A parabolic free boundary problem with double pinning

We study the Cauchy problem for the equation ∂_tu_ε−Δu_ε=−β_ε(u_ε) in (0,∞)×Rⁿ as $\text{ε→0}$, where the nonlinearity β_ε is assumed to converge to a measure concentrated at $\text{0}$. In this paper we allow for sign changes of β_ε and u_ε. The solutions are uniformly Lipschitz continuous in space and Hölder continuous in time. We show that each limit of u_ε is a solution of the free boundary problem ∂_tu−Δu=0 in {u>0}∩(0,∞)×Rⁿ,|∇u⁺|²−|∇u⁻|²=g on (∂{u>0}∪∂{u<0})∩((0,∞)×Rⁿ) in the sense of domain variations. Depending on the structure of the nonlinearity $\text{β}{}_{\mathrm{\epsilon }}\text{,}$ the function g in the condition on the free boundary need not be a constant.     

Positive solutions for slightly super-critical elliptic equations in contractible domainsSolutions positives pour l'équation Δu+u(n+2)/(n−2)+ε=0 en ouverts contractibles

We give examples of bounded domains $\text{Ω}$, even contractible, having the following property: there exists $\text{k}\text{?}\text{(Ω)}$ such that, for every integer $\text{k?}\text{k}\text{?}\text{(Ω)}$, problem $\text{P(ε,Ω)}$ below, for ε>0 small enough, has at least one solution blowing up as ε→0 at exactly k points. We also prove that the blow-up points tend to some points of $\text{?Ω}$ as k→∞. To cite this article: R. Molle, D. Passaseo, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 459–462.     

On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical rangeSur les asymptotiques des solutions globales des équations paraboliques sémi-linéaires d'ordre supérieur dans le cas surcritique

We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear 2mth order parabolic equation u_t=?(?Δ)^mu+|u|^p in R^N×R₊, where m>1, p>1, with bounded integrable initial data u₀. We prove that in the supercritical Fujita range p>p_F=1+2m/N any small global solution with nonnegative initial mass, $\text{∫u}{}_0\text{dx?0}$, exhibits as t→∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case $\text{p∈}\text{]1,p}{}_{\mathrm{F}}\text{]}$ where solutions can blow-up for any arbitrarily small initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents $\text{{p}{}_{\mathrm{l}}\text{=1+2m/(l+N),}\text{l=0,1,2,…}}$, where p₀=p_F, are discussed. To cite this article: Yu.V. Egorov et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 805–810.     

Universal bounds for global solutions of a forced porous medium equation

We show that in a smooth bounded domain $\text{Ω⊂}\text{R}{}^{\mathrm{n}}$, n⩾2, all global nonnegative solutions of u_t−Δu^m=u^p with zero boundary data are uniformly bounded in $\text{Ω×(τ,∞)}$ by a constant depending on $\text{Ω,p}$ and τ but not on u₀, provided that 1<m<p<[(n+1)/(n−1)]m. Furthermore, we prove an a priori bound in $\text{L}{}^{\mathrm{\infty }}\text{(Ω×(0,∞))}$ depending on $\text{||u}{}_0\text{||}{}_{\mathrm{L\infty (\Omega )}}$ under the optimal condition 1<m<p<[(n+2)/(n−2)]m.     

Lifting of BV functions with values in S1

We show that for every $\text{u∈}\text{BV}\text{(Ω;S}{}^1\text{)}$, there exists a bounded variation function $\text{?∈}\text{BV}\text{(Ω;}\text{R}\text{)}$ such that u=e^i? a.e. on $\text{Ω}$ and |?|_BV?2|u|_BV. The constant 2 is optimal in dimension n>1. To cite this article: J. Dávila, R. Ignat, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Uniqueness of the blow-up boundary solution of logistic equations with absorbtion

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. Assume f∈C¹[0,∞) is a non-negative function such that f(u)/u is increasing on (0,∞). Let a be a real number and let b?0, $\text{b}\text{/≡0}$ be a continuous function such that b≡0 on $\text{?Ω}$. We study the logistic equation Δu+au=b(x)f(u) in $\text{Ω}$. The special feature of this work is the uniqueness of positive solutions blowing-up on $\text{?Ω}$, in a general setting that arises in probability theory. To cite this article: F.-C. C??rstea, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 447–452.     

