关于无界域上非自治无穷维动力系统解的长时间行为
摘要
在本博士论文中,首先我们在无界区域上考察了下面非自治反应扩散方程解的渐近行为这里(?)是一个N×N的实矩阵,并且具有正的对称项(?)(a + a~*)≥βI,β> 0, a~*表示a的转置,u = u(x,t) = (u_1,...,u_N),g=g(x,t)=(g_1,...,g_N),f=f(u,t)=(f_1,...f_N)..
我们假定外力项g = g(x, t)∈L_b~2(R; H),非线性项f = f(u, t)∈C(R~N×R;R~N)满足下列条件C是一正的常数,在不同行,不同列代表不同的常数。我们主要以方程(1)在无界域上一致吸引子的存在性和结构两个方面来考虑解的渐近行为,分别证明了方程(1)在空间L~2(R~N),L~p(R~N),p>2中一致吸引子的存在性,并且同时得到了它们的结构。为了证明一致吸引子在空间L~p(R~N)中的存在性,我们运用了C.Zhong,M.Yang,C.Sun在文献[42]中提出的渐近先验估计的方法.为了描述一致吸引子在空间L~p(R~N)上的结构,我们需要相应的过程族在空间L~p(R~N)上的某种连续性。如果对指数p不加任何限制的话,过程族在空间L~p(R~N)中没有任何的连续性,即使强弱连续也没有,这是因为空间L~q(R~N)和L~p(R~N)当p≠q时没有任何的嵌套关系。在本博士论文中,我们用过程族在空间L~2(R~N)中的连续性去代替它在空间L~p(R~N)中的连续性,从而得到一致吸引子在空间L~p(R~N)上的结构,详细的细节可参看第三章。
然后,我们在无界域上考察下面的非线性,非自治反应扩散方程正解的渐近行为这里u_0∈E = L~q(Ω), 1 < q <∞,Ω(?)R~N是无界的光滑区域,E是定义了序≤的Banach空间,f: R×Ω×R→R是具有合适光滑性的函数,并且满足f(t,x,u)≥0,和(?)是关于u≥0非增的函数. (6)
我们的主要目的是在文献[1],[5],[65]思想的基础上,运用非自治无穷维动力系统在无界域上的理论,证明方程(5)在无界域上的拉回吸引子和向前吸引子的存在性。在对非线性项额外的假设下,并且假定对应于方程(5)的过程族{U(t, s)}_(t≥s)在空间E上保序.运用比较原理、上下解方法、算子的单调性、过程族{U(t, s)}_(t≥s)在空间E上的连续性,过程族在空间E上的指数稳定性,证明了方程(5)的极小完全轨道(?)_m(t)≥0和极大完全轨道(?)_m(x)的存在性,并且它们是渐近稳定的,同时得到序区间[(?)_m(t),(?)_m(t)]的正不变性。为了证明极小完全非退化轨道的存在性,我们运用了轨道逼近的方法,先找到有界域上的极小完全非退化轨道,然后通过区域逼近,从而得到在无界区域上的极小完全非退化轨道。同时证明了过程族{U(t, s)}_(t≥s)在空间L~q(Ω), 1 < q <∞,H_D~((2α),q)(Ω),α∈[-1, +1]上的拉回吸引子Α_1和向前吸引子Α_2的存在性,并且有Α_1 (?) [(?)_m(t),(?)_m(t)],Α_2 (?) [(?)_m(t),(?)_m(t)].为了证明过程族{U(t, s)}_(t≥s)在空间L~q(Ω),1 < q <∞中的紧性,我们运用截断函数的方法,用有界域去逼近无界域,在有界域上用紧的Sobolev嵌入,在无界域上让解的L~q(Ω)范数很小。为了得到过程族{U(t, s)}_(t≥s)在空间H_D~((2α),q)(Ω)中的紧性,主要用解的常数变异公式再结合能量估计得到,具体的细节和更进一步的讨论可参看第四章。
作为一个具体的例子,在无界域上我们考察下面的非自治Logistic方程正解的渐近行为Ω(?) R~N是一无界的光滑区域,p > 1, b(t)∈C~1(R),β,λ∈R.b(t)还满足下面的条件:假设存在正的常数B_0,对所有的t∈R满足
当β≥λ时,方程(7)正解的渐近行为比较简单,我们可证明对应的过程族{U(t, s)}_(t≥s)在空间E上存在拉回吸引子Α_1和向前吸引子Α_2,并且有Α_1={0},Α_2={0}.
