几类具超临界源项的非线性双曲方程解的存在及爆破性研究

发布时间：2018-05-23 13:18

本文选题：阻尼项 + 源项　；参考：《吉林大学》2017年博士论文

【摘要】：本文研究了几类具有阻尼项和源项的双曲方程解的性质.主要讨论了非线性弱阻尼项、强阻尼项、次临界源项和超临界源项对方程解的存在和爆破性的影响.本文内容共分为四章.第一章为绪论.第二章,考虑如下方程其中T0,Ω为Rn(n≥1)中的有界Lipschitz区域,(?)Ω为Ω的边界,指数m和p均为实数且满足当 n3 时,m1,1p+∞;当n≥3时,m1,1p≤2*△= 2n/n-2,pm+1/m2*.我们得到了上述问题解的爆破性,同时给出了爆破时间的上界和下界估计,并将结论推广到了超临界源项情形.我们记T*为解的爆破时间.在这一章,我们称解u(x,t)在有限时间T*处爆破,是指下式成立对于上述系统,当源项|u|p-1u真满足超临界条件,即pn/n-2时,嵌入H10(Ω)→L2p(Ω)不再成立,这使得在研究具有次临界源项的方程时所应用的方法不再适用.我们定义了一个新的能量泛函,并运用能量方法来克服这个困难.问题(1)解的爆破性和爆破时间上界估计的主要结论为下面的定理.定理1.假设指数p和m均为实数,满足p≥m1,并且当 n ≥ 3 时,pn+2/n-2;当 n3 时,p+∞,初值满足‖%絬0‖22β1,E(0)d,那么问题(1)的解u(x,t)在有限时间内爆破,并且其爆破时间T*满足那么问题(1)的解在有限时间内爆破,并且其爆破时间T*满足其中正常数M1和M2依赖于指数m,p,初始能量E(0),空间维度n和区域Ω,这里常数B为满足‖u‖2*≤B‖%絬‖2的嵌入系数,常数ρ0充分小使得F(0)0.下面的定理为解的爆破时间的下界估计.定理2.如果定理1中假设条件都成立,并且当n≥3时,指数p进一步满足p≤n2 + 2n—4/n(n-2),那么爆破时间T*具有如下形式的下界估计(?)其中正常数C7-C11依赖于指数p,空间维度n,区域Ω和初始能量E(0),同时定理2中常数t满足t1,常数σ满足当n ≥ 3时,因此,爆破时间下界估计中不等式右端的积分都是有限的.第三章,我们考虑具有变指数非线性阻尼项和变指数源项的双曲方程其中T0,Ω为Rn(n≥1)中的有界Lipschitz区域,(?)Ω为Ω的边界,QT = Ω×[0,T],初值满足系数 a(x,t),b(x,t),c(x,t)和指数 p(x,t),q(x,t)在 QT 上连续,且满足这里w(r)满足具有变指数阻尼项或变指数源项的双曲方程能够更实际地描述扩散过程.目前,对于其解的性质的研究还比较少,变指数的存在给我们的研究工作带来了很大的困难.而当方程中的源项还满足超临界条件时,情况就会更加复杂.尤其是在解的存在性证明以及爆破时间的下界估计中,超临界源项为我们关注的重点所在.在第三章中,我们首先运用极大单调算子理论,得到了一个具有更一般源项的系统解的局部存在性.其中函数f满足下列条件注意到,函数f(x,t,u)=b(x,t)|u|p(x,t)-1u满足所有假设.因此,由问题(5)解的存在性自然就可以得到问题(2)解的存在性.问题(5)解的存在性的主要结论为下面的定理.定理中对于指数p的条件涵盖了超临界情形.定理3.如果系数a(x,t),c(x,t)和指数p(x,t),q(x,t)满足(4),并且那么问题(5)存在弱解u(x,t)满足其中解的存在时间T依赖于初值u0和u1.然后,我们针对指数p和q是否依赖于t,分别对问题(2)解的爆破性以及爆破时间的上界和下界估计进行了讨论,其中难点主要来源于非标准增长条件和超临界源项的存在.第三章中,我们称解u(x,t)在有限时间T*处爆破,是指下式成立我们充分利用变指数函数空间的性质,定义了几个新的能量泛函,并运用能量方法得到了以下三个定理.第一个定理给出了当p和q与t无关时,解的爆破性和爆破时间的上界估计.定理4.假设那么问题(2)-(4)的解u(x,t)在有限时间T*处爆破,并且T*满足其中正常数M1和M2依赖于系数a,b,c,指数p,q,空间维度n,初始能量E(0),F(0)和区域Ω,并且这里常数B为满足‖u‖p(·)+1≤B‖%絬‖2的嵌入系数,常数ε0充分小使得F(0)0.第二个定理给出了当p和q与t无关时,解的爆破时间的下界估计.定理5.如果定理4中所有假设条件都成立,并且当n ≥ 3时,p还满足那么问题(2)-(4)解u(x,t)的爆破时间T*满足其中正常数C18-C24依赖于系数a,b,指数p,空间维度n,区域Ω和初始能量E(0),同时显然,定理5中δ1,0γ1,2*/2σσ,2*/2σ1.于是,爆破时间下界估计中的广义积分均是有限的.第三个定理给出了当p和q与t有关时,解的爆破性以及爆破时间的上界和下界估计.定理6.假设如下条件成立那么问题(2)一(4)的解u(x,t)在有限时间T*处爆破,且T*满足进一步,若n3或n≥3,p+≤n/n-2,那么(?)在第三章的最后,我们利用一个例子说明了前面所有结果的正确性.这个例子满足定理6中所有假设条件.通过数值模拟,我们得到了其解以及能量泛函的变化图.从图像中可以看出,解在一段时间内存在,而到达某一时刻时,解就发生了爆破.第四章,考虑了一类具有强阻尼项的双曲方程解的爆破性.其中T0,Ω是Rn(n ≥ 1)中的有界Lipschitz区域,(?)Ω为Ω的边界,并且初值u0,u1满足常数ω和μ满足这里λ1是算子-△在Dirichlet边界条件下的第一特征值.常数p为实数并满足我们知道,在双曲方程中,阻尼项对解的爆破起到抑制作用.而相比于弱阻尼项,强阻尼项的影响更加剧烈.因此强阻尼项△ut的存在使得爆破性的证明以及爆破时间的估计更为困难.在这一章中,我们称解u在有限时间T*处爆破,是指我们的主要结论为下面的两个定理.定理7.假设u为问题(6)-(9)在[0,T]上的唯一解,如果存在常数t ∈[0,T*)使得并且初值满足那么解u在有限时间T*处爆破,且T*满足其中定理8.如果(8),(9)成立且假设那么问题(6),(7)的解在有限时间T*处爆破,且T*满足其中这里常数C1为满足‖u‖2n/n-2≤C1‖%絬‖2的嵌入系数.直接计算则可得到定理8中q1,从而广义积分(?)收敛.(?)
[Abstract]:This paper deals with the properties of solutions of several kinds of hyperbolic equations with damping term and source term. The existence of nonlinear weak damping term, strong damping term, subcritical source term and supercritical source term solution are discussed. The contents of this paper are divided into four chapters. The first chapter is introduction. The second chapter considers the following equations T0, Omega Rn (n > 1). The bounded Lipschitz region, (?) Omega is the boundary of Omega, the exponent m and P are real and satisfy the N3, m1,1p+ infinity; when n > 3, m1,1p < 2* delta = 2n/n-2, pm+1/m2*. we get the blasting of the solution of the above problem, at the same time, the upper and lower bounds of the blasting time are given, and the conclusion is extended to the case of the supercritical source term. We remember the T* for the case. In this chapter, we call the solution U (x, t) blasting at a