具有记忆项和非局部非线性项的板方程的整体吸引子
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摘要
本文研究具有记忆项和非局部非线性项的板方程.首先利用近似的Faedo-Galerkin方法证得方程在初边值条件下解的适定性定理;其次通过先验估计并结合常用不等式证明该系统存在有界吸收集;最后利用Sobolev紧嵌入和收缩函数的方法证得解半群的渐近紧性,从而得到该系统整体吸引子的存在性.
In this paper, we investigate a plate equation with memory and nonlocal nonlinearities.Firstly, by using Faedo-Galerkin method, we prove the well-posedness of solutions under the initial and boundary value conditions. Secondly, by a priori estimate method and some inequality, the existence of the boundary absorbing set is obtained. Lastly, by using sobolev compact embedding and contraction function,the asymptotic compactness of semigroups is proved. Thus, we get the existence of global attractor of the system.
In this paper, we investigate a plate equation with memory and nonlocal nonlinearities.Firstly, by using Faedo-Galerkin method, we prove the well-posedness of solutions under the initial and boundary value conditions. Secondly, by a priori estimate method and some inequality, the existence of the boundary absorbing set is obtained. Lastly, by using sobolev compact embedding and contraction function,the asymptotic compactness of semigroups is proved. Thus, we get the existence of global attractor of the system.
引文
[1] SILVA M A J D, NARCISO V. Long-time behavior for a plate equation with nonlocal weak damping[J]. Differential Integral Equations, 2014, 27(9-10):931-948.
[2] CAVALCANTI M M, DOMINGOS CAVALCANTI V N, SORIANO J A. Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation[J]. Communications in Contemporary Mathematics, 2004, 5(6):705-731.
[3] YANG L. Uniform attractor for non-autonomous plate equation with a localized damping and critical nonlinearity[J]. Mathematical Analysis and Appliancations, 2008, 338:1243-1254.
[4] MA T F, NARCISO V, PELICER M L. Long-time behavior of a model of extensible beams with nonlinear boundary dissipations[J]. Mathematical Analysis and Applications, 2012, 2(396):694-703.
[5] NARCISO V. Long-time behavior of a nonlinear viscoelastic beam equation with past history[J].Mathematical Methods in the Applied Sciences, 2015, 4(38):775-784.
[6] LANGE H, MENZALA G P. Rates of decay of a nonlocal beam equation[J]. Differenial Integral Equations, 1997, 10(6):1075-1092.
[7] BARBOSA A R A, MA T F. Long-time dynamics of an extensible plate equation with thermal memory[J]. Math. Anal. Appl., 2014, 1(416):143-165.
[8] NARCISO V. Attractors for a plate equation with nonlocal nonlinearities[J]. Mathematical Methods in the Applied Science, 2017, 1(40):937-954.
[9] NAKAO M. Global attractors for wave equations with nonlinear dissipative terms[J]. Differential Equations, 2006, 1(227):204-229.
[10] CHUESHOV I, LASIECKA I. Long-time behavior of second order evolution equations with nonlinear damping[J]. Memoirs of the American Mathematical Society, 2008,(195):67-99.
[11] CHUESHOV I, LASIECKA I. Von Karman Evolution Equation:Well-Posedness and Long-Time Dynamics[M]. Springer Monographs in Mathematics. New York:Springer, 2010.
[2] CAVALCANTI M M, DOMINGOS CAVALCANTI V N, SORIANO J A. Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation[J]. Communications in Contemporary Mathematics, 2004, 5(6):705-731.
[3] YANG L. Uniform attractor for non-autonomous plate equation with a localized damping and critical nonlinearity[J]. Mathematical Analysis and Appliancations, 2008, 338:1243-1254.
[4] MA T F, NARCISO V, PELICER M L. Long-time behavior of a model of extensible beams with nonlinear boundary dissipations[J]. Mathematical Analysis and Applications, 2012, 2(396):694-703.
[5] NARCISO V. Long-time behavior of a nonlinear viscoelastic beam equation with past history[J].Mathematical Methods in the Applied Sciences, 2015, 4(38):775-784.
[6] LANGE H, MENZALA G P. Rates of decay of a nonlocal beam equation[J]. Differenial Integral Equations, 1997, 10(6):1075-1092.
[7] BARBOSA A R A, MA T F. Long-time dynamics of an extensible plate equation with thermal memory[J]. Math. Anal. Appl., 2014, 1(416):143-165.
[8] NARCISO V. Attractors for a plate equation with nonlocal nonlinearities[J]. Mathematical Methods in the Applied Science, 2017, 1(40):937-954.
[9] NAKAO M. Global attractors for wave equations with nonlinear dissipative terms[J]. Differential Equations, 2006, 1(227):204-229.
[10] CHUESHOV I, LASIECKA I. Long-time behavior of second order evolution equations with nonlinear damping[J]. Memoirs of the American Mathematical Society, 2008,(195):67-99.
[11] CHUESHOV I, LASIECKA I. Von Karman Evolution Equation:Well-Posedness and Long-Time Dynamics[M]. Springer Monographs in Mathematics. New York:Springer, 2010.