几类具p-Laplace算子的椭圆型和抛物型方程解的研究

发布时间：2018-06-17 06:08

本文选题：p-Kirchhoff型方程 + 薄膜方程　；参考：《吉林大学》2017年博士论文

【摘要】：拟线性椭圆型方程和抛物型方程是两类重要的偏微分方程,比较典型的例子有流体力学中的p-Laplace方程[1],多孔介质方程[2],非线性弹性反应扩散方程[3]等.近年来,随着人们对偏微分方程的研究更加地深入和广泛,所讨论的微分算子的形式也越来越复杂化,关于具p-Laplace算子的拟线性偏微分方程的研究得到了国内外数学家们的广泛关注.二十世纪初苏联数学家Sobolev在文[4,5]中引入了 Sobolev空间的概念,这类空间对偏微分方程的研究具有重要和广泛的应用价值,尤其是对p-Laplace方程的研究起着非常关键的作用.本文主要对几类具p-Laplace算子的椭圆型和抛物型方程解的性质进行研究.包括解的存在性、唯一性、正则性、爆破、熄灭以及解的长时间渐近行为等.全文共分为五第一章为绪论.介绍本文所研究的主要内容,研究现状及本文所研究问题需要克服的典型困难和使用的主要方法等.在第二章中,我们研究一类p-Laplace奇异椭圆方程的Dirichlet边值问题其中,Ω(?)R~N(N ≥ 1)是具有光滑边界的有界区域,是标准的p-Laplace算子,p1,γ1,h(x)是L1(Ω)中的正函数(即h(x0几乎处处于Ω上).首先由于方程(1)具有强奇性(γ1),经典的变分法(临界点理论),上下解方法以及不动点定理等常用方法对于问题(1)具有一定的限制性;又由于p-Laplace算子的存在,我们一般不能从un→u于W01,p(Ω)中弱收敛直接得到于Lp/p-1(Ω,R~N)中弱收敛,这也为我们解决问题(1)解的存在性提出了挑战;此外,求解奇异椭圆方程时,方程右端项h(x)起着至关重要的作用,它的性态和形式往往会决定求解的方法和复杂程度.问题(1)中的非齐次项h(x)是L1(Ω)中函数,具有较弱的正则性,也为我们求解奇异椭圆问题带来了很大的困难.为了克服以上困难,我们通过构造W01,p(Ω)中适当的集合(包含Nehari流形作为特殊情形),将问题(1)限制在此集合上来保证奇异项的可积性,然后考虑相应奇异能量泛函在此集合上的约束极小问题.借助Ekeland变分原理和一些分析技巧我们得到了问题(1)存在W01,p(Ω)解的充分必要条件.此外,借助p-Laplace算子的单调性证明了问题(1)W01,p(Ω)解的唯一性.这一章的主要结果如下:定理1.设Ω(?)R~N ≥ 1)是具有光滑边界的有界区域,p1,γ1,∈ 1(h是正函数(即h(x)0几乎处处于Ω上),则问题(1)存在唯一的W01,p(Ω)解当且仅当存在函数u0∈W01,p(Ω)使得在第三章中,我们研究一类p-Kirchhoff型非线性奇异椭圆方程的Dirichlet边值问题其中,Ω(?)(R~N ≥ 1)是具有光滑边界的有界区域,p1,0≤g≤p-1,γ1是实数,B:R+ → R+ 是具有正下界的 C1 函数,-△pu =-div(|%絬|p-2%絬),h(x)∈ L1(Ω)是一正函数(即h(x)0几乎处处于Ω上),k(x∈ L∞(Ω)是非负函数.对于方程(2),除了具有强奇性(71)外,它的另一个显著特点就是二阶项的系数与∫|%絬|pdx有关,因此方程(2)本身不再是一个逐点意义下的等式.通常B(1/p∫Ω|%絬|pdx)被称为非局部项,方程(2)因此也被称为非局部方程.正是由于非局部项的存在,我们一般不能从un→a u于W01,p(Ω)中弱收敛直接得到B(1/p∫Ω|%絬n|pdx)→B(1/p∫Ω|%絬|pdx),这也是解决非局部问题的最大困难所在.