本文選題：p-Kirchhoff型方程 + 薄膜方程��； 參考：《吉林大學(xué)》2017年博士論文

【摘要】：擬線性橢圓型方程和拋物型方程是兩類重要的偏微分方程,比較典型的例子有流體力學(xué)中的p-Laplace方程[1],多孔介質(zhì)方程[2],非線性彈性反應(yīng)擴(kuò)散方程[3]等.近年來,隨著人們對偏微分方程的研究更加地深入和廣泛,所討論的微分算子的形式也越來越復(fù)雜化,關(guān)于具p-Laplace算子的擬線性偏微分方程的研究得到了國內(nèi)外數(shù)學(xué)家們的廣泛關(guān)注.二十世紀(jì)初蘇聯(lián)數(shù)學(xué)家Sobolev在文[4,5]中引入了 Sobolev空間的概念,這類空間對偏微分方程的研究具有重要和廣泛的應(yīng)用價(jià)值,尤其是對p-Laplace方程的研究起著非常關(guān)鍵的作用.本文主要對幾類具p-Laplace算子的橢圓型和拋物型方程解的性質(zhì)進(jìn)行研究.包括解的存在性、唯一性、正則性、爆破、熄滅以及解的長時(shí)間漸近行為等.全文共分為五第一章為緒論.介紹本文所研究的主要內(nèi)容,研究現(xiàn)狀及本文所研究問題需要克服的典型困難和使用的主要方法等.在第二章中,我們研究一類p-Laplace奇異橢圓方程的Dirichlet邊值問題其中,Ω(?)R~N(N ≥ 1)是具有光滑邊界的有界區(qū)域,是標(biāo)準(zhǔn)的p-Laplace算子,p1,γ1,h(x)是L1(Ω)中的正函數(shù)(即h(x0幾乎處處于Ω上).首先由于方程(1)具有強(qiáng)奇性(γ1),經(jīng)典的變分法(臨界點(diǎn)理論),上下解方法以及不動(dòng)點(diǎn)定理等常用方法對于問題(1)具有一定的限制性;又由于p-Laplace算子的存在,我們一般不能從un→u于W01,p(Ω)中弱收斂直接得到于Lp/p-1(Ω,R~N)中弱收斂,這也為我們解決問題(1)解的存在性提出了挑戰(zhàn);此外,求解奇異橢圓方程時(shí),方程右端項(xiàng)h(x)起著至關(guān)重要的作用,它的性態(tài)和形式往往會(huì)決定求解的方法和復(fù)雜程度.問題(1)中的非齊次項(xiàng)h(x)是L1(Ω)中函數(shù),具有較弱的正則性,也為我們求解奇異橢圓問題帶來了很大的困難.為了克服以上困難,我們通過構(gòu)造W01,p(Ω)中適當(dāng)?shù)募?包含Nehari流形作為特殊情形),將問題(1)限制在此集合上來保證奇異項(xiàng)的可積性,然后考慮相應(yīng)奇異能量泛函在此集合上的約束極小問題.借助Ekeland變分原理和一些分析技巧我們得到了問題(1)存在W01,p(Ω)解的充分必要條件.此外,借助p-Laplace算子的單調(diào)性證明了問題(1)W01,p(Ω)解的唯一性.這一章的主要結(jié)果如下:定理1.設(shè)Ω(?)R~N ≥ 1)是具有光滑邊界的有界區(qū)域,p1,γ1,∈ 1(h是正函數(shù)(即h(x)0幾乎處處于Ω上),則問題(1)存在唯一的W01,p(Ω)解當(dāng)且僅當(dāng)存在函數(shù)u0∈W01,p(Ω)使得在第三章中,我們研究一類p-Kirchhoff型非線性奇異橢圓方程的Dirichlet邊值問題其中,Ω(?)(R~N ≥ 1)是具有光滑邊界的有界區(qū)域,p1,0≤g≤p-1,γ1是實(shí)數(shù),B:R+ → R+ 是具有正下界的 C1 函數(shù),-△pu =-div(|%絬|p-2%絬),h(x)∈ L1(Ω)是一正函數(shù)(即h(x)0幾乎處處于Ω上),k(x∈ L∞(Ω)是非負(fù)函數(shù).對于方程(2),除了具有強(qiáng)奇性(71)外,它的另一個(gè)顯著特點(diǎn)就是二階項(xiàng)的系數(shù)與∫|%絬|pdx有關(guān),因此方程(2)本身不再是一個(gè)逐點(diǎn)意義下的等式.通常B(1/p∫Ω|%絬|pdx)被稱為非局部項(xiàng),方程(2)因此也被稱為非局部方程.正是由于非局部項(xiàng)的存在,我們一般不能從un→a u于W01,p(Ω)中弱收斂直接得到B(1/p∫Ω|%絬n|pdx)→B(1/p∫Ω|%絬|pdx),這也是解決非局部問題的最大困難所在.同第二章一樣,為了克服奇異項(xiàng)以Ω及非局部項(xiàng)所帶來了困難,我們需要構(gòu)造W01,p(Ω)中特殊的集合來保證奇異項(xiàng)的可積性.通過考慮問題(2)相應(yīng)的奇異能量泛函在特殊構(gòu)造的集合上的約束極小問題,借助Ekeland變分原理及一些分析技巧我們得到了極小化序列在W01,p(Ω)中強(qiáng)收斂,給出了問題(2)存在W01,p(Ω)解的充分必要條件.在第三章第二節(jié)中,我們首先就問題(2)中p = 2,k(x 三0的特殊情形進(jìn)行了討論.此時(shí)假設(shè)B還滿足如下條件(B1)B'(s)0,(?)s0.(B2)存在常數(shù)α0,β0,M0,使得B(s)βsα,(?)sM,其中B(s)= ∫0s B(τ)dτ.這部分的主要結(jié)果如下:定理2.設(shè)Ω(?)R~N(N ≥ 1)是具有光滑邊界的有界區(qū)域,p = 2,γ1,k(x)三0,h(x)∈ LQ)是一正函數(shù)數(shù)(即h((x)0 幾乎處處處于Ω上),,B:R+→ R+是具有正下界的C1函數(shù)且滿足假設(shè)條件(B1)和(B2),則問題(2)存在唯一的H01(Ω)解當(dāng)且僅當(dāng)存在函數(shù)u0∈H01(Ω),使得∫Ωh(x)|u0|1-γdx+∞.在第三章第三節(jié)中,我們將問題(2)推廣到了 p1且具非線性增長項(xiàng)的一般情形.此時(shí)假設(shè)B還滿足如下條件(B3)B'(s)0,(?)s0.(B4)存在常數(shù)α ≥ 1+q/p,β0,M0,使得B(s)βsα,(?)sM,其中,B(s)= ∫0s B(τ)dτ.特別地,當(dāng)α = 1+q/p時(shí),我們要求,其中S0是從W01,p(Ω)到Lq+1(Ω)的嵌入常數(shù).這部分的主要結(jié)果如下:定理3.設(shè)Ω(?)R~N(N ≥ 1)是具有光滑邊界的有界區(qū)域,p1,0 ≤ g≤p-1,γ1.,h(x)∈ L1(Ω)是一正函數(shù)(即 h(x)0 幾乎處處于 上),k(x)∈L∞(Ω)是一非負(fù)函數(shù),B:R+ → R+是具有正下界的C1函數(shù)且滿足假設(shè)條件(B3)和(B4),則問題(2)至少存在一個(gè)W01,p(Ω)解當(dāng)且僅當(dāng)存在函數(shù)u0∈W01,p(Ω),使得定理4.