Doubly nonlinear parabolic equations with singular lower order term

In this paper we are concerned with positive solutions of the doubly nonlinear parabolic equation u_t=div(u^m−1|∇u|^p−2∇u)+Vu^m+p−2 in a cylinder $\text{Ω×(0,T)}$, with initial condition u(·,0)=u₀(·)⩾0 and vanishing on the parabolic boundary $\text{∂Ω×(0,T)}$. Here $\text{Ω⊂}\text{R}{}^{\mathrm{N}}$ (resp. $\text{H}{}^{\mathrm{n}}$) is a bounded domain with smooth boundary, $\text{V∈L}{}_{\mathrm{<mtext>loc</mtext>}}{}^1\text{(Ω)}$, $\text{m∈}\text{R}$, 1<p<N and m+p−2>0. The critical exponents $\text{q}{}^1$ are found and the nonexistence results are proved for $\text{q}{}^1\text{⩽m+p<3}$.     

Régularité du rayon hyperbolique

Let $\text{Ω?}\text{R}{}^2$ be a bounded domain of class $\text{C}{}^{\mathrm{2+\alpha }}\text{,}\text{0<α<1}$. We show that if u is the maximal solution of Δu=4exp(2u), which tends to +∞ as $\text{(x,y)→?Ω}$, then the hyperbolic radius v=exp(?u) is of class C^2+α up to the boundary. The proof relies on new Schauder estimates for Fuchsian elliptic equations. To cite this article: S. Kichenassamy, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Asymptotics for the blow-up boundary solution of the logistic equation with absorption

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. Assume that f?0 is a C¹-function on [0,∞) such that f(u)/u is increasing on (0,+∞). Let a be a real number and let b?0, b?0 be a continuous function such that b≡0 on $\text{?Ω}$. The purpose of this Note is to establish the asymptotic behaviour of the unique positive solution of the logistic problem Δu+au=b(x)f(u) in $\text{Ω}$, subject to the singular boundary condition u(x)→+∞ as $\text{dist}\text{(x,?Ω)→0}$. Our analysis is based on the Karamata regular variation theory. To cite this article: F.-C. Cîrstea, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Cyclic quartic fields and genus theory of their subfields

Let k = $\text{Q}$(√u) (u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = $\text{Q}$(√vwη), where η is fixed in k and satisfies η ? 1, (η) = $\text{U}$²√u, |$\text{U}$²| = |(√u)|, (v, u) = 1, v ∈ $\text{Z}$ is squarefree, w|u, 0 < w < √u. Thus if u ≠ a² + b², there is no K ? k. If u = a² + b² then for each fixed v there are 2^{g ? 1}K ? k, where g is the number of prime divisors of u. (2) $\text{K}\text{k}$ has a relative integral basis (RIB) (i.e., O_K is free over O_k) iff N(ε₀) = ?1 and w = 1, where ε₀ is the fundamental unit of k, (or, equivalently, iff K = $\text{Q}$(√vε₀√u), (v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc(K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that $\text{K}\text{k}$ has a RIB if u is a prime; (ii) Edgar and Peterson (J. Number Theory12 (1980), 77–83) showed that for u composite there is at least one K ? k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert (Ann. of Math.31 (1930), 381–418) are wrong.     

Bifurcations caused by perturbing the domain in an elliptic equation

We consider the Neumann problem ?Δu = λu ? u^p on a continuous family of bounded domains $\text{Ω}{}_{\mathrm{?}}$ which approach (as ? → 0) a set $\text{Ω}{}_0$ with two connected components, and analyze it as a bifurcation problem with two parameters, λ and ?. The bifurcation diagrams and the qualitative properties of the bifurcation sets for p odd and p even are obtained, and the relations between them are studied by considering the problem ?Δu = λu ? au² ? u³ for different values of a.     

