Kirchhoff型方程解的渐近行为
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摘要
本文研究具强阻尼项的Kirchhoff型方程初边值问题解的长时间行为其中M(s)=1+s~(m/2),m≥1.Ω是R~N中具有光滑边界(?)Ω的有界域.
本文证明了上述问题的局部解和整体解的存在性,唯一的解u∈C([0,+∞);H~2∩H_0~1)∩C~1([0,+∞);H_0~1).我们定义映射S(t):X→X,其中X=H~2∩H_0~1×H_0~1,S(t)(u_0,u_1)=(u,u_t),则S(t)为X上的C_0-半群.然后证明了C_0-半群S(t)在X中存在吸收集,并且用两种方法证明了连续半群S(t)在相空间X中整体吸引子的存在性,最后对抽象条件加以验证并给出具体实例.
The paper studies the long time behavior of the Kirchhoff type equation with strongdampingwhere M(s)=1+s~(m/2),m≥1.Ω(?)R~N is a bounded domain with smooth boundary (?)Ω.
It proves that the corresponding IBVP possesses an unique solution both locally andglobally in time, u∈C([0,+∞);H~2∩H_0~1)∩C~1([0,+∞);H_0~1). We define a mapping S(t) :X→X, where X=H~2∩H_0~1×H_0~1,S(t)(u_0,u_1)=(u,u_t), then S(t) is a Co-semigroupin X. It proves that continuous semigroup S(t) has an absorbing set in X. With twodifferent methords, it proves that the continuous semigroup S(t) has a global attractor inphase space X. At last, we give some examples to show the existence of the nonlinearfunctions g(x,u) and h(u_t).
本文证明了上述问题的局部解和整体解的存在性,唯一的解u∈C([0,+∞);H~2∩H_0~1)∩C~1([0,+∞);H_0~1).我们定义映射S(t):X→X,其中X=H~2∩H_0~1×H_0~1,S(t)(u_0,u_1)=(u,u_t),则S(t)为X上的C_0-半群.然后证明了C_0-半群S(t)在X中存在吸收集,并且用两种方法证明了连续半群S(t)在相空间X中整体吸引子的存在性,最后对抽象条件加以验证并给出具体实例.
The paper studies the long time behavior of the Kirchhoff type equation with strongdampingwhere M(s)=1+s~(m/2),m≥1.Ω(?)R~N is a bounded domain with smooth boundary (?)Ω.
It proves that the corresponding IBVP possesses an unique solution both locally andglobally in time, u∈C([0,+∞);H~2∩H_0~1)∩C~1([0,+∞);H_0~1). We define a mapping S(t) :X→X, where X=H~2∩H_0~1×H_0~1,S(t)(u_0,u_1)=(u,u_t), then S(t) is a Co-semigroupin X. It proves that continuous semigroup S(t) has an absorbing set in X. With twodifferent methords, it proves that the continuous semigroup S(t) has a global attractor inphase space X. At last, we give some examples to show the existence of the nonlinearfunctions g(x,u) and h(u_t).
引文
[1] M.Nakao.Y.Zhijian,Global attractors for some quasi-linear wave equations with a strong dissipation,Advan.M ath.Sci.Appl.17(2007).
[2] Y. Zhijian,Long time behavior of the Kirchhoff type equation with strong damping on R~N.J.Differential.Equation 242(2007).
[3] Fan Xiaoming,Zhou Shengfan,Kernel sections for non-autonomous strongly damped wave equations of non-degnerate Kirchhoff-type.Applied Mathematics and Computation 158(2004)253-266.
[4] Kosuke Ono.On Global Existence,Asymptotic Stability and Blowing Up of Solutions for Some Degenerate Non-linear Wave Equations of Kirchhoff Type with a Strong Dissipation.Math Meth Appl Sd,20(1997),151-177.
[5] J.M.Ghidaglia and A.Marzochi,Longtime behaviour of strongly damped wave equa-tions.Global attractors and their dimension. SIAM.J.Math.Anal.22(1991),879-895.
[6] Hosoya,M.and Yamada,Y.On some nonlinear wave equations :global existence and energy decay of solutions, J.Fac.Sci.Univ.Tokyo Sect.IA Math,38,239-250(1991).
[7] Marina Ghisi.Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term. J.Differential Equations 230(2006)128-139.