同第二章一样,为了克服奇异项以Ω及非局部项所带来了困难,我们需要构造W01,p(Ω)中特殊的集合来保证奇异项的可积性.通过考虑问题(2)相应的奇异能量泛函在特殊构造的集合上的约束极小问题,借助Ekeland变分原理及一些分析技巧我们得到了极小化序列在W01,p(Ω)中强收敛,给出了问题(2)存在W01,p(Ω)解的充分必要条件.在第三章第二节中,我们首先就问题(2)中p = 2,k(x 三0的特殊情形进行了讨论.此时假设B还满足如下条件(B1)B'(s)0,(?)s0.(B2)存在常数α0,β0,M0,使得B(s)βsα,(?)sM,其中B(s)= ∫0s B(τ)dτ.这部分的主要结果如下:定理2.设Ω(?)R~N(N ≥ 1)是具有光滑边界的有界区域,p = 2,γ1,k(x)三0,h(x)∈ LQ)是一正函数数(即h((x)0 几乎处处处于Ω上),,B:R+→ R+是具有正下界的C1函数且满足假设条件(B1)和(B2),则问题(2)存在唯一的H01(Ω)解当且仅当存在函数u0∈H01(Ω),使得∫Ωh(x)|u0|1-γdx+∞.在第三章第三节中,我们将问题(2)推广到了 p1且具非线性增长项的一般情形.此时假设B还满足如下条件(B3)B'(s)0,(?)s0.(B4)存在常数α ≥ 1+q/p,β0,M0,使得B(s)βsα,(?)sM,其中,B(s)= ∫0s B(τ)dτ.特别地,当α = 1+q/p时,我们要求,其中S0是从W01,p(Ω)到Lq+1(Ω)的嵌入常数.这部分的主要结果如下:定理3.设Ω(?)R~N(N ≥ 1)是具有光滑边界的有界区域,p1,0 ≤ g≤p-1,γ1.,h(x)∈ L1(Ω)是一正函数(即 h(x)0 几乎处处于上),k(x)∈L∞(Ω)是一非负函数,B:R+ → R+是具有正下界的C1函数且满足假设条件(B3)和(B4),则问题(2)至少存在一个W01,p(Ω)解当且仅当存在函数u0∈W01,p(Ω),使得定理4.当k(x)≡0 时,若问题(2)存在W01,p(Ω)解,则该解是唯一的.在第四章中,我们研究一类具退化强制项和自然增长条件梯度项的p-Laplace奇异椭圆方程的Dirichlet边值问题其中,Ω(?)(N ≥ p)是一有界区域,p1,γ,θ0,f是某一 Lebesgue空间Lm(Ω)(m ≥ 1)中的非负函数.注意到当u趋于无穷大时,趋于零,因此问题(3)中的微分算子A(u)=在W01,p(Ω)中不是强制的.我们使用截断方法,用非退化强制和非奇异算子分别逼近退化强制项和奇异项然后通过选取适当的检验函数得到逼近解序列{un}的一系列先验估计.最后通过极限过程得到问题(3)解的存在性以及正则性等结果.这里比较关键的是证明逼近解序列及其梯度的一些强收敛的结果,对此我们将通过选取合适的检验函数来实现.我们的主要结果如下:定理5.设0θ1，f是Lm(Ω)中的非负函数,若(?),则存在一个在Ω内沿革正的函数u(?),使得(?),且对任意(?)都有(?)定理6.设0θ1,f是Lm(Ω)中的非负函数.若N/pN-θ(N-1)mpN/pN-θ(N-p),则存在一个在Ω内严格正德函数u∈W01,σ(∈),σ=mN(p-θ)/N-θm,使得|%絬|p/uθ∈L1(Ω)且对任意φ∈01(Ω)都有定理7.设1≤ θp,γθ-1,f是Lm(Ω)中的非负函数.若且对每个紧子集ω(?)(?)Ω都有则存在一个在Ω内严格正的函数u ∈ W01,p(Ω),使得且对任意φ∈W01,p(Ω)∩L∞(Ω)都有定理8.设1 ≤ θp,γθ-1,f是Lm(Ω)中的非负函数.若N/(pN-θ(N-1))mpN/(pN-θ(N-p)),且对每个紧子集ω(?)