当β<λ时,如果过程族{U(t, s)}_(t≥s)在原点不稳定,方程(7)正解的渐近行为比较复杂。我们将会看到b(t)趋于零点的速度会极大的影响方程(7)正解的渐近行为。方程(7)存在非平凡的完全轨道u~*(t),在拉回的意义下吸引方程(7)其它的正解,在这种情况下拉回吸引子4,存在,并且有Α_1={u~*(t)}_(t∈R)。但是,当t→∞, u~*(t)可能无界,显然,向前吸引子不存在。然而,我们仍然能描述方程(7)正解的渐近行为,我们可计算u~*(t)和方程其它正解的相对误差和绝对误差,如果b(t)趋于零的速度很慢,则u~*(t)和方程其它正解的相对误差趋于零,在这种情况下,u~*(t)就可以看做是方程(7)的Forward attractor的“一阶逼近”。接下来我们还给出入和b(t)满足的区域,计算u~*(t)和方程其它正解的绝对误差,我们将会看到在某种程度下u~*(t)要么是方程(7)的Forward attractor,要么不是。但是目前我们还没有想出很好的办法在b(t)趋于零的速度很快时描述方程(7)正解的渐近行为,这也是我们接下来要做的工作,具体的细节可参看第五章。
In this doctoral dissertation, first, we consider the asymptotic behavior of solutions of following nonautonomous reaction-diffusion equations in unbounded domainwhere a = (?) is a N×N real matrix with positive symmetric part (?)(a + a~*)≥βI,β> 0, a~* is the transpose of a, u = u(x,t) = (u_1,...,u_N),g=g(x,t)=(g_1,...,g_N),f=f(u,t)=(f_1,...f_N).
We assume that g = g(x, t)∈L_b~2(R; H), f = f(u, t)∈C(R~N×R;R~N) and the following conditions hold:where letter C denotes a positive constant which may be different in each occasion throughout this paper. We mainly consider the existence and structure of uniform attractor for Eq.(11) in unbounded domain, we prove the existence of uniform attractor for Eq.(11) in spaces L~2(R~N) and L~p(R~N), p > 2, respectively, and at the same time, obtain it's structure. In order to prove the existence of uniform attractor in space L~p(R~N), we use the method that developed by C. Zhong, M. Yang, C. Sun in [42] called asymptotic a priori estimate. In order to describe the structure of uniform attractor in space L~p(R~N), we must need some continuity of associated processes in space L~p(R~N), if we don't restrict the order of p, then there isn't any continuity of processes in space L~p(R~N), even norm-to-weak continuity, since the spaces L~q(R~N) and L~p(R~N) are not nested, if p≠q. In this doctoral dissertation, we use the continuity of processes in L~2(R~N) to instead of the continuity of processes in space L~q(R~N) to obtain the structure of uniform attractor in space L~p(R~N), for more details, please see chapter three.