當(dāng)k(x)≡0 時(shí),若問題(2)存在W01,p(Ω)解,則該解是唯一的.在第四章中,我們研究一類具退化強(qiáng)制項(xiàng)和自然增長條件梯度項(xiàng)的p-Laplace奇異橢圓方程的Dirichlet邊值問題其中,Ω(?)(N ≥ p)是一有界區(qū)域,p1,γ,θ0,f是某一 Lebesgue空間Lm(Ω)(m ≥ 1)中的非負(fù)函數(shù).注意到當(dāng)u趨于無窮大時(shí),趨于零,因此問題(3)中的微分算子A(u)=在W01,p(Ω)中不是強(qiáng)制的.我們使用截?cái)喾椒?用非退化強(qiáng)制和非奇異算子分別逼近退化強(qiáng)制項(xiàng) 和奇異項(xiàng) 然后通過選取適當(dāng)?shù)臋z驗(yàn)函數(shù)得到逼近解序列{un}的一系列先驗(yàn)估計(jì).最后通過極限過程得到問題(3)解的存在性以及正則性等結(jié)果.這里比較關(guān)鍵的是證明逼近解序列及其梯度的一些強(qiáng)收斂的結(jié)果,對此我們將通過選取合適的檢驗(yàn)函數(shù)來實(shí)現(xiàn).我們的主要結(jié)果如下:定理5.設(shè)0θ1，f是Lm(Ω)中的非負(fù)函數(shù),若(?),則存在一個(gè)在Ω內(nèi)沿革正的函數(shù)u(?),使得(?),且對任意(?)都有(?)定理6.設(shè)0θ1,f是Lm(Ω)中的非負(fù)函數(shù).若N/pN-θ(N-1)mpN/pN-θ(N-p),則存在一個(gè)在Ω內(nèi)嚴(yán)格正德函數(shù)u∈W01,σ(∈),σ=mN(p-θ)/N-θm,使得|%絬|p/uθ∈L1(Ω)且對任意φ∈01(Ω)都有定理7.設(shè)1≤ θp,γθ-1,f是Lm(Ω)中的非負(fù)函數(shù).若且對每個(gè)緊子集ω(?)(?)Ω都有則存在一個(gè)在Ω內(nèi)嚴(yán)格正的函數(shù)u ∈ W01,p(Ω),使得 且對任意φ∈W01,p(Ω)∩L∞(Ω)都有定理8.設(shè)1 ≤ θp,γθ-1,f是Lm(Ω)中的非負(fù)函數(shù).若N/(pN-θ(N-1))mpN/(pN-θ(N-p)),且對每個(gè)緊子集ω(?)(?)Ω都有則存在一個(gè)在Ω內(nèi)嚴(yán)格正的函數(shù)且對任意φ∈C01Ω 都有在第五章中,我們借助位勢井族理論定性地研究幾類具p-Laplace算子的薄膜方程.我們首先研究一類具非局部源項(xiàng) 的p-Laplace型薄膜方程其中,Ω是R中的有界開區(qū)間,T ∈((0,+∞],p1,gmmax{1,p-1},u0 ∈H.這我們首先構(gòu)造問題(4)對應(yīng)的Lyapunov泛函J(u)和Nehari泛函Ⅰ(u),引進(jìn)改進(jìn)的位勢井族.通過分析相應(yīng)泛函和位勢井族,并結(jié)合Galerkin逼近方法及凸方法,我們得到了問題(4)在具次臨界初始能量時(shí),即J(u0)d時(shí)(d為問題(4)相應(yīng)的位勢井的井深),弱解整體存在、有限時(shí)間爆破、有限時(shí)間熄滅的門檻結(jié)果.對于具臨界初始能量J(u0)= d的情形,通過對初值進(jìn)行擾動(dòng),我們也得到了相應(yīng)的結(jié)果.此外,我們也給出了問題(4)弱解的唯一性的證明并對整體弱解的漸近性進(jìn)行了刻畫.最后我們給出了問題(4)的解在有限時(shí)間爆破的數(shù)值模擬.這部分的主要結(jié)果如下:定理 9.設(shè) p1,gmax{1,p-1} u0∈H.若J(u0)d,I(u0)0,問題(4題存在唯一的整體弱解u ∈ L∞(0,∞;H2(Ω)),ut∈ L2(0,∞;L2(Ω)).此外,u不會(huì)在有限時(shí)間熄滅,且定理10.設(shè)p1,qmax{1,P-1},u0∈H,u是問題(4)的弱解.若J(u0)d,I(u00,則存在有限時(shí)間T,使得u在T時(shí)刻爆破,即定理 11.設(shè)p1,qmax{1,p-1},u0 ∈.若J(u0)=d,I(u0)≥ 0,問題(4存在唯一的整體弱解 u ∈ L∞(0,∞;H2(Ω)),ut ∈ L2(0,∞;L2(Ω)),并且 I(u)≥ 0.此外,若對任意t0都有I(u)0,則解不會(huì)在有限時(shí)間熄滅,且否則,解在有限時(shí)間熄滅.定理12.設(shè)p1,qmax{1,p-1},u0∈H,u是問題(4)的弱解,若J(u0)=d，，I(u0)0,則存在有限時(shí)間T,使得u在T時(shí)刻爆破，即接下來,我們將問題(4)的結(jié)果推廣到具非局部源項(xiàng)(?)的p-Kirchhoff型薄膜方程其中，Ω(?)R是有界開區(qū)間,T∈(0,+∞),p1,q2p-1,a0,b0,u0∈H.同樣地，構(gòu)造問題(5)對應(yīng)的Lyapunov泛函J(u)和Nehari泛函I(u),借助位勢井族理論我們得到了與問題(4)平行的結(jié)果.用d表示問題(5)相應(yīng)的位勢井井深，這部分的主要結(jié)果如下：定理 13.設(shè) p1,q2p-1,u0∈H.若J(u0)d,I(u0)0,則問題(5)存在唯一的整體弱解u∈L∞(0,∞;H2(Ω)),ut∈L2(0,∞;L2(Ω)).此外,u不會(huì)在有限時(shí)間熄滅，且定理14.設(shè)p1，q2p-1,u0∈H,u是問題(5)的弱解.若J(u0)d,I(u0)0,則存在有限時(shí)間T,使得u在T時(shí)刻爆破,即定理 15.設(shè)p1,g2p-1,u0 ∈ H.若 J(u0)= d,I(u0)≥ 0,則問題(5)存在唯一的整體弱解 u ∈L∞(0,∞;H2(Ω)),∈L2((0,∞;;L2(Ω)),并且I(u)≥ 0此外,若對任意t0都有I(u)0,則解不會(huì)在有限時(shí)間熄滅,且否則,解在有限時(shí)間熄滅.定理16.設(shè)p1,g-1,u0∈H,u是問題(5)的弱解.若J(u0)= d,I(u0)0,則存在有限時(shí)間T,使得u在T時(shí)刻爆破,即。
