共查询到20条相似文献,搜索用时 15 毫秒
1.
一个山路引理的应用 总被引:5,自引:0,他引:5
本文主要考虑如下形式的Dirichlet问题-△u(x)=f(x,u),x∈Ω,∈H01(Ω),其中f(x,t)∈C(Ω×R),f(x,t)/t关于t单调不减,并且当t∈R时关于x∈Ω一致趋向于某个L∞函数q(x)(此时,称f(x,t)关于t在无穷远处是渐近线性的).显然,在该条件下常用的Ambrosetti-Rabinowitz型条件,即关于所有的|s|>M和x∈Ω,0<θF(x,s)2,M>0为常数, F(x,s)=∫0s f(x,t)dt. 众所周知,条件(AR)在山路引理的应用中起着非常重要的作用.本文通过应用一种改进了的山路引理在没有条件(AR)的情况下来证明上面Dirichlet问题(P)也有正解存在。此方法也适用于f(x,t)关于t在无穷远处是超线性,即q(x)≡+∞的情形. 相似文献
2.
Cauchy Problem for Semilinear Wave Equations in Four Space Dimensions with Small Initial Data 下载免费PDF全文
Yi Zhou 《偏微分方程(英文版)》1995,8(2):135-144
In this paper, we consider the Cauchy problem ◻u(t,x) = |u(t,x)|^p, (t,x) ∈ R^+ × R^4 t = 0 : u = φ(x), u_t = ψ(x), x ∈ R^4 where ◻ = ∂²_t - Σ^4_{i=1}∂²_x_i, is the wave operator, φ, ψ ∈ C^∞_0 (R^4). We prove that for p > 2 the problem has a global solution provided tile initial data is sufficiently small. 相似文献
3.
In this paper we study the initial boundary value problem of GBBM equations on unbounded domain u_t - Δu_t = div f(u) u(x,0) = u_0(x) u|_{∂Ω} = 0 and corresponding Cauchy problem. Under the conditions: f( s) ∈ C^sup1 and satisfies (H)\qquad |f'(s)| ≤ C|s|^ϒ, 0 ≤ ϒ ≤ \frac{2}{n-2} if n ≥ 3; 0 ≤ ϒ < ∞ if n = 2 u_0(x) ∈ W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω)(W^{2,p}(R^n) ∩ W^{2,2}(R^n) for Cauchy problem), 2 ≤ p < ∞, we obtain the existence and uniqueness of global solution u(x, t) ∈ W^{1,∞}(0, T; W^{2,p}(Ω) ∩ W^{2,2}(Ω) ∩ W^{1,p}_0(Ω))(W^{1,∞}(0, T; W^{2,p}(R^n) ∩ W^{2,2} (R^n)) for Cauchy problem), so the results of [1] and [2] are generalized and improved in essential. 相似文献
4.
刘衍胜 《应用泛函分析学报》2004,6(3):193-199
考虑下述奇异半线性反应扩散方程初值问题(()-1-t△u=ut+f(x),t>0,x∈RN
lim u(t,x)=0,x∈RN t→0=)其中r>0,△=∑( )/( )x2i,f(x)非负且f(x)∈L∞(RN).首先利用增算子不动点定理,重新证明了IVP在(0,+∞)上至少存在一个非负解,并给出了IVP解的迭代逼近序列.其次获得了一个有关IVP(1)正解的无限增长性的结果.最后,证明了当r>1时,去掉条件1/r-1≥n/2,IVP的正解u(t)同样会产生爆破.研究结果表明情形limut→+∞(t,x)=+∞不会出现. 相似文献
5.
<正> Cauchy问题的唯一性是偏微分方程的基本问题之一.经典的Cauchy-Kowalewski定理断言,解析方程或方程组的解析解是唯一的.1901年,Holmgren证明了,线性的解析方程或方程组的光滑解的唯一性.在取消关于系数的解析性的假设这个方向上的第一个结果是由Carleman在1939年给出的,他证明了两个自变量的相应结果,其中假设方程的主部的系数是实的,以及特征根是单重的,因而特征根的虚部如果不恒为零则总不为零. 相似文献
6.
