Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain

We consider a boundary-value problem for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ω _ε that is ε-periodically perforated by small holes. The holes are split into two ε-periodic sets depending on the boundary interaction via their boundary surfaces. Therefore, two different nonlinear boundary conditions σ _ε (u _ε ) + εκ _m (u _ε ) = εg _ε ^(m) , m = 1, 2, are given on the corresponding boundaries of the small holes. The asymptotic analysis of this problem is performed as ε → 0, namely, the convergence theorem for both the solution and the energy integral is proved without using an extension operator, asymptotic approximations for the solution and the energy integral are constructed, and the corresponding approximation error estimates are obtained.     

Existence of Solutions to a Singular Elliptic Equation

We study the equation ${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}}${{-{\Delta}u = (-\frac{1}{u^{\beta}}+\lambda{u}^{p})\chi\{u >0 }\}} in Ω with Dirichlet boundary condition, where 0 < p < 1 and 0 < β < 1. We regularize the term 1/u ^β near u ~ 0 by using a function g _ε (u) which pointwisely tends to 1/u ^β as ε → 0. When the parameter λ > 0 is large enough, the corresponding energy functional has critical points u _ε . Letting ε → 0, then u _ε converges to a solution of the original problem, which is nontrivial, nonnegative and vanishes at some portion of Ω. There are two nontrivial solutions.     

Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems     

The blow-up rate for a system of heat equations with neumann boundary conditions

This paper deals with the blow-up properties of solutions to a system of heat equations u _t=Δu, v _t=Δv in B _R×(0, T) with the Neumann boundary conditions εu/εη=e ^v, εv/εη=e ^u on S _R×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary. This work is supported by the National Natural Science Foundation of China     

Complex ordering and stochastic oscillations in a class of reaction-diffusion systems with small diffusion

Summary We consider four models of partial differential equations obtained by applying a generalization of the method of normal forms to two-component reaction-diffusion systems with small diffusionu _t=εDu _xx+(A+εA ₁)u+F(u),u ∈ ℝ². These equations (quasinormal forms) describe the behaviour of solutions of the original equation forε → 0. One of the quasinormal forms is the well-known complex Ginzburg-Landau equation. The properties of attractors of the other three equations are considered. Two of these equations have an interesting feature that may be called asensitivity to small parameters: they contain a new parameterϑ(ε)=−(aε ^−1/2 mod 1) that influences the behaviour of solutions, but changes infinitely many times whenε → 0. This does not create problems in numerical analysis of quasinormal forms, but makes numerical study of the original problem involvingε almost impossible.     

Problem with periodicity conditions for a degenerating equation of mixed type

For the equation K(t)u _xx + u _tt − b ² K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| ^m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u _x (0, t) = u _x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.     

Existence of solutions with turning points for nonlinear singularly perturbed boundary-value problems

We consider the following singularly perturbed boundary-value problem:

Uniqueness results and asymptotic behavior of solutions with boundary blow-up for logistic-type porous media equations

Under the proper structure conditions on the nonlinear term f(u) and weight function b(x), the paper shows the uniqueness and asymptotic behavior near the boundary of boundary blow-up solutions to the porous media equations of logistic type −Δu = a(x)u ^1/m − b(x)f(u) with m > 1.     

<Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> Estimates of solution for <Emphasis Type="Italic">m</Emphasis>-Laplacian parabolic equation with a nonlocal term

In this paper, we consider the global existence, uniqueness and L ^∞ estimates of weak solutions to quasilinear parabolic equation of m-Laplacian type u _t − div(|∇u| ^m−2∇u) = u|u| ^β−1 ∫_Ω |u| ^α dx in Ω × (0,∞) with zero Dirichlet boundary condition in tdΩ. Further, we obtain the L ^∞ estimate of the solution u(t) and ∇u(t) for t > 0 with the initial data u ₀ ∈ L ^q (Ω) (q > 1), and the case α + β < m − 1.     

On homogenization of the mixed boundary-value problem for the heat equation in a domain whose part contains channels arranged in a periodic way

In this paper, we study the asymptotic behavior of the solutionsu _ε (ε is a small parameter) of boundaryvalue problems for the heat equation in the domain Ω_ε=Ω⁻∪Ω _ε ⁺ ∪γ one part of which (Ω _ε ⁺ ) contains ε-periodically situated channels with diameters of order ε and the other part of which (Ω⁺) is a homogeneous medium; γ=∂Ω _ε ⁺ ∩∂Ω⁺. On the boundary of the channels the Neumann boundary condition is posed, and on ∂Ω_ε∩∂Ω the Dirichlet boundary condition is prescribed. The homogenized problem is the Dirichlet problem in Ω with the transmission condition on γ. The estimates for the difference betweenu _ε and the solution of the homogenized problem are obtained. Bibliography: 14 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 20, pp. 27–47, 1997.     

