共查询到20条相似文献,搜索用时 203 毫秒
1.
Qing Liu Yao 《数学学报(英文版)》2014,30(2):361-370
This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u″(t) + λ(t)u(t) = f(t,u(t)),a.e.t ∈ [0,2π],u(0) = u(2π),u′(0) = u′(2π),where f(t,u) is a local Carath′eodory function.This shows that the problem is singular with respect to both the time variable t and space variable u.By applying the Leggett–Williams and Krasnosel'skii fixed point theorems on cones,an existence theorem of triple positive solutions is established.In order to use these theorems,the exact a priori estimations for the bound of solution are given,and some proper height functions are introduced by the estimations. 相似文献
2.
Qi WANG 《数学年刊B辑(英文版)》2018,39(1):129-144
This paper deals with the electrostatic MEMS-device parabolic equation u_t-?u =λf(x)/(1-u)~p in a bounded domain ? of R~N,with Dirichlet boundary condition,an initial condition u0(x) ∈ [0,1) and a nonnegative profile f,where λ 0,p 1.The study is motivated by a simplified micro-electromechanical system(MEMS for short) device model.In this paper,the author first gives an asymptotic behavior of the quenching time T*for the solution u to the parabolic problem with zero initial data.Secondly,the author investigates when the solution u will quench,with general λ,u0(x).Finally,a global existence in the MEMS modeling is shown. 相似文献
3.
《数学物理学报(B辑英文版)》2020,(2)
In higher dimension, there are many interesting and challenging problems about the dynamics of non-autonomous Chafee-Infante equation. This article is concerned with the asymptotic behavior of solutions for the non–autonomous Chafee-Infante equation (?u)/(?t)-?u =λ(t)(u-u~3) in higher dimension, where λ(t) ∈ C~1[0, T ] and λ(t) is a positive, periodic function.We denote λ_1 as the first eigenvalue of-?? = λ?, x ∈ ?; ? = 0, x ∈ ??. For any spatial dimension N ≥ 1, we prove that if λ(t) ≤λ_1, then the nontrivial solutions converge to zero,namely, ■ u(x, t) = 0, x ∈ ?; if λ(t) λ_1 as t → +∞, then the positive solutions t→+∞are "attracted" by positive periodic solutions. Specially, if λ(t) is independent of t, then the positive solutions converge to positive solutions of-?U = λ(U-U~3). Furthermore,numerical simulations are presented to verify our results. 相似文献
4.
This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n + : u tt u + u t + divf (u) = 0, t > 0, x = (x 1 , x ′ ) ∈ R n + := R + × R n 1 , u(0, x) = u 0 (x) → u + , as x 1 → + ∞ , u t (0, x) = u 1 (x), u(t, 0, x ′ ) = u b , x ′ = (x 2 , x 3 , ··· , x n ) ∈ R n 1 . (I) For the non-degenerate case f ′ 1 (u + ) < 0, it was shown in [10] that the above initialboundary value problem (I) admits a unique global solution u(t, x) which converges to the corresponding planar stationary wave φ(x 1 ) uniformly in x 1 ∈ R + as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u(t, x) for t → + ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f ′ (u) < 0. The main purpose of this paper isdevoted to discussing the case of f ′ 1 (u b ) ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates. 相似文献
5.
《数学物理学报(B辑英文版)》2017,(5)
We study the existence of multiple positive solutions for a Neumann problem with singular φ-Laplacian{-(φ(u′))′= λf(u), x ∈(0, 1),u′(0) = 0 = u′(1),where λ is a positive parameter, φ(s) =s/(1-s~2)~(1/2), f ∈ C~1([0, ∞), R), f′(u) 0 for u 0, and for some 0 β θ such that f(u) 0 for u ∈ [0, β)(semipositone) and f(u) 0 for u β.Under some suitable assumptions, we obtain the existence of multiple positive solutions of the above problem by using the quadrature technique. Further, if f ∈ C~2([0, β) ∪(β, ∞), R),f′′(u) ≥ 0 for u ∈ [0, β) and f′′(u) ≤ 0 for u ∈(β, ∞), then there exist exactly 2 n + 1 positive solutions for some interval of λ, which is dependent on n and θ. Moreover, We also give some examples to apply our results. 相似文献
6.