设u(x,t)=(SΩf)(x,t)是一般色散初值问题(?)tu-iΩ(D)u=0,u(x,0)=f(x),(x,t)∈Rn×R的解,SΩ*f,SΩ**f是它的局部和整体极大算子.本文给出它们范数的若干估计. 相似文献
7.
在本文中,我们讨论了u∧(-Δu+λ(u,n)n)=0,|u|=1,x∈B1和u=(x1,x2,0)x∈B1的轴对称解uλ的渐近行为,其中B1是R2中单位圆,u=(u1,u2,u3).我们证明了uλu∞在H2B1\Bε,R3中. 相似文献
8.
半线性椭圆型问题爆炸解的存在性与渐近行为 总被引:1,自引:0,他引:1
设Ω是RN(N≥3)中的C2有界区域,f是单调非减的非负连续可微函数满足f'(a)∫a∞1/f(s)ds≤C0, a>0.应用一种新型的非线性变换w(x)=∫u(x)∞ ds/f(s)将爆炸解问题△u=k(x)f(u),u>0,x∈Ω,u| Ω=∞转化成等价的带奇异项的Dirichlet问题,不仅得到了爆炸解问题解的最小爆炸速度,而且揭示了两类典型非线性爆炸解问题基本上是相同的.应用摄动方法,上下解方法得到了爆炸解的存在性.特别允许非线性项的系数不仅在Ω的内部子区域恒为零而且在Ω上可适当无界.随后再应用摄动方法,将所得结果推广到无界区域,得到了整体爆炸解的存在性以及在无穷远附近的最小爆炸速度(有关文献参见[1-33]). 相似文献
9.
研究具有阻尼的半线性波动方程的初边值问题u_(tt)-△u+βu_t=|u|~(p-1)u,x∈Ω,t>0u(x,0)=u_0(x),u_t(x,0)=u_1(x),x∈Ωu|_((?)Ω)=0,t≥0其中γ为正常数,Ω■R~n为有界域,当n≥3时,1
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10.
本文研究快速扩散方程ut-Δum +| u|p =0的柯西问题 ,其中m ,p∈ ( 0 ,1) .对于 0
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11.
Ning Zhu 《偏微分方程(英文版)》1996,9(2):129-138
In this paper, we consider the Cauchy problem \frac{∂u}{∂t} = Δφ(u) in R^N × (0, T] u(x,0} = u_0(x) in R^N where φ ∈ C[0,∞) ∩ C¹(0,∞), φ(0 ) = 0 and (1 - \frac{2}{N})^+ < a ≤ \frac{φ'(s)s}{φ(s)} ≤ m for some a ∈ ((1 - \frac{2}{n})^+, 1), s > 0. The initial value u_0 (z) satisfies u_0(x) ≥ 0 and u_0(x) ∈ L¹_{loc}(R^N). We prove that, under some further conditions, there exists a weak solution u for the above problem, and moreover u ∈ C^{α, \frac{α}{2}}_{x,t_{loc}} (R^N × (0, T]) for some α > 0. 相似文献
12.
REN Guangbin Department of Mathematics University of Science Technology of China Hefei China 《中国科学A辑(英文版)》2005,48(Z1)
Let Ωbe a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ωwhich are Dunkl polyharmonic, i.e. (△h)nf =0 for some integer n. Here △h=∑j=1N Dj2 is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, where kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x) |x|2f1(x) … |x|2(n-1)fn-1(x),(?)x∈Ω, where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition. 相似文献
13.
