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Nonlinear Evolution Equations and Elliptic Equations

非线性演化方程和椭圆方程

基本信息

批准号：
09440070
负责人：
OTANI Mitsuharu
金额：
$ 7.49万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

Elliptic Equations (1) Concerning the equation (E) - △u = |u|ィイD1q-2ィエD1u x ∈Ω, u(x) = 0 x ∈∂Ω we obtained the following results.Let Ω = RィイD1NィエD1＼BィイD2R1ィエD2, BィイD2RィエD2 = {x ∈ IRィイD1NィエD1 ; |x|【less than or equal】 R }, 2ィイD1*ィエD1<q< +∞ (2ィイD1*ィエD1 is the critical exponent for Sobolev's embedding HィイD31(/)0ィエD3 (Ω) ⊂ LィイD1qィエD1 (Ω) ), then (E) admits a radially symmetric solution in HィイD11ィエD1 (Ω) ∩ LィイD1qィエD1 (Ω). This fact has been conjectured from the duality between bounded domains and exterior domains.(II) Consider the equation : (E)ィイD2λィエD2 -△u = λu + |u|ィイD1q-2ィエD1u x ∈Ω, u(x) = 0 x ∈∂Ω (1) Let Ω = ΩィイD2dィエD2 × λRィイD1N-dィエD1, (ΩィイD2dィエD2 is a bounded domain in IRィイD1dィエD1), q = 2ィイD1*ィエD1, d【greater than or equal】 1, N 【greater than or equal】 4, then for all λ ∈ (0, λィイD21ィエD2), λィイD21ィエD2 = infィイD2v∈HィイD31(/)0ィエD3 (Ω)ィエD2‖∇ィイD2uィエD2‖LィイD42ィエD4ィイD12ィエD1/‖u‖LィイD42ィエD4ィイD22ィエD2 > 0, (E)ィイD2λィエD2 has a nontrivial solution, which gives a generalization of the well-known result of … More Brezis-Nirenberg to unbounded cylinders. (2) Let Ω = ΩィイD2dィエD2 x RィイD1N-dィエD1 and let ΩィイD2dィエD2 be a d-dimensional annulus. ・ If q 【greater than or equal】 NィイD2dィエD2 = 2 (N -d+1)/(N-d+1-2) , then (E)ィイD2λィエD2 admits no nontrivial weak solution.・ If q < NィイD2dィエD2, then (E)ィイD2λィエD2 admits a nontrivial weak solution.These results reveal the fact that the d-dimensional symmetry reduces the effective dimension by (d-1).(III) Consider (E)ィイD21ィエD2 -Δu + u = a(x) |u|ィイD1q-2ィエD1u + f(x) x ∈ IRィイD1NィエD1, 2 < q < 2ィイD1*ィエD1 o < a(x), |a(x) - 1| 【less than or equal】 CeィイD1λ|x|ィエD1, λ > 0 It is shown that if ‖f‖ィイD2H-1(RィイD1NィエD1)ィエD2 is sufficiently small, then (E)ィイD21ィエD2 has at least two positive solutions. Furthermore, we found that for the case where f = 0 and q < 2ィイD1*ィエD1 is close enugh to 2ィイD1*ィエD1,the multiplicity of positive solutions depends upon the topological property (su as category) of the set {x ∈Ω ; u(x) = maxィイD2x∈ΩィエD2 }.The analysis of this phenomenon will be an interesting subject to study in future.Parabolic Equations (I) It has been well known that weak solutions of porous medium equations enjoy the Holder continuity. However, the existence of smooth (local) solutions has been left as an open problem for long time. Otani-Sugiyama gave an affirmative answer to this open problem, by developing the LィイD1∞ィエD1-energy method, which was introduce by themselves to show the local existence of WィイD11,∞ィエD1-solutions for more general doubly nonlinear parabolic equations. This is the most fascinating result among our results obtained in this reseach project.(II) It was left as an unsolved problem to determine the asymptotic behabiour of solutions of (P) uィイD2tィエD2, -Δu = |u|ィイD12ィイD1*ィエD1-2ィエD1u x∈Ω, u(x) = 0 x∈∂Ω. To this problem, the following partial answer was obtained. 「Let Ω = {x ∈ RィイD1nィエD1 : |x|< 1 } and the solution u (x.t) be positive, radially symmetric and monotone decreasing with respect to r = |x|. Then u blows up in a finite time or becomes a global solution and satisfies the following property : 「There exists a sequence {tィイD2nィエD2 } such that |∇u (x,tィイD2nィエD2)|ィイD12ィエD1 - CoィイD1δィエD1(0) (u - x), |u (x,tィイD2nィエD2)|ィイD12ィエD1 - CoィイD1σィエD1(0) (u - x). 」 This result give some information about the problem above to some extent. However, since strong technical condtions are assumed. We need further in vestigation to solve this problem in a natural setting. Less