[Abstract]:Quasilinear elliptic equations and parabolic equations are two important partial differential equations. The typical examples are p-Laplace equation [1], porous medium equation [2], nonlinear elastic reaction diffusion equation [3], and so on. In recent years, with the research of partial differential equations more deeply and widely, the differential operators are discussed. The form of the p-Laplace operator is becoming more and more complex, and the research on the quasi linear partial differential equation with the operator is widely concerned by the mathematicians at home and abroad. In the early twentieth Century, the Soviet mathematician Sobolev introduced the concept of Sobolev space in text [4,5]. This kind of space has important and extensive application value for the study of partial differential equations. Especially, it plays a very important role in the study of the p-Laplace equation. This paper mainly studies the properties of solutions of several elliptic and parabolic equations with p-Laplace operators, including the existence, uniqueness, regularity, blasting, extinguishing and the long time asymptotic behavior of solutions. The full text is divided into five chapters as introduction. In the second chapter, we study the Dirichlet boundary value problems of a class of p-Laplace singular elliptic equations, in which omega (?) R~N (N > 1) is a bounded domain with a smooth boundary, a standard p-Laplace operator, P1, gamma 1, H (H). X) is a positive function in L1 (i. e. H (x0 is almost on omega). First, due to the strong singularity (1) of the equation (gamma 1), the classical variational method (critical point theory), the upper and lower solutions and the fixed point theorems have some restrictions on the problem (1); and because of the existence of the p-Laplace operator, we can not generally be from UN to W01, P (omega). The weak convergence of the medium and weak convergence is directly obtained in Lp/p-1 (omega, R~N), which also challenges the existence of the solution of the problem (1). In addition, the right end term H (x) plays a vital role in solving the singular elliptic equation, and its state and form often determine the method and complexity of the solution. The nonhomogeneous term H (x) in question (1) is L The function in 1 (omega) has a weak regularity, which also brings great difficulty for solving the singular elliptic problem. In order to overcome the above difficulties, we restrict the problem (1) to the integrability of the singular terms by constructing the appropriate set (including the Nehari manifold) in the W01, P (omega), and then consider the corresponding singularity. The constrained minimum problem on this set of energy functional. By means of the Ekeland variational principle and some analytical techniques we have obtained the sufficient and necessary conditions for the existence of W01, P (omega) solutions. In addition, the