On singular diffusion equations with applications to self-organized criticality

We consider solutions of the singular diffusion equation _t, = (u^m?1 u_x)_x, m ≦ 0, associated with the flux boundary condition lim_x→?∞ (u^m?1u_x)_x = λ > 0. The evolutions defined by this problem depend on both m and λ. We prove existence and stability of traveling wave solutions, parameterized by λ. Each traveling wave is stable in its appropriate evolution. These traveling waves are in L¹ for ?1 < m ≦ 0, but have non-integrable tails for m ≦ ?1. We also show that these traveling waves are the same as those for the scalar conservation law u_t = ?[f(u)]_x + u_xx for a particular nonlinear convection term f(u) = f(u;m, λ). © 1993 John Wiley & Sons, Inc.     

On Cesáro summability of fourier series with respect to double complete orthonormal systems

Let {ϕn(x), n = 1, 2,...} be an arbitrary complete orthonormal system on the interval I:= [0, 1]which consists of a.e. bounded functions. Suppose that E ₀ ⊂ I ² is any Lebesgue measurable set such that μ₂ E ₀ > 0, and φ, φ(0) = 0, is an increasing continuous function on [0, ∞) with φ(u) = o(u ln u) as u → ∞. Then there exist a function f ∈ L¹(I ²) and a set E ₀ ^′ , ⊂ E ₀, μ₂ E ₀ ^′ > 0, such that

Continuation of Time Bounds for a Regularized Boussinesq System

We study the periodic solution of a perturbed regularized Boussinesq system (Bona et al., J. Nonlinear Sci. 12:283–318, 2002, Bona et al., Nonlinearity 17:925–952, 2004), namely the system η _t +u _x +β(−η _xxt +u _xxx )+α((ηu) _x +ηη _x +uu _x )=0,u _t +η _x +β(η _xxx −u _xxt )+α((ηu) _x +ηη _x +uu _x )=0, with 0<α,β≤1. We prove that the solution, starting from an initial datum of size ε, remains smaller than ε for a time scale of order (ε ⁻¹ α ⁻¹ β)², whereas the natural time is of order ε ⁻¹ α ⁻¹ β.     

Anharmonic Oscillators in the Complex Plane, $\boldsymbol{\mathcal{PT}}$-symmetry,and Real Eigenvalues

For integers m ≥ 3 and 1 ≤ ℓ ≤ m − 1, we study the eigenvalue problems − u ^″(z) + [( − 1)^ℓ(iz) ^m  − P(iz)]u(z) = λu(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays argz=-\fracp2±\frac(l+1)pm+2\arg z=-\frac{\pi}{2}\pm \frac{(\ell+1)\pi}{m+2} in the complex plane, where P is a polynomial of degree at most m − 1. We provide asymptotic expansions of the eigenvalues λ _n . Then we show that if the eigenvalue problem is PT\mathcal{PT}-symmetric, then the eigenvalues are all real and positive with at most finitely many exceptions. Moreover, we show that when gcd(m,l)=1\gcd(m,\ell)=1, the eigenvalue problem has infinitely many real eigenvalues if and only if one of its translations or itself is PT\mathcal{PT}-symmetric. Also, we will prove some other interesting direct and inverse spectral results.     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

Existence and iteration of positive solutions for a singular two-point boundary value problem with a <Emphasis Type="Italic">p</Emphasis>-Laplacian operator

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation (ϕ _p (u′))′+q(t)f(u) = 0, 0 < t < 1, where ϕ _p (s):= |s| ^p−2 s, p > 1, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient q(t) may be singular at t = 0; 1.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

A Uniqueness Result for a Model for Mixtures in the Absence of External Forces and Interaction Momentum

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ϱ _i of the fluids and their velocity fields u ⁽ⁱ⁾ are prescribed at infinity: ϱ _i |_∞ = ϱ _i∞ > 0, u ⁽ⁱ⁾|∞ = 0. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ϱ _i ≡ ϱ _i∞, u ⁽ⁱ⁾ ≡ 0, i = 1, 2. This work was supported by the SFB 611 at the University of Bonn and the European HYKE network (contract no. HPRN-CT-2002-00282). The third author was also supported by the project CSF 201/03/0934, and by MSM 0021620839.     

Large time behavior of solutions to a class of doubly nonlinear parabolic equations

We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation u _t = div(u ^m−1|Du| ^p−2 Du) − u ^q with an initial condition u(x, 0) = u ₀(x). Here the exponents m, p and q satisfy m + p ⩾ 3, p > 1 and q > m + p − 2. The paper was supported by NSF of China (10571144), NSF for youth of Fujian province in China (2005J037) and NSF of Jimei University in China.