Then, we consider the asymptotic behavior of the positive solutions of the following nonlinear、nonautonomous reaction-diffusion equations in unbounded domainwhere u_0∈E = L~q(Ω), 1 < q <∞, E is a Banach space that defined order≤,Ω(?) R~N is an unbounded smooth domain, f : R×Ω×R→R is suitable smooth function, and satisfies f(t,x,u)≥0,
Our goal is that based on the ideal in papers [1,5,65], and combining with the theory of infinite dimensional dynamical system in unbounded domain, prove the existence of pullback attractor and forward attractor for Eq.(15) in unbounded domain. Under other assumptions on nonlinear force and suppose that the processes {U(t, s)}_(t≥s) associated to Eq.(15) is order preserving. Using comparison principle, method of sub-super solutions, monotony of operators, the continuity of processes {U(t, s)}_(t≥s) in space E, the property of exponential stable of processes {U(t, s)}_(t≥s) in space E, we prove that there exist two extremal equilibria (?)_m(t)≥0 and (?)_m(t), minimal and maximal, respectively, and they are asymptotic stable. Furthermore, we obtain that the order interval [(?)_m(t),(?)_m(t)] is forward invariant respect to processes {U(t, s)}_(t≥s). In order to prove the existence of non-degenerate minimal equilibria, we use the method of domain approximation, first we find non-degenerate minimal equilibria in bounded domain, then by domain approximation, we obtain the non-degenerate minimal equilibria in whole unbounded domain. At the end, we obtain the existence of pullback attractorΑ_1 and forward attractorΑ_2 in spaces L~q(Ω), 1 < q <∞, and H_D~((2α),q)(Ω),α∈[-1, +1], respectively, and at the same time, we obtain thatΑ_1 (?) [(?)_m(t),(?)_m(t)],Α_2 (?) [(?)_m(t),(?)_m(t)].
In order to prove the compactness of the processes {U(t, s)}_(t≥s) in space L~q(Ω), 1 < q <∞, we use the method of cutoff function, decomposing the whole space into two parts, one is bounded, another it's complement, in bounded domain we use compact sobolev embedding, in another part, let the L~q-norm of solution very small. In order to obtain the compactness of processes {U(t, s)}_(t≥s) in space H_D~((2α),q)(Ω), we use the formula of constant variation and energy estimate, for more details and further talks, please see chapter four.
As a concrete example, we consider the asymptotic behavior of positive solutions of following nonautonomous Logistic equationwhereΩ(?) R~N is unbounded smooth domain, p > 1, b(t)∈C~1(R),β,λ∈R. Also assume b(t) satisfies following conditions: there exist positive constant B_0, such that for all t∈RWhenβ≥λ, the asymptotic behavior of positive solutions of Eq.(17) is simple, there exists only one complete obit, it is 0, such that pullback attractorΑ_1 = {0} and forward attractorΑ_2 = {0}.
Whenβ<λ, if the processes {U(t, s)}_(t≥s) is unstable at point 0, then the asymptotic behavior of positive solutions of Eq.(17) is complicate. We will show that the asymptotic behavior of positive solutions is affected by the velocity of b(t) go to 0 completely. There exists non-trivial complete obit u~*(t) for Eq.(17), attracts other positive solutions in pullback sense, the pullback attractorΑ_1 exists, andΑ_1 = {u~*(t)}_(t∈R). But when t→∞, u~*(t) maybe unbounded, obviously, the forward attractor nonexist. However, in this situation, we can still describe the asymptotic behavior of positive solutions of Eq.(17). By compute absolute error and relative error between u~*(t) and other positive solutions, if the velocity of b(t) go to 0 is slowly, the relative error between u~*(t) and other positive solutions of Eq.(17) go to 0, then u~*(t) is the ''first order approximation" of forward attractor, in this situation, u~*(t) can be considered as a forward attractor. We also give a regain forλand b(t) and compute the absolute error between u~*(t) and other positive solutions of Eq.(17), we will see that in one situation u~*(t) is forward attractor, in another one it is not, for more details, please see chapter five.