limited time T*, which means the lower form is set up for the above system. When the source term |u|p-1u really satisfies the supercritical condition, that is, pn/n-2, the embedded H10 (omega) - L2p (omega) is no longer established. This makes the method applied in the study of the equation with the subcritical source term no longer applicable. We define the method. A new energy functional is proposed and the energy method is used to overcome this difficulty. The main conclusion of the problem (1) the main conclusion of the blasting and blasting time upper bounds is the following theorem. Theorem 1. the exponent P and m are real numbers, which satisfy P > M1, and when n is 3, pn+2/n-2; when N3, p+ infinity satisfies the percentage of 22 beta 1, E (0) d, so The solution (1) of solution U (x, t) blasting in a finite time, and its blasting time T* satisfies the problem (1) of the solution in a finite time, and its blasting time T* satisfies the normal number M1 and M2 depends on the exponential m, P, the initial energy E (0), the spatial dimension N and the region Omega, and the constant B is a constant of constant B to satisfy the embedding coefficient of 2. The theorem of F (0) 0. is sufficient to make the theorem under F (0) 0. as the lower bound of the blasting time. Theorem 2. if the hypothesis of Theorem 1 is established, and when n > 3, the exponential P further satisfies the P < N2 + 2n 4/n (n-2), then the blasting time T* has the following form of lower bounds (?) and the normal number C7-C11 depends on the exponential P, the space dimension N, and the region Omega and initial energy E (0), while theorem 2 constant T satisfies T1, constant Sigma is satisfied when n is more than 3, so the integral of the right end of inequality in the estimation of the lower bounds of blasting time is limited. The third chapter, we consider the hyperbolic square path with variable exponential and variable exponential source term in which T0, Omega is a bounded Lipschitz region in Rn (n > 1), (?) QT = Omega boundary, QT = Omega [0, T], the initial value satisfies the coefficient a (x, t), B (x, t), C (x, t) and exponentially exponential source terms can be more practical to describe the diffusion process. At present, there are few studies on the properties of the solutions and the existence of variable exponents. When the source term in the equation satisfies the supercritical condition, the situation will be more complex. Especially in the existence proof of the solution and the estimation of the lower bounds of the blasting time, the supercritical source term is the focus of our attention. In the third chapter, we first use the maximal monotone operator theory. The local existence of a system solution with more general source terms is obtained. In which the function f satisfies the following conditions that the function f (x, t, U) =b (x, t) |u|p (x, t) -1u satisfies all hypotheses. Therefore, the existence of the problem (2) is naturally obtained by the existence of the problem (5) solution. The main conclusion of the existence of the problem (5) is the following definite In theorem 3., the condition of exponential P covers the supercritical condition. Theorem 3. if the coefficient a (x, t), C (x, t) and exponential P (x, t), q (x,) satisfy (4), and then the problem (5) satisfies the existence time of the solution The upper and lower bounds of the blasting time are discussed. The difficulties are mainly derived from the nonstandard growth conditions and the existence of the supercritical source term. In the third chapter, we