(?)Ω都有则存在一个在Ω内严格正的函数且对任意φ∈C01Ω 都有在第五章中,我们借助位势井族理论定性地研究几类具p-Laplace算子的薄膜方程.我们首先研究一类具非局部源项的p-Laplace型薄膜方程其中,Ω是R中的有界开区间,T ∈((0,+∞],p1,gmmax{1,p-1},u0 ∈H.这我们首先构造问题(4)对应的Lyapunov泛函J(u)和Nehari泛函Ⅰ(u),引进改进的位势井族.通过分析相应泛函和位势井族,并结合Galerkin逼近方法及凸方法,我们得到了问题(4)在具次临界初始能量时,即J(u0)d时(d为问题(4)相应的位势井的井深),弱解整体存在、有限时间爆破、有限时间熄灭的门槛结果.对于具临界初始能量J(u0)= d的情形,通过对初值进行扰动,我们也得到了相应的结果.此外,我们也给出了问题(4)弱解的唯一性的证明并对整体弱解的渐近性进行了刻画.最后我们给出了问题(4)的解在有限时间爆破的数值模拟.这部分的主要结果如下:定理 9.设 p1,gmax{1,p-1} u0∈H.若J(u0)d,I(u0)0,问题(4题存在唯一的整体弱解u ∈ L∞(0,∞;H2(Ω)),ut∈ L2(0,∞;L2(Ω)).此外,u不会在有限时间熄灭,且定理10.设p1,qmax{1,P-1},u0∈H,u是问题(4)的弱解.若J(u0)d,I(u00,则存在有限时间T,使得u在T时刻爆破,即定理 11.设p1,qmax{1,p-1},u0 ∈.若J(u0)=d,I(u0)≥ 0,问题(4存在唯一的整体弱解 u ∈ L∞(0,∞;H2(Ω)),ut ∈ L2(0,∞;L2(Ω)),并且 I(u)≥ 0.此外,若对任意t0都有I(u)0,则解不会在有限时间熄灭,且否则,解在有限时间熄灭.定理12.设p1,qmax{1,p-1},u0∈H,u是问题(4)的弱解,若J(u0)=d，，I(u0)0,则存在有限时间T,使得u在T时刻爆破，即接下来,我们将问题(4)的结果推广到具非局部源项(?)的p-Kirchhoff型薄膜方程其中，Ω(?)R是有界开区间,T∈(0,+∞),p1,q2p-1,a0,b0,u0∈H.同样地，构造问题(5)对应的Lyapunov泛函J(u)和Nehari泛函I(u),借助位势井族理论我们得到了与问题(4)平行的结果.用d表示问题(5)相应的位势井井深，这部分的主要结果如下：定理 13.设 p1,q2p-1,u0∈H.若J(u0)d,I(u0)0,则问题(5)存在唯一的整体弱解u∈L∞(0,∞;H2(Ω)),ut∈L2(0,∞;L2(Ω)).此外,u不会在有限时间熄灭，且定理14.设p1，q2p-1,u0∈H,u是问题(5)的弱解.若J(u0)d,I(u0)0,则存在有限时间T,使得u在T时刻爆破,即定理 15.设p1,g2p-1,u0 ∈ H.若 J(u0)= d,I(u0)≥ 0,则问题(5)存在唯一的整体弱解 u ∈L∞(0,∞;H2(Ω)),∈L2((0,∞;;L2(Ω)),并且I(u)≥ 0此外,若对任意t0都有I(u)0,则解不会在有限时间熄灭,且否则,解在有限时间熄灭.定理16.设p1,g-1,u0∈H,u是问题(5)的弱解.若J(u0)= d,I(u0)0,则存在有限时间T,使得u在T时刻爆破,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.
【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

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