我们假定外力项g = g(x, t)∈L_b~2(R; H),非线性项f = f(u, t)∈C(R~N×R;R~N)满足下列条件C是一正的常数,在不同行,不同列代表不同的常数。我们主要以方程(1)在无界域上一致吸引子的存在性和结构两个方面来考虑解的渐近行为,分别证明了方程(1)在空间L~2(R~N),L~p(R~N),p>2中一致吸引子的存在性,并且同时得到了它们的结构。为了证明一致吸引子在空间L~p(R~N)中的存在性,我们运用了C.Zhong,M.Yang,C.Sun在文献[42]中提出的渐近先验估计的方法.为了描述一致吸引子在空间L~p(R~N)上的结构,我们需要相应的过程族在空间L~p(R~N)上的某种连续性。如果对指数p不加任何限制的话,过程族在空间L~p(R~N)中没有任何的连续性,即使强弱连续也没有,这是因为空间L~q(R~N)和L~p(R~N)当p≠q时没有任何的嵌套关系。在本博士论文中,我们用过程族在空间L~2(R~N)中的连续性去代替它在空间L~p(R~N)中的连续性,从而得到一致吸引子在空间L~p(R~N)上的结构,详细的细节可参看第三章。
然后,我们在无界域上考察下面的非线性,非自治反应扩散方程正解的渐近行为这里u_0∈E = L~q(Ω), 1 < q <∞,Ω(?)R~N是无界的光滑区域,E是定义了序≤的Banach空间,f: R×Ω×R→R是具有合适光滑性的函数,并且满足f(t,x,u)≥0,和(?)是关于u≥0非增的函数. (6)
我们的主要目的是在文献[1],[5],[65]思想的基础上,运用非自治无穷维动力系统在无界域上的理论,证明方程(5)在无界域上的拉回吸引子和向前吸引子的存在性。在对非线性项额外的假设下,并且假定对应于方程(5)的过程族{U(t, s)}_(t≥s)在空间E上保序.运用比较原理、上下解方法、算子的单调性、过程族{U(t, s)}_(t≥s)在空间E上的连续性,过程族在空间E上的指数稳定性,证明了方程(5)的极小完全轨道(?)_m(t)≥0和极大完全轨道(?)_m(x)的存在性,并且它们是渐近稳定的,同时得到序区间[(?)_m(t),(?)_m(t)]的正不变性。为了证明极小完全非退化轨道的存在性,我们运用了轨道逼近的方法,先找到有界域上的极小完全非退化轨道,然后通过区域逼近,从而得到在无界区域上的极小完全非退化轨道。同时证明了过程族{U(t, s)}_(t≥s)在空间L~q(Ω), 1 < q <∞,H_D~((2α),q)(Ω),α∈[-1, +1]上的拉回吸引子Α_1和向前吸引子Α_2的存在性,并且有Α_1 (?) [(?)_m(t),(?)_m(t)],Α_2 (?) [(?)_m(t),(?)_m(t)].为了证明过程族{U(t, s)}_(t≥s)在空间L~q(Ω),1 < q <∞中的紧性,我们运用截断函数的方法,用有界域去逼近无界域,在有界域上用紧的Sobolev嵌入,在无界域上让解的L~q(Ω)范数很小。为了得到过程族{U(t, s)}_(t≥s)在空间H_D~((2α),q)(Ω)中的紧性,主要用解的常数变异公式再结合能量估计得到,具体的细节和更进一步的讨论可参看第四章。
作为一个具体的例子,在无界域上我们考察下面的非自治Logistic方程正解的渐近行为Ω(?) R~N是一无界的光滑区域,p > 1, b(t)∈C~1(R),β,λ∈R.b(t)还满足下面的条件:假设存在正的常数B_0,对所有的t∈R满足
当β≥λ时,方程(7)正解的渐近行为比较简单,我们可证明对应的过程族{U(t, s)}_(t≥s)在空间E上存在拉回吸引子Α_1和向前吸引子Α_2,并且有Α_1={0},Α_2={0}.