call the solution U (x, t) blasting at the limited time T*, which means that the lower formula establishes the properties of the variable exponential function space and defines several new energy functionals. Three theorems are obtained by the energy method. The first theorem gives the blasting property of the solution and the upper bound of the blasting time when P and Q are independent of t. Theorem 4. suppose that the solution (2) - (4) of the solution (x, t) explode at a finite time T*, and T* satisfies the normal number M1 and M2 depends on the coefficient a, B, exponential, spatial dimension, initial energy The amount of E (0), F (0) and region Omega, and the constant B to satisfy the embedding coefficient of u p (.) +1 < B% 2, the constant epsilon 0 makes F (0) 0. second theorems when P and Q and T are irrelevant to the lower bounds of the blasting time. Theorem 5. if all the hypothesis conditions in Theorem 4 are established, and when n is more than 3, it also satisfies that question. Question (2) - (4) the blasting time for solving U (x, t) T* satisfies the normal number C18-C24 depends on the coefficient a, B, exponential P, the spatial dimension N, the region omega and the initial energy E (0). At the same time, it is obvious that the Delta 1,0 1,2*/2 Sigma and sigma (1.) in Theorem 5, therefore, the generalized integral in the lower bounds of the blasting time are finite. The third theorems give the solution when it is related to it. The upper and lower bounds of blasting and blasting time. Theorem 6. assume that the following conditions are assumed to be established. (2) a (4) solution U (x, t) blasting at a limited time T*, and T* satisfies further, if N3 or n is equal to 3, p+ is less than n/n-2, then (?) at the end of the third chapter, we use an example to illustrate the correctness of all the previous results. All the hypothesis conditions in theorem 6 are satisfied. Through numerical simulation, we get the solution and the change diagram of the energy functional. It can be seen from the image that the solution exists in a period of time and at a certain time, the solution exploded. The fourth chapter, considering the explosion of the solution of a kind of hyperbolic equation with strongly hindrance. T0, Omega is Rn (n The bounded Lipschitz region (> 1), (?) Omega is the boundary of Omega, and the initial value U0, U1 satisfies the constant omega and Mu satisfying the first eigenvalue under the boundary condition of Dirichlet. The constant P is the real number and we know that in the hyperbolic equation, the damping term plays an inhibitory effect on the blasting of the solution. The effect of the term is more intense. Therefore, the existence of the strong damping term, delta UT, makes the proof of blasting and the estimation of the time of blasting more difficult. In this chapter, we call the solution of u blasting at a limited time T*. It means that our main conclusion is the following two theorems. Theorem 7. assumes that u is a question (6) - (9) the unique solution on [0, T], if there is a constant The number T [0, T*) makes and the initial value satisfies the solution of the solution u at a finite time T*, and T* satisfies theorem 8. if (8), (9), and assumes that the problem (6), (7) is blasting in a finite time T*, and T* satisfies the constant C1 to satisfy the embedding coefficient of the 2n/n-2 u 2n/n-2 < < C1% > 2. The direct calculation can be obtained in theorem 8. Q1, so that the generalized integral (?) converges. (?)
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

【相似文献】