On linear integral operators with nonnegative kernels

Let K be an eventually compact linear integral operator on $\text{L}{}^{\mathrm{p}}\text{(Ω, μ), 1 ? p < ∞}$, with nonnegative kernel k(x, y), where the underlying measure μ is totally σ-finite on the domain set Ω when p = 1. In considering the equation λf = Kf + g for given nonnegative $\text{g ? L}{}^{\mathrm{p}}\text{(Ω, μ), λ > 0}$, P. Nelson, Jr. provided necessary and sufficient conditions, in terms of the support of g, such that a nonnegative solution $\text{f ? L}{}^{\mathrm{p}}\text{(Ω, μ)}$ was attained. Such conditions led to generalizing some of the graph-theoretic ideas associated with the normal form of a nonnegative reducible matrix. The purpose of this paper is to show that the analysis by Nelson can be enlarged to provide a more complete generalization of the normal form of a nonnegative matrix which can be used to characterize the distinguished eigenvalues of K and K¹, and to describe sets of support for the eigenfunctions and generalized eigenfunctions of both K and K¹ belonging to the spectral radius of K.     

The effect of perturbations on the first eigenvalue of the p-LaplacianL'éffet des pérturbations sur la première valeur propre du p-Laplacien

Let $\text{Ω}$ be a domain with Lipschitzian boundary of a compact Riemannian manifold (M,g) and p>1. We prove that we can make the volume of M arbitrarily close to the volume of $\text{(Ω,g)}$ while the first eigenvalue of the p-Laplacian on M remains uniformly bounded from below in terms of the the first eigenvalue of the Neumann problem for the p-Laplacian on $\text{(Ω,g)}$. To cite this article: A.-M. Matei, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 255–258.     

Non existence de solutions globales de certaines équations d'ondes non linéairesNonexistence of global solutions to some nonlinear wave equations

We study the semilinear wave equation u_tt?Δu=p^?k|u|^m in $\text{R}\text{×}\text{R}{}^{\mathrm{n}}$, where p is a conformal factor approaching 0 at infinity. We prove that the solutions blow-up in finite time for small powers m, while having an arbitrarily long life-span for large m. Furthermore, we study the finite time blow-up of solutions for the class of quasilinear wave equations u_tt?Δu=p^?k|Lu|^m in $\text{R}\text{×}\text{R}{}^{\mathrm{n}}$. To cite this article: M. Aassila, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 961–966.     

Uniform estimates for solutions of nonlinear Klein-Gordon equations

Uniform estimates in $\text{H}{}_0{}^1\text{(Ω)}$ of global solutions to nonlinear Klein-Gordon equations of the form $\text{u}{}_{\mathrm{tt}}\text{? Δu + mu = g(u)}\text{in}\text{Ω, u = 0}\text{in}\text{?Ω}$, where Ω is an open subset of $\text{R}$^N, m > 0, and g satisfies some growth conditions are established.     

Null Lagrangians,weak continuity,and variational problems of arbitrary order

We consider the problem of minimizing integral functionals of the form $\text{I(u) = ∝}{}_{\mathrm{\Omega }}\text{F(x, ▽}{}^{\mathrm{[k]}}\text{u(x)) dx}$, where Ω ?$\text{R}$^p, u:ω →$\text{R}$ and ▽^[k]u denotes the set of all partial derivatives of u with orders ?k. The method is based on a characterization of null Lagrangians L(▽^ku) depending only on derivatives of order k. Applications to elasticity and other theories of mechanics are given.     

Remarks on an S-shaped bifurcation curve

We study the non-negative solutions of the boundary value problem $\text{?Δu = λ [}\text{exp}\text{αu}\text{(α + u)}\text{]; x ? Ω. u = 0; x ? ?Ω,}\text{where}\text{α > 0, λ ? 0, Ω ? R}{}^{\mathrm{n}}$ is bounded with smooth boundary ?Ω.This problem arises in the theory of combustion. We study the estimates on the supremum norm of the solutions and estimates on the critical values of λ.     

Uniqueness of solutions of nonlinear Dirichlet problems

In this paper, we consider the uniqueness of radial solutions of the nonlinear Dirichlet problem $\text{Δu + ?(u) = 0}\text{in}\text{Ω}\text{with}\text{u = 0}\text{on}\text{?Ω,}\text{where}\text{Δ = ∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{?}{}^2\text{?x}{}_{\mathrm{i}}{}^2\text{,?}$ satisfies some appropriate conditions and Ω is a bounded smooth domain in $\text{R}$ⁿ which possesses radial symmetry. Our uniqueness results apply to, for instance, $\text{?(u) = u}{}^{\mathrm{p}}\text{, p > 1}$, or more generally λu + ∑_{i = 1}^ka_iu^pi, λ ? 0, a_i > 0 and p_i > 1 with appropriate upper bounds, and Ω a ball or an annulus.     

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).