[8] J.Limaco,H.R.Clark,L.A.Medeiros.On damped Kirchhoff equation with variable coefficients. J.Math.Anal.Appl.307(2005)641-655.
[9] A.V.Babin.M.I.Vishik,Attractors of Evolution Equations,North-Holland,Amsterdam,1992.
[10] J,K.Hale,Asymptotic Behavior of Dissipative Systems,AMS.Providence,RI,1988.
[11] R.Temam,Infinite Dimensional Dynamical System in Mathematics and Physics,Sringer, New York, 1997.
[12] F.-X.Chen,B.-L.Guo,P.Wang.Long time behavior of strongly damped nonlinear wave equations. J.Differental Equatons 147(1998)231-241.
[13] N.I.Karachalois,N.M.Stavrakakis,Existencc of a global attractor for semilinear dissipa- tive wave equation on R~N, J.Differential Equations,157(1999)183-205.
[14] S.Zhou,Attractors for strongly damped wave equations with critical exponents,Appl.Math. Lett. 16(2003) 1307-1314.
[15] T.Matsuyama,R.Ikehata,On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term,J.Math.Anal.204(1996)729-753.
[16] K.Ono,Global existence,decay,and blow up of solutions for some mildly degenerate nonlinear Kirchhoff stings.J.Differential Equations 137(1997)273-301.
[17] K.Ono,Global existence,asymptotic behavior,and global non-existence of solutons for damped non-linear wave equations of Kirchhoff type in the whole space.Math.Meth. Appl.Sci.23(2000)535-560.
[2] Y. Zhijian,Long time behavior of the Kirchhoff type equation with strong damping on R~N.J.Differential.Equation 242(2007).
[3] Fan Xiaoming,Zhou Shengfan,Kernel sections for non-autonomous strongly damped wave equations of non-degnerate Kirchhoff-type.Applied Mathematics and Computation 158(2004)253-266.
[4] Kosuke Ono.On Global Existence,Asymptotic Stability and Blowing Up of Solutions for Some Degenerate Non-linear Wave Equations of Kirchhoff Type with a Strong Dissipation.Math Meth Appl Sd,20(1997),151-177.
[5] J.M.Ghidaglia and A.Marzochi,Longtime behaviour of strongly damped wave equa-tions.Global attractors and their dimension. SIAM.J.Math.Anal.22(1991),879-895.
[6] Hosoya,M.and Yamada,Y.On some nonlinear wave equations :global existence and energy decay of solutions, J.Fac.Sci.Univ.Tokyo Sect.IA Math,38,239-250(1991).
[7] Marina Ghisi.Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term. J.Differential Equations 230(2006)128-139.
[8] J.Limaco,H.R.Clark,L.A.Medeiros.On damped Kirchhoff equation with variable coefficients. J.Math.Anal.Appl.307(2005)641-655.
[9] A.V.Babin.M.I.Vishik,Attractors of Evolution Equations,North-Holland,Amsterdam,1992.
[10] J,K.Hale,Asymptotic Behavior of Dissipative Systems,AMS.Providence,RI,1988.
[11] R.Temam,Infinite Dimensional Dynamical System in Mathematics and Physics,Sringer, New York, 1997.
[12] F.-X.Chen,B.-L.Guo,P.Wang.Long time behavior of strongly damped nonlinear wave equations. J.Differental Equatons 147(1998)231-241.
[13] N.I.Karachalois,N.M.Stavrakakis,Existencc of a global attractor for semilinear dissipa- tive wave equation on R~N, J.Differential Equations,157(1999)183-205.
[14] S.Zhou,Attractors for strongly damped wave equations with critical exponents,Appl.Math. Lett. 16(2003) 1307-1314.
[15] T.Matsuyama,R.Ikehata,On global solution and energy decay for the wave equation of Kirchhoff type with nonlinear damping term,J.Math.Anal.204(1996)729-753.
[16] K.Ono,Global existence,decay,and blow up of solutions for some mildly degenerate nonlinear Kirchhoff stings.J.Differential Equations 137(1997)273-301.
[17] K.Ono,Global existence,asymptotic behavior,and global non-existence of solutons for damped non-linear wave equations of Kirchhoff type in the whole space.Math.Meth. Appl.Sci.23(2000)535-560.