uniqueness of the problem (1) W01, P (omega) solution is proved by the monotonicity of the p-Laplace operator. The main results of this chapter are as follows: theorem 1. set omega (?) R~N > 1) It is a bounded region with a smooth boundary, P1, gamma 1, 1 (H is a positive function (H (x) 0 almost on omega), then the problem (1) exists only W01, P (omega) solution if and only if there is a function U0 W01, P (omega) makes the Dirichlet boundary value problem of a class of p-Kirchhoff Nonlinear Singular Elliptic Equations in the third chapter, omega (?) (R~N > 1) It is a bounded region with a smooth boundary, p1,0 < g < P-1, gamma 1 is real, B:R+ to R+ is a C1 function with positive and lower bounds, delta Pu =-div, H (x) L1 (omega) is a positive function (i.e. 2) is a non negative function. For the equation (2), besides the strong singularity (71), it is another remarkable special. The point is that the coefficient of the two order is related to the |pdx, so the equation (2) itself is no longer an equation in point by point. Generally, the B (1/p) is called a non local term, so the equation (2) is also called a non local equation. It is because of the existence of a non local term that we can not generally be directly obtained from the weak convergence of UN to a U in W01 and P (omega). To the second chapter, to overcome the difficulty of the singular term in order to overcome the difficulty of Omega and non local terms, we need to construct a special set of W01, P (omega) to guarantee the integrability of the singular terms in the second chapter, as in the second chapter. By considering the singular energy of the problem (2), the corresponding singular energy is considered. With the help of the Ekeland variational principle and some analytical techniques, we have obtained the strong convergence of the minimized sequence in W01, P (omega) with the help of the Ekeland variational principle and some analytical techniques. We give the sufficient and necessary conditions for the problem (2) the existence of W01, P (omega). In the third chapter second, we first have a special case of P = 2 in the problem (2), K (x three 0). It is assumed that B also satisfies the following conditions (B1) B'(s) 0, (?) s0. (B2) exists constant alpha 0, beta 0, M0, which makes B (s) beta s alpha and sM, and the main results of this part are as follows: theorem 2. set omega (?) is a bounded region with smooth boundaries, 2, gamma 1, three 0, 0 It is on omega) that B:R+ - R+ is a C1 function with positive and lower bounds and satisfies the hypothesis (B1) and (B2), then the problem (2) exists the only H01 (omega) solution if and only if there is a function U0 H01 (omega), which makes the H (x) |u0|1- gamma dx+ infinity. In the third chapter third, we generalize the problem (2) to the general case of nonlinear growth. It is assumed that B also satisfies