REGULARIZATION OF AN ILL-POSED HYPERBOLIC PROBLEM AND THE UNIQUENESS OF THE SOLUTION BY A WAVELET GALERKIN METHOD 下载免费PDF全文
We consider the problem K(x)u xx = u tt , 0 < x < 1, t ≥ 0, with the boundary condition u(0,t) = g(t) ∈ L 2 (R) and u x (0, t ) = 0, where K(x) is continuous and 0 < α≤ K (x) < +∞. This is an ill-posed problem in the sense that, if the solution exists, it does not depend continuously on g. Considering the existence of a solution u(x, ) ∈ H 2 (R) and using a wavelet Galerkin method with Meyer multiresolution analysis, we regularize the ill-posedness of the problem. Furthermore we prove the uniqueness of the solution for this problem. 相似文献
14.
在本文中我们考虑下列非线性扩散方程在时间充分长时的性态ut=(φ(u))xx+φ(u),(x∈R,t∈R+=(0,+∞))其中函数φ(u)和φ(u)允许此方程具有行波解.首先我们给出该方程柯西问题的广义解的存在性、唯一性和一些比较原理.然后给定φ(u)的某些条件,我们证明了一些阀值效应.由这些结果我们可以看到在这些假设条件下,静态解u=a稳定的,而u=0或u=1是不稳定的,等等. 相似文献
15.
Fuxia Cheng 《偏微分方程(英文版)》1997,10(3):275-283
M. Bertsch & R. Dal Passo proved the existence and uniqueness of the Cauchy problem for u_t = (φ(u),ψ(u_x))_x, where φ > 0, ψ is a strictly increasing function with lim_{s → ∞}ψ(s) = ψ_∞ < ∞. The regularity of the solution has been obtained under the condition φ" < 0 or φ = const. In the present paper, under the condition φ" ≤ 0, we give some regularity results. We show that the solution can be classical after a finite time. Further, under the condition φ" ≤ -α_0 (where -α_0 is a constant), we prove the gradient of the solution converges to zero uniformly with respect to x as t → +∞. 相似文献
16.
Huashui ZHAN 《数学年刊B辑(英文版)》2012,33(5):767-782
Consider the following Cauchy problem:u_t = div(|▽u ~m |~ p-2▽u~m),(x,t) ∈ST=R~N ×(0,T),u(x,0) = μ,x ∈R~N,where 1
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17.
非线性Klein-Gordon方程柯西问题解的整体存在性与Blow-up 总被引:2,自引:0,他引:2
研究非线性Klein-Gordon方程的柯西问题u_(tt)-Δu+u=u|u|~(p-1),x∈R~n,t>0;u(x,0)=u_0(x),u_t(x,0)=u_1(x),x∈R~n.通过引进一族位势井,得到了解的整体存在性与不存在的门槛结果. 相似文献
18.
本文考虑 Lienard方程 x″+f (x) x′+g(x) =e(t) ,我们得到 :当 -∞ 0且 0 相似文献
19.
Daomin Cao 《偏微分方程(英文版)》1995,8(3):261-272
In this paper, we obtain the existence of positive solution of {-Δu = b(x)(u - λ)^p_+,\qquad x ∈ R^N λ > 0, |∇ u| ∈ L² (R^N),\qquad u ∈ L\frac{2N}{N-2} (R^N) under the assumptions that 1 < p < \frac{N+2}{N-2}, N ≥ 3, b(x) satisfies b(x) ∈ C(R^N), b(x) > 0 in R^N b(x) →_{|x|→∞}b^∞ and b(x) > \frac{4}{p+3}b^∞ for x ∈ R^N 相似文献
20.
V. M. Babich 《Journal of Mathematical Sciences》1985,28(5):628-632
One considers the problem of the asymptotic behavior for K→+∞ of the solution of the Cauchy problem $$u_{tt} - u_{xx} + \kappa ^2 u = 0; u|_{t = 0} = \theta (x), u_t |_{t = 0} = 0 (t > 0 - fixed)$$ Hereθ(x) is the Heaviside function. In the neighborhood of the characteristics x=±t function u(x,t)?2 oscillates exceptionally fast (the wavelength is of order k?2). Near the t axis the asymptotics of u(x,t) contains the Fresnel integral. 相似文献