共查询到20条相似文献,搜索用时 31 毫秒
1.
The following reaction-diffusion system in spatially non-homogeneous almost-periodic media is considered in a bounded domain
: (1) ∂tu=Au−f(u)+g, u|∂Ω=0. Here u=(u1,…,uk) is an unknown vector-valued function, f is a given nonlinear interaction function and the second order elliptic operator A has the following structure: where aijl(y) are given almost-periodic functions. We prove that, under natural assumptions on the nonlinear term f(u), the longtime behavior of solutions of (1) can be described in terms of the global attractor
of the associated dynamical system and that the attractors
, 0<<01, converge to the attractor
of the homogenized problem (1) as →0. Moreover, in the particular case of periodic media, we give explicit estimates for the distance between the non-homogenized
and the homogenized
attractors in terms of the parameter . 相似文献
2.
Carmen Cortzar Manuel del Pino Manuel Elgueta 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2002,19(6):71
We study the blow-up phenomenon for the porous-medium equation in RN, N1, ut=Δum+um, m>1, for nonnegative, compactly supported initial data. A solution u(x,t) to this problem blows-up at a finite time
. Our main result asserts that there is a finite number of points x1,…,xkRN, with |xi−xj|2R* for i≠j, such that Here w*(|x|) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation
in RN and R* is the radius of its support. Moreover u(x,t) remains uniformly bounded up to its blow-up time on compact subsets of
. The question becomes reduced to that of proving that the ω-limit set in the problem
consists of a single point when its initial condition is nonnegative and compactly supported. 相似文献
3.
For any −1<m<0, positive functions f, g and u0≥0, we prove that under some mild conditions on f, g and u0 as R→∞ the solution uR of the Dirichlet problem ut=(um/m)xx in (−R,R)×(0,∞), u(R,t)=(f(t)|m|R)1/m, u(−R,t)=(g(t)|m|R)1/m for all t>0, u(x,0)=u0(x) in (−R,R), converges uniformly on every compact subset of R×(0,T) to the solution of the equation ut=(um/m)xx in R×(0,T), u(x,0)=u0(x) in R, which satisfies some mass loss formula on (0,T) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) [15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains. 相似文献
4.
The authors consider the semilinear SchrSdinger equation
-△Au+Vλ(x)u= Q(x)|u|γ-2u in R^N,
where 1 〈 γ 〈 2* and γ≠ 2, Vλ= V^+ -λV^-. Exploiting the relation between the Nehari manifold and fibrering maps, the existence of nontrivial solutions for the problem is discussed. 相似文献
-△Au+Vλ(x)u= Q(x)|u|γ-2u in R^N,
where 1 〈 γ 〈 2* and γ≠ 2, Vλ= V^+ -λV^-. Exploiting the relation between the Nehari manifold and fibrering maps, the existence of nontrivial solutions for the problem is discussed. 相似文献
5.
We consider nonlinear elliptic differential equations of second order in two variables
. Supposing analyticity of F, we prove analyticity of the real solution z=z(x,y) in the open set Ω. Furthermore, we show that z may be continued as a real analytic solution for F=0 across the real analytic boundary arc Γ∂Ω, if z satisfies one of the boundary conditions z= or zn=ψ(x,y,z,zt) on Γ with real analytic functions and ψ, respectively (zn denotes the derivative of z w.r.t. the outer normal n on Γ and zt its derivative w.r.t. the tangent). The proof is based on ideas of H. Lewy combined with a uniformization method. Studying quasilinear equations, we get somewhat better results concerning the initial regularity of the given solution and a little more insight. 相似文献
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6.
We show that for ε small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ2 and f grows superlinearly. (A typical f(u) is f(u)= a1 u+p – a1 u-p, a1, a2 >0, p, q>1.) More precisely, for any positive integer K, there exists εK>0 such that for 0<ε<εK, the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed. 相似文献
7.
In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u + λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k(Ω) in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞). 相似文献
8.
The authors prove the existence of nontrivial solutions for the SchrSdinger equation -△u + V(x)u =λf(x, u) in R^N, where f is superlinear, subcritical and critical at infinity, respectively, V is periodic. 相似文献
9.
陈泽乾 《数学物理学报(B辑英文版)》2009,29(5):1225-1232
In this article, we are concerned with the Dirichlet problem of the stationary von Neumann-Landau wave equation:
{(-△x+△y)φ(x,y)=0,x,y∈Ω
φ|δΩxδΩ=f
where Ω is a bounded domain in R^n. By introducing anti-inner product spaces, we show the existence and uniqueness of the generalized solution for the above Dirichlet problem by functional-analytic methods. 相似文献
{(-△x+△y)φ(x,y)=0,x,y∈Ω
φ|δΩxδΩ=f
where Ω is a bounded domain in R^n. By introducing anti-inner product spaces, we show the existence and uniqueness of the generalized solution for the above Dirichlet problem by functional-analytic methods. 相似文献
10.
In this article, we get non-selfsimilar elementary waves of the conservation laws in another kind of view, which is different from the usual self-similar transformation. The solution has different global structure. This article is divided into three parts. The first part is introduction. In the second part, we discuss non-selfsimilar elementary waves and their interactions of a class of twodimensional conservation laws. In this case, we consider the case that the initial discontinuity is parabola with u+ 〉 0, while explicit non-selfsirnilar rarefaction wave can be obtained. In the second part, we consider the solution structure of case u+ 〈 0. The new solution structures are obtained by the interactions between different elementary waves, and will continue to interact with other states. Global solutions would be very different from the situation of one dimension. 相似文献
11.