当β<λ时,如果过程族{U(t, s)}_(t≥s)在原点不稳定,方程(7)正解的渐近行为比较复杂。我们将会看到b(t)趋于零点的速度会极大的影响方程(7)正解的渐近行为。方程(7)存在非平凡的完全轨道u~*(t),在拉回的意义下吸引方程(7)其它的正解,在这种情况下拉回吸引子4,存在,并且有Α_1={u~*(t)}_(t∈R)。但是,当t→∞, u~*(t)可能无界,显然,向前吸引子不存在。然而,我们仍然能描述方程(7)正解的渐近行为,我们可计算u~*(t)和方程其它正解的相对误差和绝对误差,如果b(t)趋于零的速度很慢,则u~*(t)和方程其它正解的相对误差趋于零,在这种情况下,u~*(t)就可以看做是方程(7)的Forward attractor的“一阶逼近”。接下来我们还给出入和b(t)满足的区域,计算u~*(t)和方程其它正解的绝对误差,我们将会看到在某种程度下u~*(t)要么是方程(7)的Forward attractor,要么不是。但是目前我们还没有想出很好的办法在b(t)趋于零的速度很快时描述方程(7)正解的渐近行为,这也是我们接下来要做的工作,具体的细节可参看第五章。
In this doctoral dissertation, first, we consider the asymptotic behavior of solutions of following nonautonomous reaction-diffusion equations in unbounded domainwhere a = (?) is a N×N real matrix with positive symmetric part (?)(a + a~*)≥βI,β> 0, a~* is the transpose of a, u = u(x,t) = (u_1,...,u_N),g=g(x,t)=(g_1,...,g_N),f=f(u,t)=(f_1,...f_N).
We assume that g = g(x, t)∈L_b~2(R; H), f = f(u, t)∈C(R~N×R;R~N) and the following conditions hold:where letter C denotes a positive constant which may be different in each occasion throughout this paper. We mainly consider the existence and structure of uniform attractor for Eq.(11) in unbounded domain, we prove the existence of uniform attractor for Eq.(11) in spaces L~2(R~N) and L~p(R~N), p > 2, respectively, and at the same time, obtain it's structure. In order to prove the existence of uniform attractor in space L~p(R~N), we use the method that developed by C. Zhong, M. Yang, C. Sun in [42] called asymptotic a priori estimate. In order to describe the structure of uniform attractor in space L~p(R~N), we must need some continuity of associated processes in space L~p(R~N), if we don't restrict the order of p, then there isn't any continuity of processes in space L~p(R~N), even norm-to-weak continuity, since the spaces L~q(R~N) and L~p(R~N) are not nested, if p≠q. In this doctoral dissertation, we use the continuity of processes in L~2(R~N) to instead of the continuity of processes in space L~q(R~N) to obtain the structure of uniform attractor in space L~p(R~N), for more details, please see chapter three.
Then, we consider the asymptotic behavior of the positive solutions of the following nonlinear、nonautonomous reaction-diffusion equations in unbounded domainwhere u_0∈E = L~q(Ω), 1 < q <∞, E is a Banach space that defined order≤,Ω(?) R~N is an unbounded smooth domain, f : R×Ω×R→R is suitable smooth function, and satisfies f(t,x,u)≥0,
Our goal is that based on the ideal in papers [1,5,65], and combining with the theory of infinite dimensional dynamical system in unbounded domain, prove the existence of pullback attractor and forward attractor for Eq.(15) in unbounded domain. Under other assumptions on nonlinear force and suppose that the processes {U(t, s)}_(t≥s) associated to Eq.(15) is order preserving. Using comparison principle, method of sub-super solutions, monotony of operators, the continuity of processes {U(t, s)}_(t≥s) in space E, the property of exponential stable of processes {U(t, s)}_(t≥s) in space E, we prove that there exist two extremal equilibria (?)_m(t)≥0 and (?)_m(t), minimal and maximal, respectively, and they are asymptotic stable. Furthermore, we obtain that the order interval [(?)_m(t),(?)_m(t)] is forward invariant respect to processes {U(t, s)}_(t≥s). In order to prove the existence of non-degenerate minimal equilibria, we use the method of domain approximation, first we find non-degenerate minimal equilibria in bounded domain, then by domain approximation, we obtain the non-degenerate minimal equilibria in whole unbounded domain. At the end, we obtain the existence of pullback attractorΑ_1 and forward attractorΑ_2 in spaces L~q(Ω), 1 < q <∞, and H_D~((2α),q)(Ω),α∈[-1, +1], respectively, and at the same time, we obtain thatΑ_1 (?) [(?)_m(t),(?)_m(t)],Α_2 (?) [(?)_m(t),(?)_m(t)].
