Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

Blow-up and propagation of disturbances for fast diffusion equations

This paper is concerned with the Cauchy problem for the fast diffusion equation _ut−Δu^m=αu^_p1$u_t-\Delta u^m=\alpha u^{p_1}$ in R^N${\mathbb{R}}^N$ (N≥1$N\ge 1$), where m∈(0,1)$m\in (0,1)$, _p1>1$p_1>1$ and α>0$\alpha >0$. The initial condition _u0$u_0$ is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value _u0$u_0$ so that u(t,x)$u(t,x)$ blows up in finite time, and we show how to get estimates on the profile of u(t,x)$u(t,x)$ for small enough values of t>0$t>0$.     

Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(_Xt,α)$b(X_t,\alpha )$ and diffusion coefficient εa(_Xt,β)$\epsilon a(X_t,\beta )$ where α$\alpha$ and β$\beta$ are two unknown parameters, while ε$\epsilon$ is known. For a high frequency sample of observations of the diffusion at the time points k/n$k/n$, k=1,…,n$k=1,\dots ,n$, we propose a class of contrast functions and thus obtain estimators of (α,β)$(\alpha ,\beta )$. The estimators are shown to be consistent and asymptotically normal when n→∞$n\to \infty$ and ε→0$\epsilon \to 0$ in such a way that ε⁻¹n^−ρ${\epsilon }^{-1}{n}^{-\rho }$ remains bounded for some ρ>0$\rho >0$. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.     

Nonparametric estimation for stochastic differential equations with random effects

We consider N$N$ independent stochastic processes (_Xj(t),t∈[0,T])$(X_j\left(t\right),t\in [0,T])$, j=1,…,N$j=1,\dots ,N$, defined by a one-dimensional stochastic differential equation with coefficients depending on a random variable _?j${?}_j$ and study the nonparametric estimation of the density of the random effect _?j${?}_j$ in two kinds of mixed models. A multiplicative random effect and an additive random effect are successively considered. In each case, we build kernel and deconvolution estimators and study their L²$L^2$-risk. Asymptotic properties are evaluated as N$N$ tends to infinity for fixed T$T$ or for T=T(N)$T=T\left(N\right)$ tending to infinity with N$N$. For T(N)=N²$T\left(N\right)=N^2$, adaptive estimators are built. Estimators are implemented on simulated data for several examples.     

Large deviations of infinite intersections of events in Gaussian processes

Consider events of the form {_Zs≥ζ(s),s∈S}$\{Z_s\ge \zeta \left(s\right),s\in S\}$, where Z$Z$ is a continuous Gaussian process with stationary increments, ζ$\zeta$ is a function that belongs to the reproducing kernel Hilbert space R$R$ of process Z$Z$, and S⊂R$S\subset \mathbb{R}$ is compact. The main problem considered in this paper is identifying the function β^∗∈R${\beta }^{\ast }\in R$ satisfying β^∗(s)≥ζ(s)${\beta }^{\ast }\left(s\right)\ge \zeta \left(s\right)$ on S$S$ and having minimal R$R$-norm. The smoothness (mean square differentiability) of Z$Z$ turns out to have a crucial impact on the structure of the solution. As examples, we obtain the explicit solutions when ζ(s)=s$\zeta \left(s\right)=s$ for s∈[0,1]$s\in [0,1]$ and Z$Z$ is either a fractional Brownian motion or an integrated Ornstein–Uhlenbeck process.     

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.     

Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s)$x\left(s\right)$, s∈R^d$s\in R^d$ be a Gaussian self-similar random process of index H$H$. We consider the problem of log-asymptotics for the probability _pT$p_T$ that x(s)$x\left(s\right)$, x(0)=0$x\left(0\right)=0$ does not exceed a fixed level in a star-shaped expanding domain T⋅Δ$T\cdot \Delta$ as T→∞$T\to \infty$. We solve the problem of the existence of the limit, θ?lim(−log_pT)/(logT)^D$\theta ?lim(-log{p}_T)/{(logT)}^D$, T→∞$T\to \infty$, for the fractional Brownian sheet x(s)$x\left(s\right)$, s∈[0,T]²$s\in {[0,T]}^2$ when D=2$D=2$, and we estimate θ$\theta$ for the integrated fractional Brownian motion when D=1$D=1$.     