王贺元 《高等学校计算数学学报(英文版)》2001,10(2)
1 IntroductionConsider the parameter dependent equationu"+ (λ+ s(μ) ) f( u) -μsinx =0 in ( 0 ,π)u( 0 ) =u(π) =0 ( 1 .1 )whereλ,μ∈R are parameters and f:R→R and S:R→R are smooth odd functions anda) f′( 0 ) =1 , b) f ( 0 )≠ 0 , c) s( 0 ) =0 , d) s′( 0 ) =1 . ( 1 .2 )Let S:u( x)→ u(π-x) ,Γ ={ S,I} ,then ( 1 .1 ) isΓ -equivariant.The equality ( 1 .2 a) isjust a normalization of f at x=0 .Otherwise,one may reseek the parameter x to ensure( 1 .2 a) .To simplify an… 相似文献
7.
刘斌 《高校应用数学学报(英文版)》2002,17(2):135-144
§ 1 IntroductionWe are interested in the existence ofthree-solutions ofthe following second-order dif-ferential equations with nonlinear boundary value conditionsx″=f( t,x,x′) , t∈ [a,b] ,( 1 .1 )g1 ( x( a) ,x′( a) ) =0 , g2 ( x( b) ,x′( b) ) =0 ,( 1 .2 )where f:[a,b]×R1 ×R1 →R1 ,gi:R1 ×R1 →R1 ( i=1 ,2 ) are continuous functions.The study ofthe existence of three-solutions ofboundary value prolems forsecond or-der differential equations was initiated by Amann[1 ] .In[1 … 相似文献
8.
《数学物理学报(B辑英文版)》2017,(5)
In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|~(N-2) u + f(x, u), x ∈ ?,u ∈ W_0~(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R~N(N 2), f(x, u) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ λ_1, λ = λ_?(? = 2, 3, ···), and λ_? is the eigenvalues of the operator(-?_N, W_0~(1,N)(?)),which is defined by the Z_2-cohomological index. 相似文献
9.
《数学物理学报(B辑英文版)》2010,(6)
In this paper, we prove the existence of at least one positive solution pair (u, v) ∈ H 1 (R N ) × H 1 (R N ) to the following semilinear elliptic system{-u + u = f(x, v), x ∈RN ,-v + v = g(x,u), x ∈ R N ,(0.1) by using a linking theorem and the concentration-compactness principle. The main con-ditions we imposed on the nonnegative functions f, g ∈ C 0 (R N × R 1 ) are that, f (x, t) and g(x, t) are superlinear at t = 0 as well as at t = +∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245(2008), 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem{-u + u = f(x, u), x ∈Ω,u ∈H10(Ω)where ΩRN is bounded and a result of Li and Yang [G. Li and J. Yang: Communications in P.D.E. Vol. 29(2004) Nos.5 6.pp.925–954, 2004] concerning (0.1) when f and g are asymptotically linear. 相似文献
10.
Harun Karsli 《分析论及其应用》2010,26(2):140-152
In the present paper we state some approximation theorems concerning pointwise convergence and its rate for a class of non-convolution type nonlinear integral operators of the form:Tλ (f;x) = B A Kλ (t,x, f (t))dt , x ∈< a,b >, λ∈Λ. In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 of f as (x,λ ) → (x0,λ0) in L1 < A,B >, where < a,b > and < A,B > are is an arbitrary intervals in R, Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology. The results of the present paper generalize several ones obtained previously in the papers [19]-[23]. 相似文献
11.
Qingliu Yao 《Applications of Mathematics》2011,56(6):543-555
We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions
$\left\{ \begin{gathered}
u^{(4)} (t) = f(t,u(t),u'(t),u'(t),u'(t)),a.e.t \in [0,1], \hfill \\
u(0) = a,u'(0) = b,u(1) = c,u'(1) = d, \hfill \\
\end{gathered} \right.
$\left\{ \begin{gathered}
u^{(4)} (t) = f(t,u(t),u'(t),u'(t),u'(t)),a.e.t \in [0,1], \hfill \\
u(0) = a,u'(0) = b,u(1) = c,u'(1) = d, \hfill \\
\end{gathered} \right.
相似文献
12.
In this paper we deal with the four-point singular boundary value problem
|