具衰退记忆的四阶拟抛物方程的长时间行为
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摘要
该文主要研究带衰退记忆和临界非线性的四阶拟抛物方程的长时间行为.在过去历史框架下,利用解算子半群的分解技巧和紧性转移定理证明了对应的动力系统的整体吸引子存在性.
This paper is devoted to study the long-time dynamical behavior of fourth order pseudo-parabolic equations with memory when nonlinearity is critical. By using the decompose techniques of solution semigroup and compactness transitivity theorem, we show the existence of global attractors within the past history framework.
This paper is devoted to study the long-time dynamical behavior of fourth order pseudo-parabolic equations with memory when nonlinearity is critical. By using the decompose techniques of solution semigroup and compactness transitivity theorem, we show the existence of global attractors within the past history framework.
引文
[1] Borini S, Pata V. Uniform attractors for a strongly damped wave equations with linear memory. Asymptot Anal, 1999, 20(3/4):263-277
[2] Pata V, Zucchi A. Attractors for a damped hyperbolic equation with linear memory. Adv Math Sci Appl,2001, 11(2):505-529
[3] Gatti C, Miranville A, Pata V, et al. Attractors for semilinear equations of viscoelasticity with very low dissipation. R Mountain J Math, 2008, 38(4):1117-1138
[4]Al'shin A B, Korpusov M O, Siveshnikov A G. Blow up in Nonlinear Sobolev Type Equations. Berlin:Walter De Gruyter, 2011
[5] Kabanin V A. Solution of mixed problem for fourth order equation. Differ Uravn, 1972, 8(1):54-61
[6] Bakiyevich N I, Shadrin G A. Cauchy problem for an equation in filtration theory. Sb Trudov Mosgospedinstituta, 1978, 7:47-63
[7] Zhao H, Xuan B. Existence and convergence of solutions for the generalized BBM-Burgers equations with dissipative term. Nonlinear Anal, 1997, 28(11):1835-1849
[8] Khudaverdiyev K I, Farhadova G M. On global existence for generalized solution of one-dimensional nonselfadjoint mixed problem for a class of fourth order semilinear pseudo-parabolic equations. Proc Inst Math Mech Natl Acad Sci Azerb, 2009, 31:119-134
[9] Qu C Y, Cao Y. Global existence of solutions for a viscous Cahn-Hilliard equation with gradient dependent potentials and sources. Proc Math Sci, 2013, 123(4):499-513
[10] Song L Y, Zhang Y D, Ma T. Global attractor of a modified Swift-Hohenberg equation in Hk spaces.Nonlinear Anal, 2010, 72(1):183-191
[11] Xu R Z, Chen T L, Liu C M, Ding Y H. Global well-posedness and Global attractor of fourth order semilinear parabolic equation. Math Methods Appl Sci, 2015, 38(8):1515-1529
[12] Wang X, Zhong C K. Attractors for the non-autonomous nonclassical diffusion equations with fading memory. Nonlinear Anal, 2009, 71(11):5733-5746
[13] Wang X, Ju W C, Zhong C K. Strong attractors for non-autonomous nonclassical diffusion equations with fading memory. Chinese Ann Math Ser A, 2013, 34(6):671-688
[14] Wang X, Yang L, Zhong C K. Attractors for the nonclassical diffusion equations with fading memory. J Math Anal Appl, 2010, 362(2):327-337
[15] Chen T, Chen Z, Tang Y B. Global attractors for a non-classical reaction-diffusion equation with memory.Dyn Contin Discrete Impuls Syst Ser A, 2011, 18(5):569-578
[16] Chen T, Chen Z, Tang Y B. Finite dimensionality of global attractors for a non-classical reaction-diffusion equation with memory. Appl Math Lett, 2012, 25(3):357-362
[17] Conti M, Marchini E M, Pata V. Nonclassical diffusion with memory. Math Methods Appl Sci, 2015,38(5):948-958
[18] Conti M, Marchini E M. A remark on nonclassical diffusion equations with memory. Appl Math Optim,2016, 73(1):1-21
[19] Di H F, Shang Y D, Zheng X X. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete Contin Dyn Syst B, 2016,21(3):781-801
[20] Defamos C M. Asymptotic stability in viscoelasticity. Arch Rational Mech Anal, 1970, 37:297-308
[21] Arrieta J, Carvalho A N, Hale J K. A damped hyperbolic equation with critical exponent. Comm Partial Differential Equations, 1992, 17(5/6):841-866
