一类非线性耦合梁方程解的动力学行为
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摘要
为了研究一类非线性耦合梁方程的初边值问题,对该方程解的存在唯一性及整体吸引子的存在性进行讨论;关于该方程解的存在唯一性,利用Galerkin方法和常微分方程理论证明该方程存在局部解,运用Sobolev空间理论并结合对局部解的一致先验估计得到该方程整体解的存在性,利用Gronwall引理得到方程整体解的唯一性;关于该方程整体吸引子的存在性,利用Hölder不等式、Young不等式证明方程有界吸收集的存在性,以克服非线性项及积分项带来的计算困难,并利用验证紧性的方法证明该方程解半群的紧致性。结果表明,该方程在解存在的情况下,存在整体吸引子。
To study the initial-boundary value problem of a class of nonlinear coupled beam equations, the existence and uniqueness of equation solution and the existence of global attractor were discussed. With regard to the existence and uniqueness of equation solution, the existence of local solutions of the equations was proved by using Galerkin method and the theory of ordinary differential equation. The existence of the global solution of the equations was obtained by using Sobolev space theory combined with the prior estimates of local solution, and the uniqueness of the global solution of the equations was gotten by using Gronwall Lemma. On the existence of the global attractor of the equations, the bounded absorbing set of the equations was obtained by using Hölder inequalities and Young inequalities to overcome the computational difficulties caused by the nonlinear term and integral term. The compactness of the solution semigroup of the equation was proved by taking the method for compact property verification. The results show that there exists a global attractor in the existence of the solution.
To study the initial-boundary value problem of a class of nonlinear coupled beam equations, the existence and uniqueness of equation solution and the existence of global attractor were discussed. With regard to the existence and uniqueness of equation solution, the existence of local solutions of the equations was proved by using Galerkin method and the theory of ordinary differential equation. The existence of the global solution of the equations was obtained by using Sobolev space theory combined with the prior estimates of local solution, and the uniqueness of the global solution of the equations was gotten by using Gronwall Lemma. On the existence of the global attractor of the equations, the bounded absorbing set of the equations was obtained by using Hölder inequalities and Young inequalities to overcome the computational difficulties caused by the nonlinear term and integral term. The compactness of the solution semigroup of the equation was proved by taking the method for compact property verification. The results show that there exists a global attractor in the existence of the solution.
引文
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[15] CHUESHOV I, LASIECKA I. Von Karman evolution equations: well-posedness and long-time dynamics[M]//Springer monographs in mathematics. New York: Springer, 2010.
[2] BALL J M. Initial-boundary value problem for an extensible beam[J]. Journal of Mathematical Analysis and Applications, 1973, 42(1): 61-90.
[3] FEIREISL E. Nonzero time periodic solutions to an equation of Petrovsky type with nonlinear boundary conditions: slow oscillations of beams on elastic bearings[J]. Annali Della Scuola Normale Superiore Di Pisa Classe Di Scienze, 1993, 20(1): 133-146.
[4] FECKAN M. Free vibrations of beams on bearings with nonlinear elastic responses[J]. Journal of Differential Equations, 1999, 154(1): 55-72.
[5] MA T F, NARCISO V. Global attractor for a model of extensible beam with nonlinear damping and source terms[J]. Nonlinear Analysis, 2010, 73(10): 3402-3412.
[6] YANG Z J. On an extensible beam equation with nonlinear damping and source terms[J]. Journal of Differential Equations, 2013, 254(9): 3903-3927.
[7] LEE M J. Global attractor for some beam equation with nonlinear source and damping terms[J]. East Asian Mathematical Journal, 2016, 32(3): 377-385.
[8] VANDO N. Long-time behavior of a nonlinear viscoelastic beam equation with past history[J]. Mathematical Methods in the Applied Sciences, 2015, 38(4): 775-784.
[9] GIORGI C, NASO M G, PATA V, et al. Global attractors for the extensible thermoelastic beam system[J]. Journal of Different-ial Equations, 2009, 246(9): 3496-3517.
[10] LI H Y, LIANG H W. The existence of uniform attractors for a non-autonomous extensible beam equation with thermal effects[J]. 河南大学学报(自然科学版), 2015, 45(1): 6-14.
[11] BARBOSA A R A, MA T F. Long-time dynamics of an exten-sible plate equation with thermal memory[J]. Journal of Mathematical Analysis and Applications, 2014, 416(1): 143-165.
[12] WANG D X, ZHANG J W. Long-time dynamics of N-dimensional structure equations with thermal memory[J]. Boundary Value Problems, 2017, 2017: 136.
[13] MA T F, NARCISO V, PELICER M L. Long-time behavior of a model of extensible beams with nonlinear boundary dissipations[J]. Journal of Mathematical Analysis and Applications, 2012, 396(2): 694-703.
[14] CHUESHOV I, LASIECKA I. Long-time behavior of second order evolution equations with nonlinear damping[M]//Memoirs of the American Mathematical Society: Vol 195. Providence: Amer-ican Mathematical Society, 2008.
[15] CHUESHOV I, LASIECKA I. Von Karman evolution equations: well-posedness and long-time dynamics[M]//Springer monographs in mathematics. New York: Springer, 2010.