In order to prove the compactness of the processes {U(t, s)}_(t≥s) in space L~q(Ω), 1 < q <∞, we use the method of cutoff function, decomposing the whole space into two parts, one is bounded, another it's complement, in bounded domain we use compact sobolev embedding, in another part, let the L~q-norm of solution very small. In order to obtain the compactness of processes {U(t, s)}_(t≥s) in space H_D~((2α),q)(Ω), we use the formula of constant variation and energy estimate, for more details and further talks, please see chapter four.
As a concrete example, we consider the asymptotic behavior of positive solutions of following nonautonomous Logistic equationwhereΩ(?) R~N is unbounded smooth domain, p > 1, b(t)∈C~1(R),β,λ∈R. Also assume b(t) satisfies following conditions: there exist positive constant B_0, such that for all t∈RWhenβ≥λ, the asymptotic behavior of positive solutions of Eq.(17) is simple, there exists only one complete obit, it is 0, such that pullback attractorΑ_1 = {0} and forward attractorΑ_2 = {0}.
Whenβ<λ, if the processes {U(t, s)}_(t≥s) is unstable at point 0, then the asymptotic behavior of positive solutions of Eq.(17) is complicate. We will show that the asymptotic behavior of positive solutions is affected by the velocity of b(t) go to 0 completely. There exists non-trivial complete obit u~*(t) for Eq.(17), attracts other positive solutions in pullback sense, the pullback attractorΑ_1 exists, andΑ_1 = {u~*(t)}_(t∈R). But when t→∞, u~*(t) maybe unbounded, obviously, the forward attractor nonexist. However, in this situation, we can still describe the asymptotic behavior of positive solutions of Eq.(17). By compute absolute error and relative error between u~*(t) and other positive solutions, if the velocity of b(t) go to 0 is slowly, the relative error between u~*(t) and other positive solutions of Eq.(17) go to 0, then u~*(t) is the ''first order approximation" of forward attractor, in this situation, u~*(t) can be considered as a forward attractor. We also give a regain forλand b(t) and compute the absolute error between u~*(t) and other positive solutions of Eq.(17), we will see that in one situation u~*(t) is forward attractor, in another one it is not, for more details, please see chapter five.
引文
[1] J. C. Robinson, A. Rodriguez-Bernal, A. Vidal-Lopez, Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems, J. Differential Equations 238 (2007) 289-337.
[2] J. A. Langa, J. C. Robinson, A. Rodriguez-bernal, A. Suarez, A. Vidal-Lopez, Existence and non-existence of unbounded forward attractor for a class of non-autonomous reaction diffusion equations, Discrete Contin. Dyn. Syst., Vol 18, Num 2&3, 483-497 (2007).
[3] J. A. Langa & Antonio Suarez, Pullback permanence for non-autonomous partial differential equations, Electronic Journal of Differential Equations, 2002 (2002) 1-20.
[4] A. Rodriguez-Bernal, A. Vidal-Lopez, Extremal equilibria for reaction-diffusion equations in bounded domains and applications, J. Differential Equations 244 (2008) 2983-3030.
[5] A. Rodriguez-Bernal, A. Vidal-Lopez, Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems, Discrete Cont. Dynam. Systems 18 (2007) 537-567.
[6] A. Rodriguez-Bernal, A. Vidal-Lopez, Extremal equilibria and asymptotic behavior of parabolic nonlinear reaction-diffusion equations, Progress in Nonlinear Differential Equations and Their Aoolications, 64 (2005) 509-516.
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