[22] Pata V, Squassina M. On the strongly damped wave equation. Comm Math Phys, 2005, 253(3):511-533
[23] Babin A V, Vishik M I. Attractors of Evolution Equations. Amsterdam:North-Holland, 1992
[24] Temam R. Infinite Dimensional Dynamical Systems in Mechanics and Physics. 2nd ed. New York:Springer-Verlag, 1997
[25] Sell G R,You Y C. Dynamics of Evolutionary Equations. New York:Springer-Verlag,2002
[2] Pata V, Zucchi A. Attractors for a damped hyperbolic equation with linear memory. Adv Math Sci Appl,2001, 11(2):505-529
[3] Gatti C, Miranville A, Pata V, et al. Attractors for semilinear equations of viscoelasticity with very low dissipation. R Mountain J Math, 2008, 38(4):1117-1138
[4]Al'shin A B, Korpusov M O, Siveshnikov A G. Blow up in Nonlinear Sobolev Type Equations. Berlin:Walter De Gruyter, 2011
[5] Kabanin V A. Solution of mixed problem for fourth order equation. Differ Uravn, 1972, 8(1):54-61
[6] Bakiyevich N I, Shadrin G A. Cauchy problem for an equation in filtration theory. Sb Trudov Mosgospedinstituta, 1978, 7:47-63
[7] Zhao H, Xuan B. Existence and convergence of solutions for the generalized BBM-Burgers equations with dissipative term. Nonlinear Anal, 1997, 28(11):1835-1849
[8] Khudaverdiyev K I, Farhadova G M. On global existence for generalized solution of one-dimensional nonselfadjoint mixed problem for a class of fourth order semilinear pseudo-parabolic equations. Proc Inst Math Mech Natl Acad Sci Azerb, 2009, 31:119-134
[9] Qu C Y, Cao Y. Global existence of solutions for a viscous Cahn-Hilliard equation with gradient dependent potentials and sources. Proc Math Sci, 2013, 123(4):499-513
[10] Song L Y, Zhang Y D, Ma T. Global attractor of a modified Swift-Hohenberg equation in Hk spaces.Nonlinear Anal, 2010, 72(1):183-191
[11] Xu R Z, Chen T L, Liu C M, Ding Y H. Global well-posedness and Global attractor of fourth order semilinear parabolic equation. Math Methods Appl Sci, 2015, 38(8):1515-1529
[12] Wang X, Zhong C K. Attractors for the non-autonomous nonclassical diffusion equations with fading memory. Nonlinear Anal, 2009, 71(11):5733-5746
[13] Wang X, Ju W C, Zhong C K. Strong attractors for non-autonomous nonclassical diffusion equations with fading memory. Chinese Ann Math Ser A, 2013, 34(6):671-688
[14] Wang X, Yang L, Zhong C K. Attractors for the nonclassical diffusion equations with fading memory. J Math Anal Appl, 2010, 362(2):327-337
[15] Chen T, Chen Z, Tang Y B. Global attractors for a non-classical reaction-diffusion equation with memory.Dyn Contin Discrete Impuls Syst Ser A, 2011, 18(5):569-578
[16] Chen T, Chen Z, Tang Y B. Finite dimensionality of global attractors for a non-classical reaction-diffusion equation with memory. Appl Math Lett, 2012, 25(3):357-362
[17] Conti M, Marchini E M, Pata V. Nonclassical diffusion with memory. Math Methods Appl Sci, 2015,38(5):948-958
[18] Conti M, Marchini E M. A remark on nonclassical diffusion equations with memory. Appl Math Optim,2016, 73(1):1-21
[19] Di H F, Shang Y D, Zheng X X. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete Contin Dyn Syst B, 2016,21(3):781-801
[20] Defamos C M. Asymptotic stability in viscoelasticity. Arch Rational Mech Anal, 1970, 37:297-308
[21] Arrieta J, Carvalho A N, Hale J K. A damped hyperbolic equation with critical exponent. Comm Partial Differential Equations, 1992, 17(5/6):841-866
[22] Pata V, Squassina M. On the strongly damped wave equation. Comm Math Phys, 2005, 253(3):511-533
[23] Babin A V, Vishik M I. Attractors of Evolution Equations. Amsterdam:North-Holland, 1992
[24] Temam R. Infinite Dimensional Dynamical Systems in Mechanics and Physics. 2nd ed. New York:Springer-Verlag, 1997
[25] Sell G R,You Y C. Dynamics of Evolutionary Equations. New York:Springer-Verlag,2002