对于方程(E) -△u = |u| x∈Ω， u(x) = x∈∂Ω，我们得到如下结果。让Ω= RィイD1NィエD1 \ BィイD2R1ィエD2, BィイD2RィエD2 = {x∈红外ィイD1NィエD1;|x|【小于或等于】R}， 2 D1* D1<q< +∞(2 D1* D1是Sobolev嵌入H D31（/)0 D3 （Ω） L D1q D1 （Ω））的临界指数，那么(E)在H D11 D1 （Ω）∩L D1q D1 （Ω）中承认一个径向对称解。这一事实是由有界域与外域之间的对偶性推测出来的。（二）考虑如下等式：(E)ィイD2λィエD2 -△u =λu + | |ィイD1q-2ィエD1u x∈Ω,u (x) = 0 x∈∂Ω(1)让Ω=ΩィイD2dィエD2×λRィイD1N-dィエD1,(ΩィイD2dィエD2是红外的有限域ィイD1dィエD1), q = 2ィイD1 *ィエD1, d【大于或等于】1,N【大于或等于】4,然后对所有λ∈(0,λィイD21ィエD2),λィイD21ィエD2 = infィイD2v∈HィイD31(/) 0ィエD3(Ω)ィエD2为∇ィイD2uィエD2为LィイD42ィエD4ィイD12ィエD1 /为u为LィイD42ィエD4ィイD22摊位ィエD2 > 0, (E)ィイD2λィエD2非平凡解,它将著名的…More Brezis-Nirenberg结果推广到无界圆柱体。(2)设Ω = Ω D2d D2 × R D1N-d D1，设Ω D2d D2为d维环。·如果q【大于等于】N D2d D2 = 2 (N-d+1)/(N-d+1-2)，则(E) D2λ D2不存在非平凡弱解。·如果q < N μ u \ \ D2d μ u \ D2，则(E) μ u \ D2λ μ u \ D2存在一个非平凡弱解。这些结果揭示了d维对称性使有效维数减少（d-1）的事实。(3)考虑(E)ィイD21ィエD2 -Δu + u = u (x) | |ィイD1q-2ィエD1u + f (x) x∈红外ィイD1NィエD1, 2 < q < 2ィイD1 *ィエD1 o < (x), | (x) - 1 |【小于或等于】CeィイD1λx | |ィエD1,λ> 0表明如果为f为ィイD2H-1 (RィイD1NィエD1)ィエD2是足够小,那么(E)ィイD21ィエD2至少有两个正解。进一步地，我们发现对于f = 0且q < 2 D1* D1足够接近于2 D1* D1的情况，正解的多重性依赖于集合{x∈Ω；u(x) = max D2x∈Ω D2}。对这一现象的分析将是未来一个有趣的研究课题。抛物方程（一）多孔介质方程的弱解具有Holder连续性。然而，光滑（局部）解的存在性长期以来一直是一个悬而未决的问题。对于这个开放问题，Otani-Sugiyama给出了一个肯定的答案，他提出了一种能量法，用于证明更一般的双非线性抛物方程的W′′_(_)_(_)_(_)_(_)_(_)_(_)_(_)_(_)_（）解的局部存在性。这是我们在这个研究项目中得到的最令人着迷的结果。(2)它是一个尚未解决的问题(P)的确定解的渐近behabiour uィイD2tィエD2 -Δu = | |ィイD12ィイD1 *ィエD1-2ィエD1u x∈Ω,u (x) = 0 x∈∂Ω。对于这个问题，得到了以下部分答案。令Ω = {x∈R γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ。然后u炸毁在一个有限的时间或成为一个全球性的解决方案,满足以下属性:“存在一个序列{tィイD2nィエD2}这样|∇u (x, tィイD2nィエD2) |ィイD12ィエD1 - CoィイD1δィエD1 (0) (u - x) | u (x, tィイD2nィエD2) |ィイD12ィエD1 - CoィイD1σィエD1 (0) (u - x)”。这一结果给出一些信息上面的问题在某种程度上。但是，由于假定有很强的技术条件。为了在自然环境中解决这个问题，我们需要进一步的调查。少