the following conditions (B3) B'(s) 0, (?) s0. (B4) exists constant alpha > 1+q/p, beta 0, M0, which makes B (s) beta s alpha, (?) sM. The bounded region of the slippery boundary, p1,0 < g < P-1, gamma 1., H (x) L1 (omega) is a positive function (i.e. H (x) 0), K (x), L infinity (omega) is a nonnegative function. Theorem 4. when K (x) 0, if the problem (2) exists W01, P (omega) solution, then the solution is unique. In the fourth chapter, we study a class of Dirichlet boundary value problems of a class of p-Laplace Singular Elliptic Equations with degenerate coercion and natural growth condition gradient term, which is a bounded region, P1, gamma, theta 0, f is a Lebesgue space Lm (omega) The non negative function in more than 1. Notice that when u tends to infinity, it tends to zero, so the differential operator A (U) = in the problem (3) is not mandatory in W01, P (omega). We use the truncation method to approximate the degenerate coercion and singular terms with the non degenerate coercive and non singular operators, and obtain the approximate solution sequence by selecting the appropriate test function. A series of prior estimates of {un}. Finally, the existence and regularity of the problem (3) are obtained by the limit process. The key is to prove that some strong convergence results of the approximation solution sequence and its gradient are proved, and we will realize the results by selecting the appropriate test function. Our main results are as follows: theorem 5. set 0 theta 1, f is The non negative function in Lm (?), if (?), there is a function U (?) in the inner evolution of Omega, making (?), and having (?) theorem 6. set 0 theta 1, f is a non negative function in Lm (omega). If N/pN- theta (N-1) mpN/pN- theta (N-p), there is a strictly positive moral function u W01, sigma, sigma =mN (p- theta) and theta (p- theta) in Omega. Meaning 01 (omega) has theorem 7. set 1 less than theta P, gamma theta -1, f is a non negative function in Lm (omega). If and for every compact subset omega (?) Omega there is a strictly positive function of u W01, P (omega) in Omega, so that there is a theorem 8. for W01, P (omega) L infinity (omega). N/ (pN- theta (N-p)), and every compact subset omega (?) Omega has a strictly positive function in Omega and there are fifth chapters on arbitrary C01 Omega. By means of the potential well family theory, we qualitatively study several kinds of thin film equations with p-Laplace operators. We first study a class of p-Laplace thin film equations with non local source terms. Omega is a bounded open interval in R, T ((0, + infinity], P1, gmmax{1, p-1}, U0 H.). We first construct a problem (4) the corresponding Lyapunov functional J (U) and Nehari functional I (U), and introduce the improved potential well family. By analyzing the corresponding functional and potential well family, and junction