Asymptotic integration of some evolution problems with large high-frequency terms

We present results concerning the justification of the averaging method, the construction of a complete justified asymptotics, and the time stability of solutions of semilinear parabolic equations and Navier-Stokes systems with polynomial nonlinearities and large rapidly oscillating terms.     

Bifurcation for a class of singular elliptic problems with quadratic convection term

We study the bifurcation problem ?Δu=g(u)+λ|?u|²+μ in $\text{Ω,}\text{u=0}$ on $\text{?Ω}$, where λ,μ?0 and $\text{Ω}$ is a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. The singular character of the problem is given by the nonlinearity g which is assumed to be decreasing and unbounded around the origin. In this Note we prove that the above problem has a positive classical solution (which is unique) if and only if λ(a+μ)<λ₁, where a=lim_t→+∞g(t) and λ₁ is the first eigenvalue of the Laplace operator in $\text{H}{}^1{}_0\text{(Ω)}$. We also describe the decay rate of this solution, as well as a blow-up result around the bifurcation parameter. To cite this article: M. Ghergu, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Equivalence estimates for a class of singular perturbation problems

We give some equivalence estimates on the solution of a singular perturbation problem that represents, among other models, the Koiter and Naghdi shell models. Two of the estimates apply to intermediate shell problems and the third is for membrane/shear dominated shells. From these equivalences, many known and some new sharp estimates on the solutions of the singular perturbation problems easily follow. To cite this article: S. Zhang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Multi-peak solutions for some singular perturbation problems

We consider the problem where is a smooth domain in , not necessarily bounded, is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses a solution that concentrates, as approaches zero, at a maximum of the function , the distance to the boundary. We obtain multi-peak solutions of the equation given above when the domain presents a distance function to its boundary d with multiple local maxima. We find solutions exhibiting concentration at any prescribed finite set of local maxima, possibly degenerate, of d. The proof relies on variational arguments, where a penalization-type method is used together with sharp estimates of the critical values of the appropriate functional. Our main theorem extends earlier results, including the single peak case. We allow a degenerate distance function and a more general nonlinearity. Received September 3, 1998 / Accepted February 29, 1999     

Lower bounds of error estimates for singular perturbation problems with Jacobi-Spectral approximations

With weighted orthogonal Jacobi polynomials, we study spectral approximations for singular perturbation problems on an interval. The singular parameters of the model are included in the basis functions, and then its stiff matrix is diagonal. Considering the estimations for weighted orthogonal coefficients, a special technique is proposed to investigate the a posteriori error estimates. In view of the difficulty of a posteriori error estimates for spectral approximations, we employ a truncation projection to study lower bounds for the models. Specially, we present the lower bounds of a posteriori error estimates with two different weighted norms in details.     

Wegner estimates and localization for continuum Anderson models with some singular distributions

We give a simple geometric proof of Wegner's estimate which leads to a variety of new results on localization for multi-dimensional random operators.     

Asymptotic stability of nonlinear fractional neutral singular systems

In this paper, fractional calculus has been introduced into neutral singular systems. The (asymptotical) stability and (generalized) Mittag-Leffler stability of nonlinear fractional neutral singular systems with Caputo and Riemann-Liouville derivatives are studied, respectively. Several sufficient conditions guaranteeing stability of such systems are established by using the Lyapunov direct method.     

New a posteriori error estimates for singular boundary value problems

In this paper, we discuss the asymptotic properties and efficiency of several a posteriori estimates for the global error of collocation methods. Proofs of the asymptotic correctness are given for regular problems and for problems with a singularity of the first kind. We were also strongly interested in finding out which of our error estimates can be applied for the efficient solution of boundary value problems in ordinary differential equations with an essential singularity. Particularly, we compare estimates based on the defect correction principle with a strategy based on mesh halving. AMS subject classification 65L05Supported in part by the Austrian Research Fund (FWF) grant P-15072-MAT and SFB Aurora.     

Lack of natural weighted estimates for some singular integral operators

We show that the classical Hörmander condition, or analogously the -Hörmander condition, for singular integral operators is not sufficient to derive Coifman's inequality

where , is the Hardy-Littlewood maximal operator, is any weight and is a constant depending upon and the constant of . This estimate is well known to hold when is a Calderón-Zygmund operator.

As a consequence we deduce that the following estimate does not hold:

where and where is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever is a Calderón-Zygmund operator.

One of the main ingredients of the proof is a very general extrapolation theorem for weights.

     

Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth

Let be a ball centered at the origin with radius R. We investigate the asymptotic behavior of positive solutions for the Dirichlet problem in on ∂B_R when ɛ→⁺ for suitable positive numbers μ Mathematics Subject Classification (2000) 35J60, 35B33     

Asymptotic stability in singular perturbation problems. II: Problems having matched asymptotic expansion solutions

The stability of systems of ordinary differential equations of the form

$\text{dx}\text{dt}\text{= f(t, x, y, ?), ?}\text{dy}\text{dt}\text{= g(t, x, y, ?)}$

, where ? is a real parameter near zero, is studied. It is shown that if the reduced problem

$\text{dx}\text{dt}\text{= f(t, x, y, 0), 0 = g(t, x, y, 0)}$

, is stable, and certain other conditions which ensure that the method of matched asymptotic expansions can be used to construct solutions are satisfied, then the full problem is asymptotically stable as t → ∞, and a domain of stability is determined which is independent of ?. Moreover, under certain additional conditions, it is shown that the solutions of the perturbed problem have limits as t → ∞. In this case, it is shown how these limits can be calculated directly from the equations

$\text{f(∞, x, y, ?) = 0 g(∞, x, y, ?) = 0}$

as expansions in powers of ?.     

On a class of sublinear singular elliptic problems with convection term

We establish several results related to existence, nonexistence or bifurcation of positive solutions for the boundary value problem −Δu+K(x)g(u)+a|∇u|=λf(x,u) in Ω, u=0 on ∂Ω, where Ω⊂RN(N?2) is a smooth bounded domain, 0<a?2, λ is a positive parameter, and f is smooth and has a sublinear growth. The main feature of this paper consists in the presence of the singular nonlinearity g combined with the convection term a|∇u|. Our approach takes into account both the sign of the potential K and the decay rate around the origin of the singular nonlinearity g. The proofs are based on various techniques related to the maximum principle for elliptic equations.     

Boundary value problems for some nonlinear systems with singular \phi-laplacian

Systems of differential equations of the form