Journal of Differential Equations, Vol. 147, No. 2, 1998, pp. 231-241. doi:10.1006/jdeq.1998.3447

[2] N. I. Karachalios and N. M. Staurakakis, “Global Existence and Blow up Results for Some Nonlinear Wave Equations on ,” Advances in Differential Equations, Vol. 6, No. 2, 2001, pp. 155-174.

[3] A. O. Celebi and D. Ugurlu, “Determining Functionals for the Strongly Damped Nonlinear Wave Equation,” Journal of Dynamical Systems and Geometric Theories, Vol. 5, No. 2, 2007, pp. 105-116. doi:10.1080/1726037X.2007.10698530

[4] I. D. Chueshov, “Theory of Functionals That Uniquely Determine Long-Time Dynamics of Infinitive Dimensional Dissipative Systems,” Russian Mathematical Surveys, Vol. 53, No. 4, 1998, pp. 1-58. doi:10.1070/RM1998v053n04ABEH000057

[5] I. D. Chueshov and V. K. Kalantarov, “Determining Functionals for Nonlinear Damped Wave Equations,” Matematicheskaya Fizika, Analiz, Geometriya, Kharkovskii Matematicheskii Zhurnal, Vol. 8, No. 2, 2001, pp. 215-227.

[6] B. Cockburn, D. A. Jones and E. S. Titi, “Determining Degrees of Freedom for Nonlinear Dissipative Systems,” Comptes Rendus de I’Académie des Sciences Paris Série I Mathématique, Vol. 321, 1995, pp. 563-568.

[7] J. Duan, E. S. Titi and P. Holmes, “Regularity, Approximation and Asymptotic Dynamics for a Generalized Ginzburg-Landou Equation,” Nonlinearty, Vol. 6, No. 6, 1993, pp. 915-933. doi:10.1088/0951-7715/6/6/005

[8] D. Henry, “Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics,” Springer-Verlag, New York, 1981.

[9] P. Massatt, “Limiting Behavior for Strongly Damped Nonlinear Wave Equations,” Journal of Differential Equations, Vol. 48, No. 3, 1983, pp. 334-349. doi:10.1016/0022-0396(83)90098-0

[10] H. Takeda and S. Yoshikawa, “On the Initial Value Problem of the Semilinear Beam Equation with Weak Damping II: Asymptotic Profiles,” Journal of Differential Equations, Vol. 253, No. 11, 2012, pp. 3061-3080.