approximation method and convex method, we get the problem (4) at the subcritical initial stage. When the initial energy is J (U0) d (D as a problem (4) the well depth of the corresponding potential well), the weak solution exists as a whole, the finite time blasting, the threshold of the finite time extinguishing. For the case with critical initial energy J (U0) = D, we also get the corresponding results by disturbing the initial value. In addition, we also give the uniqueness of the problem (4) the weak solution. We describe the asymptotic property of the whole weak solution. Finally, we give the numerical simulation of the solution of the problem (4) in the finite time blasting. The main results of this part are as follows: theorem 9. set P1, gmax{1, p-1} U0 H. if J (U0) d, I (U0) 0, problem (4 problems exist only one integral weak solution L infinity L infinity (0, infinity; omega)). In addition, u will not be extinguished at a limited time, and theorem 10. set P1, qmax{1, P-1}, U0 H, u is a weak solution to the problem (4). And, and I (U) > 0. in addition, if there is I (U) 0 for any T0, then the solution will not be extinguished at finite time, and otherwise, the solution is extinguished in finite time. Theorem 12. set P1, qmax{1, p-1}, U0 H, u is the weak solution of the problem (4), then there is a finite time blasting, that is, then, we extend the result of the problem (4) to the tool. The non local source term (?) p-Kirchhoff type film equation, in which omega (?) R is a bounded open interval, T (0, + infinity), P1, q2p-1, A0, B0, U0 H., and the corresponding Lyapunov functional J (U) and the functional theorem, we get the parallel result with the problem (4) by the potential well family theory. The main results of this part are as follows: theorem 13. set up P1, q2p-1, U0 H. if J (U0) d, I (U0) 0, then the problem (5) exists the only global weak solution u L L (0, infinity; H2 (omega)). Besides, it will not be extinguished at the limited time, and it is the weak solution of the problem (5). U at T time blasting, that is, theorem 15. set P1, g2p-1, U0 H. if J (U0) = D, I (U0) > 0, then the problem (5) exists the only integral weak solution (0, infinity (omega)), and (0, infinity; (omega)), and the solution will not be extinguished at a finite time, and otherwise, the solution will be extinguished at a limited time. Otherwise, the solution is extinguished at a limited time. Theorem 16. set out, theorem 16. set out, theorem 16. set out, theorem 16. set out, theorem 16. set out, theorem 16. set out, theorem 16. set out, theorem 16. set out 1, U0, H, u are the weak solutions of the problem (5). If J (U0) = D, I (U0) 0, there is a finite time T, which makes the U blow up at the time of the T.
【學(xué)位授予單位】：吉林大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O175

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