Attractors of the reaction-diffusion systems with rapidly oscillating coefficients and their homogenization

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,…,u^k) 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 a_ij^l(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 .     

Uniqueness and stability of regional blow-up in a porous-medium equation

We study the blow-up phenomenon for the porous-medium equation in R^N, N1, u_t=Δu^m+u^m, 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,…,x_kR^N, with |x_i−x_j|2R^* for i≠j, such that Here w_*(|x|) is the unique nontrivial, nonnegative compactly supported, radially symmetric solution of the equation in R^N 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.     

Convergence of the Dirichlet solutions of the very fast diffusion equation

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 u^R of the Dirichlet problem u_t=(u^m/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 u_t=(u^m/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.     

MULTIPLE SOLUTIONS FOR SCHR(O)DINGER EQUATIONS WITH MAGNETIC FIELD

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.     

On the continuation of solutions for elliptic equations in two variables

We consider nonlinear elliptic differential equations of second order in two variables

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. 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 z_n=ψ(x,y,z,z_t) on Γ with real analytic functions and ψ, respectively (z_n denotes the derivative of z w.r.t. the outer normal n on Γ and z_t 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.     

On the number of nodal solutions to a singularly perturbed Neumann problem

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 ε_K>0 such that for 0<ε<ε_K, 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.     

EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE Hk

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, ∞）.     

NONTRIVIAL SOLUTIONS FOR SEMILINEAR SCHR(O)DINGER EQUATIONS

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.     

DIRICHLET PROBLEMS FOR STATIONARY VON NEUMANN-LANDAU WAVE EQUATIONS

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.     

NEW STRUCTURES FOR NON-SELFSIMILAR SOLUTIONS OF MULTI-DIMENSIONAL CONSERVATION LAWS

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.     

Existence of maximal solutions

Let Ω be a plane bounded region. Let U = {U_μ(P):μ(P)εL∞(Ω), u_μ ε H²_{2, 0}(Ω) and a(P, μ(P))u_μ,xx + 2b(P, μ(P))u_μ,xy + c(P, μ(P))u_μ,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 L_p(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 u_μ0 (P) u_μ(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.     

Existence of three solutions for a class of elliptic eigenvalue problems

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 W₀^1,2(Ω).     

Stability estimates in time-backwards problem for the rod equation

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).     

A Modification of the Blowup Technique for Variational Integrals with Subquadratic Growth

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 everywhereC^{1, α}-regularity.     

Localization of blow-up points for a parabolic system with a nonlinear boundary condition

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.     

A note on the behavior of blow-up solutions for one-phase Stefan problems

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On a variational problem with lack of compactness: the topological effect of the critical points at infinity

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).

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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).

     

Global existence and L ∞ estimates of solutions for a quasilinear parabolic system

In this paper, we study the global existence, L^∞ estimates and decay estimates of solutions for the quasilinear parabolic system u_t = div (|∇ u|^m ∇ u) + f(u, v), v_t = div (|∇ v|^m ∇ v) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ R^N. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation u_t = div (|∇ u|^m ∇ u) + λ |u|^{α - 1} u.     

Global existence and blow-up of solutionsfor a higher-order kirchhoff-type equation with nonlinear dissipation

This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipationu_tt+(∫_Ω׀D^mu׀²dx)^q(−Δ)^mu+u_t׀u_t׀^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 L^p+2 norm.     

On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle

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) = ∝₀¹r^{p − 1}u(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.