Let Ω be a plane bounded region. Let U = {Uμ(P):μ(P)εL∞(Ω), uμ ε H22, 0(Ω) and a(P, μ(P))uμ,xx + 2b(P, μ(P))uμ,xy + c(P, μ(P))uμ,vv = ƒ(P) for P ε Ω; here we are given a(P, X), b(P, X), c(P, X) ε L∞(Ω × E1), ƒ(P) ε Lp(Ω) with p > 2, and our partial differential equation is uniformly elliptic. The functions μ(P) are called profiles. We establish sufficient conditions—which when they apply are constructive—that there exist a μ0 ε L∞(Ω) such that uμ0 (P) uμ(P) for all P ε Ω and for each μ ε L∞(Ω). Similar results are obtained for a difference equation and convergence is proved. 相似文献
12.
B. Ricceri 《Mathematical and Computer Modelling》2000,32(11-13)
In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rn is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W01,2(Ω). 相似文献
13.
In this paper we consider the equationu
tt
+u
xxxx
=0 for a vibrating rod supported at its end-points. We study the problem of determining the shape of the rod,u(·,0), knowingu(·,T). We see that the solution is unique ifT belongs to a certain dense subsetS of (0,+∞). IfT belongs toS, u(·,0) does not depend continuously onu(·,T). In this case, with the assumption of a bounded initial energy of the rod, we get stability estimates foru(·,0). 相似文献
14.
We present a version of the blowup technique which applies to local minimizersu: Ω →
N, Ω open in
n, of strictly convex variational integralsJ(u) ∫Ω f(u) dx, wherefis of p-growth for some 1 < p < 2. This provides an alternative approach towards the partial regularity theorem of Anzellotti and Giaquinta. For two-dimensional problems we obtain everywhereC1, α-regularity. 相似文献
15.
The authors localize the blow-up points of positive solutions of the systemu
t
=Δu,v
t
=Δv with conditions
at the boundary of a bounded smooth domain Θ under some restrictions off andg and the initial data (Δu
0, Δν0>c>0).
If Θ is a ball, the hypothesis on the initial data can be removed.
Supported by Universidad de Buenos Aires under grant EX071 and CONICET. 相似文献
16.
Zhu Ning 《高校应用数学学报(英文版)》1998,13(3):241-250
ANOTEONTHEBEHAVIOROFBLOW┐UPSOLUTIONSFORONE┐PHASESTEFANPROBLEMSZHUNINGAbstract.Inthispaper,thefolowingone-phaseStefanproblemis... 相似文献
17.
On a variational problem with lack of compactness: the topological effect of the critical points at infinity 总被引:5,自引:0,他引:5
Abbas Bahri Yanyan Li Olivier Rey 《Calculus of Variations and Partial Differential Equations》1995,3(1):67-93
We study the subcritical problemsP
:–u=u
p–,u>0 on;u=0 on , being a smooth and bounded domain in N,N–3,p+1=2N/N–2 the critical Sobolev exponent and >0 going to zero — in order to compute the difference of topology that the critical points at infinity induce between the level sets of the functional corresponding to the limit case (P0).
Résumé Nous étudions les problèmes sous-critiquesP :–u=u p–,u > 0 sur;u=0 sur –où est un domaine borné et régulier de N,N–3,p + 1=2N/N –2 est l'exposant critique de Sobolev, et >0 tend vers zéro, afin de calculer la différence de toplogie induite par les points critiques à l'infini entre les ensembles de niveau de la fonctionnelle correspondant au cas limite (P0).相似文献
18.
Caisheng Chen 《Journal of Evolution Equations》2006,6(1):29-43
In this paper, we study the global existence, L∞ estimates and decay estimates of solutions for the quasilinear parabolic system ut = div (|∇ u|m ∇ u) + f(u, v), vt = div (|∇ v|m ∇ v) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ RN. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation ut = div (|∇ u|m ∇ u) + λ |u|α - 1 u. 相似文献
19.
Fucai Li 《Applied Mathematics Letters》2004,17(12):686
This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipationutt+(∫Ω׀Dmu׀2dx)q(−Δ)mu+ut׀ut׀r=׀u׀pu,xΩ,t>0,in a bounded domain, where m < 1 is a positive integer, q, p, r < 0 arepositive constants. We obtain that the solution exists globally if p ≤ r, while ifp > max r, 2q , then for any initial data with negative initial energy, the solution blowsup at finite time in Lp+2 norm. 相似文献
20.
Let u(r,θ) be biharmonic and bounded in the circular sector ¦θ¦ < π/4, 0 < r < ρ (ρ > 1) and vanish together with δu/δθ when ¦θ¦ = π/4. We consider the transform û(p,θ) = ∝01rp − 1u(r,θ)dr. We show that for any fixed θ0 u(p,θ0) is meromorphic with no real poles and cannot be entire unless u(r, θ0) ≡ 0. It follows then from a theorem of Doetsch that u(r, θ0) either vanishes identically or oscillates as r → 0. 相似文献