發(fā)布時(shí)間：2018-05-23 13:18

  本文選題：阻尼項(xiàng) + 源項(xiàng)�。� 參考：《吉林大學(xué)》2017年博士論文

【摘要】：本文研究了幾類具有阻尼項(xiàng)和源項(xiàng)的雙曲方程解的性質(zhì).主要討論了非線性弱阻尼項(xiàng)、強(qiáng)阻尼項(xiàng)、次臨界源項(xiàng)和超臨界源項(xiàng)對(duì)方程解的存在和爆破性的影響.本文內(nèi)容共分為四章.第一章為緒論.第二章,考慮如下方程其中T0,Ω為Rn(n≥1)中的有界Lipschitz區(qū)域,(?)Ω為Ω的邊界,指數(shù)m和p均為實(shí)數(shù)且滿足當(dāng) n3 時(shí),m1,1p+∞;當(dāng)n≥3時(shí),m1,1p≤2*△= 2n/n-2,pm+1/m2*.我們得到了上述問題解的爆破性,同時(shí)給出了爆破時(shí)間的上界和下界估計(jì),并將結(jié)論推廣到了超臨界源項(xiàng)情形.我們記T*為解的爆破時(shí)間.在這一章,我們稱解u(x,t)在有限時(shí)間T*處爆破,是指下式成立對(duì)于上述系統(tǒng),當(dāng)源項(xiàng)|u|p-1u真滿足超臨界條件,即pn/n-2時(shí),嵌入H10(Ω)→L2p(Ω)不再成立,這使得在研究具有次臨界源項(xiàng)的方程時(shí)所應(yīng)用的方法不再適用.我們定義了一個(gè)新的能量泛函,并運(yùn)用能量方法來克服這個(gè)困難.問題(1)解的爆破性和爆破時(shí)間上界估計(jì)的主要結(jié)論為下面的定理.定理1.假設(shè)指數(shù)p和m均為實(shí)數(shù),滿足p≥m1,并且當(dāng) n ≥ 3 時(shí),pn+2/n-2;當(dāng) n3 時(shí),p+∞,初值滿足‖%絬0‖22β1,E(0)d,那么問題(1)的解u(x,t)在有限時(shí)間內(nèi)爆破,并且其爆破時(shí)間T*滿足那么問題(1)的解在有限時(shí)間內(nèi)爆破,并且其爆破時(shí)間T*滿足其中正常數(shù)M1和M2依賴于指數(shù)m,p,初始能量E(0),空間維度n和區(qū)域Ω,這里常數(shù)B為滿足‖u‖2*≤B‖%絬‖2的嵌入系數(shù),常數(shù)ρ0充分小使得F(0)0.下面的定理為解的爆破時(shí)間的下界估計(jì).定理2.如果定理1中假設(shè)條件都成立,并且當(dāng)n≥3時(shí),指數(shù)p進(jìn)一步滿足p≤n2 + 2n—4/n(n-2),那么爆破時(shí)間T*具有如下形式的下界估計(jì)(?)其中正常數(shù)C7-C11依賴于指數(shù)p,空間維度n,區(qū)域Ω和初始能量E(0),同時(shí)定理2中常數(shù)t滿足t1,常數(shù)σ滿足當(dāng)n ≥ 3時(shí),因此,爆破時(shí)間下界估計(jì)中不等式右端的積分都是有限的.第三章,我們考慮具有變指數(shù)非線性阻尼項(xiàng)和變指數(shù)源項(xiàng)的雙曲方程其中T0,Ω為Rn(n≥1)中的有界Lipschitz區(qū)域,(?)Ω為Ω的邊界,QT = Ω×[0,T],初值滿足系數(shù) a(x,t),b(x,t),c(x,t)和指數(shù) p(x,t),q(x,t)在 QT 上連續(xù),且滿足這里w(r)滿足具有變指數(shù)阻尼項(xiàng)或變指數(shù)源項(xiàng)的雙曲方程能夠更實(shí)際地描述擴(kuò)散過程.目前,對(duì)于其解的性質(zhì)的研究還比較少,變指數(shù)的存在給我們的研究工作帶來了很大的困難.而當(dāng)方程中的源項(xiàng)還滿足超臨界條件時(shí),情況就會(huì)更加復(fù)雜.尤其是在解的存在性證明以及爆破時(shí)間的下界估計(jì)中,超臨界源項(xiàng)為我們關(guān)注的重點(diǎn)所在.在第三章中,我們首先運(yùn)用極大單調(diào)算子理論,得到了一個(gè)具有更一般源項(xiàng)的系統(tǒng)解的局部存在性.其中函數(shù)f滿足下列條件注意到,函數(shù)f(x,t,u)=b(x,t)|u|p(x,t)-1u滿足所有假設(shè).因此,由問題(5)解的存在性自然就可以得到問題(2)解的存在性.問題(5)解的存在性的主要結(jié)論為下面的定理.