一类非线性发展方程的整体吸引子
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摘要
由Sobolev嵌入定理及一些先验估计,应用Galerkin方法研究了一类非线性发展方程的解的存在性和唯一性,并通过Gronwall引理和验证条件(C)证明了该方程在空间H_0~1(Ω)×H_0~1(Ω)中整体吸引子的存在性,其中非线性项f(u)满足临界指数增长条件。
In this paper,some priori estimates are obtained by the Sobolev embedding theorem for studying the existence and uniqueness of solutions of a generalized nonlinear equation via Galerkin method.It is proved that under the natural assumptions,these equations possess the global attractors in H_0~1(Ω)×H_0~1(Ω)by using Gronwall lemma and verifying conditions(C),where the nonlinear termf(u)satisfies a critical exponential growth condition.
In this paper,some priori estimates are obtained by the Sobolev embedding theorem for studying the existence and uniqueness of solutions of a generalized nonlinear equation via Galerkin method.It is proved that under the natural assumptions,these equations possess the global attractors in H_0~1(Ω)×H_0~1(Ω)by using Gronwall lemma and verifying conditions(C),where the nonlinear termf(u)satisfies a critical exponential growth condition.
引文
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[12]秦桂香,李妍汝,李青松,等.一类非线性发展方程的全局吸引子[J].湘潭大学自然科学学报,2013,35(1):25-28.QIN G X,LI Y R,LI Q S,et al.Global attractors for a class of nonlinear evolution equations[J].Natural Science Journal of Xiangtan University,2013,35(1):25-28.
[13]XIE Y Q,ZHONG C K.Asymptotic behavior of a class of nonlinear evolution equation[J].Nonlinear Analysis,2009,71(11):5095-5105.
[14]ZHONG C K,YANG M H,SUN C Y.The existence of global attractors for the norm-to-weak continuous semigroup[J].J Differential Equations,2006,223:367-399.
[2]张宏伟,呼青英.一类非线性双曲方程整体弱解的存在性和不存在性[J].工程数学学报,2003,20(3):131-134.ZHANG H W,HU Q Y.Existence and nonexistence of solution for a class nonlinear hyperbolic equation[J].Journal of Engineering Mathematics,2003,20(3):131-134.
[3]XIE Y Q,ZHONG C K.The existence of global attractors for a class nonlinear evolution equation[J].Journal of Mathematical Analysis and Applications,2007,336(1):54-69.
[4]牛丽芳,张建文,张建国.一类带记忆项的非线性弹性杆的全局吸引子[J].数学的实践与认知,2013,43(18):262-268.NIU L F,ZHANG J W,ZHANG J G.Existence of global attractors for a class of nonlinear elastic rod equation with memory type[J].Mathematics in Practice and Theory,2013,43(18):262-268.
[5]牛丽芳,张建文.一类具有黏阻尼的非线性弹性杆方程的初边值问题[J].太原理工大学学报,2014,45(1):128-132.NIU L F,ZHANG J W.Initial-boundary value problems for a kind of nonlinear elastic rods with some suitable damped[J].Journal of Taiyuan University of Technology,2014,45(1):128-132.
[6]XIE Y Q,HE Z F,XI C,et al.Asymptotic smoothing and global attractors for a class of nonlinear evolution equations[J].ISRN Mathematical Analysis,2015,2013(4):13-19.
[7]YANG M,SUN C.Dynamics of strongly damped wave equations in locally uniform spaces:attractors and asymptotic regularity[J].Transactions of the American Mathematical Society,2009,361(2):1069-1101.
[8]SUN C,YANG M.Dynamics of the nonclassical diffusion equations[J].Asymptotical Analysis,2008,59(1/2):51-81.
[9]ZELIKS.Asymptotic regularity of solutions of a nonautonomous damped wave equationwith a critical growth exponent[J].Communications on Pure&Applied Analysis,2004,3(4):921-934.
[10]张宏伟,呼青英.一类非线性发展方程整体弱解的存在性和稳定性[J].数学物理方程,2004,24(3):329-336.ZHANG H W,HU Q Y.Existence of global weak solution and stability of a class nonlinear evolution equation[J].Acta Mathematica Scientia,2004,24(3):329-336.
[11]杨莉,谢永钦.一类非线性发展方程整体弱解的存在性[J].湖南工业大学学报,2010,24(1):36-39.YANG L,XIE Y Q.Existence of global weak solution for a class of nonlinear evolution equation[J].Journal of Hunan University of Technology,2010,24(1):36-39.
[12]秦桂香,李妍汝,李青松,等.一类非线性发展方程的全局吸引子[J].湘潭大学自然科学学报,2013,35(1):25-28.QIN G X,LI Y R,LI Q S,et al.Global attractors for a class of nonlinear evolution equations[J].Natural Science Journal of Xiangtan University,2013,35(1):25-28.
[13]XIE Y Q,ZHONG C K.Asymptotic behavior of a class of nonlinear evolution equation[J].Nonlinear Analysis,2009,71(11):5095-5105.
[14]ZHONG C K,YANG M H,SUN C Y.The existence of global attractors for the norm-to-weak continuous semigroup[J].J Differential Equations,2006,223:367-399.