定理中對(duì)于指數(shù)p的條件涵蓋了超臨界情形.定理3.如果系數(shù)a(x,t),c(x,t)和指數(shù)p(x,t),q(x,t)滿足(4),并且那么問題(5)存在弱解u(x,t)滿足其中解的存在時(shí)間T依賴于初值u0和u1.然后,我們針對(duì)指數(shù)p和q是否依賴于t,分別對(duì)問題(2)解的爆破性以及爆破時(shí)間的上界和下界估計(jì)進(jìn)行了討論,其中難點(diǎn)主要來源于非標(biāo)準(zhǔn)增長(zhǎng)條件和超臨界源項(xiàng)的存在.第三章中,我們稱解u(x,t)在有限時(shí)間T*處爆破,是指下式成立我們充分利用變指數(shù)函數(shù)空間的性質(zhì),定義了幾個(gè)新的能量泛函,并運(yùn)用能量方法得到了以下三個(gè)定理.第一個(gè)定理給出了當(dāng)p和q與t無關(guān)時(shí),解的爆破性和爆破時(shí)間的上界估計(jì).定理4.假設(shè)那么問題(2)-(4)的解u(x,t)在有限時(shí)間T*處爆破,并且T*滿足其中正常數(shù)M1和M2依賴于系數(shù)a,b,c,指數(shù)p,q,空間維度n,初始能量E(0),F(0)和區(qū)域Ω,并且這里常數(shù)B為滿足‖u‖p(·)+1≤B‖%絬‖2的嵌入系數(shù),常數(shù)ε0充分小使得F(0)0.第二個(gè)定理給出了當(dāng)p和q與t無關(guān)時(shí),解的爆破時(shí)間的下界估計(jì).定理5.如果定理4中所有假設(shè)條件都成立,并且當(dāng)n ≥ 3時(shí),p還滿足那么問題(2)-(4)解u(x,t)的爆破時(shí)間T*滿足其中正常數(shù)C18-C24依賴于系數(shù)a,b,指數(shù)p,空間維度n,區(qū)域Ω和初始能量E(0),同時(shí)顯然,定理5中δ1,0γ1,2*/2σσ,2*/2σ1.于是,爆破時(shí)間下界估計(jì)中的廣義積分均是有限的.第三個(gè)定理給出了當(dāng)p和q與t有關(guān)時(shí),解的爆破性以及爆破時(shí)間的上界和下界估計(jì).定理6.假設(shè)如下條件成立那么問題(2)一(4)的解u(x,t)在有限時(shí)間T*處爆破,且T*滿足進(jìn)一步,若n3或n≥3,p+≤n/n-2,那么(?)在第三章的最后,我們利用一個(gè)例子說明了前面所有結(jié)果的正確性.這個(gè)例子滿足定理6中所有假設(shè)條件.通過數(shù)值模擬,我們得到了其解以及能量泛函的變化圖.從圖像中可以看出,解在一段時(shí)間內(nèi)存在,而到達(dá)某一時(shí)刻時(shí),解就發(fā)生了爆破.第四章,考慮了一類具有強(qiáng)阻尼項(xiàng)的雙曲方程解的爆破性.其中T0,Ω是Rn(n ≥ 1)中的有界Lipschitz區(qū)域,(?)Ω為Ω的邊界,并且初值u0,u1滿足常數(shù)ω和μ滿足這里λ1是算子-△在Dirichlet邊界條件下的第一特征值.常數(shù)p為實(shí)數(shù)并滿足我們知道,在雙曲方程中,阻尼項(xiàng)對(duì)解的爆破起到抑制作用.而相比于弱阻尼項(xiàng),強(qiáng)阻尼項(xiàng)的影響更加劇烈.因此強(qiáng)阻尼項(xiàng)△ut的存在使得爆破性的證明以及爆破時(shí)間的估計(jì)更為困難.在這一章中,我們稱解u在有限時(shí)間T*處爆破,是指我們的主要結(jié)論為下面的兩個(gè)定理.定理7.假設(shè)u為問題(6)-(9)在[0,T]上的唯一解,如果存在常數(shù)t ∈[0,T*)使得并且初值滿足那么解u在有限時(shí)間T*處爆破,且T*滿足其中定理8.如果(8),(9)成立且假設(shè)那么問題(6),(7)的解在有限時(shí)間T*處爆破,且T*滿足其中這里常數(shù)C1為滿足‖u‖2n/n-2≤C1‖%絬‖2的嵌入系數(shù).直接計(jì)算則可得到定理8中q1,從而廣義積分(?)收斂.(?)
[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. (?)
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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