Critical exponents for non-simultaneous blow-up in a localized parabolic system

The paper deals with the radially symmetric solutions of _ut=Δu+u^m(x,t)vⁿ(0,t)$u_t=\Delta u+u^m(x,t)v^n(0,t)$, _vt=Δv+u^p(0,t)v^q(x,t)$v_t=\Delta v+u^p(0,t)v^q(x,t)$, subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u$u$ (v$v$) blows up alone if and only if m>p+1$m>p+1$ (q>n+1$q>n+1$), which means that any blow-up is simultaneous if and only if m≤p+1$m\le p+1$, q≤n+1$q\le n+1$. (ii) Any blow-up is u$u$ (v$v$) blowing up with v$v$ (u$u$) remaining bounded if and only if m>p+1$m>p+1$, q≤n+1$q\le n+1$ (m≤p+1$m\le p+1$, q>n+1$q>n+1$). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1$m>p+1$, q>n+1$q>n+1$. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.     

A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h$h$ affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)$△u=b\left(x\right)g\left(u\right)+\lambda h\left(x\right)$, u>0$u>0$ in Ω$\Omega$, u_|∂Ω=∞$u{|}_{\partial \Omega }=\infty$, where Ω$\Omega$ is a bounded domain with smooth boundary in R^N${\mathbb{R}}^N$, λ>0$\lambda >0$, g∈C¹[0,∞)$g\in C^1[0,\infty )$ is increasing on [0,∞)$[0,\infty )$, g(0)=0$g\left(0\right)=0$, g^′$g^{\prime }$ is regularly varying at infinity with positive index ρ$\rho$, the weight b$b$, which is non-trivial and non-negative in Ω$\Omega$, may be vanishing on the boundary, and the inhomogeneous term h$h$ is non-negative in Ω$\Omega$ and may be singular on the boundary.     

The relaxed Newton method derivative: Its dynamics and non-linear properties

The dynamic behaviour of the one-dimensional family of maps f(x)=_c2[(a−1)x+_c1]^{−λ/(α−1)}$f\left(x\right)=c_2{[(a-1)x+c_1]}^{-\lambda /(\alpha -1)}$ is examined, for representative values of the control parameters a,_c1$a,c_1$, _c2$c_2$ and λ$\lambda$. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a$a$. The maps f(x)$f\left(x\right)$ are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an _xn$x_n$ versus λ$\lambda$ plot, an initial exponential decay followed by a bifurcation. The value of λ$\lambda$ at which this bifurcation takes place depends on the values of the parameters a,_c1$a,c_1$ and _c2$c_2$. This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x)$f\left(x\right)$ undergoing a period doubling. For values of a$a$ higher than 1 and at higher values of λ$\lambda$ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter _c1$c_1$ between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.     

Selections and their absolutely continuous invariant measures

Let I=[0,1]$I=[0,1]$ and let P be a partition of I   into a finite number of intervals. Let _τ1${\tau }_1$, _τ2${\tau }_2$; I→I$I\to I$ be two piecewise expanding maps on P  . Let G⊂I×I$G\subset I\times I$ be the region between the boundaries of the graphs of _τ1${\tau }_1$ and _τ2${\tau }_2$. Any map τ:I→I$\tau :I\to I$ that takes values in G is called a selection of the multivalued map defined by G  . There are many results devoted to the study of the existence of selections with specified topological properties. However, there are no results concerning the existence of selection with measure-theoretic properties. In this paper we prove the existence of selections which have absolutely continuous invariant measures (acim). By our assumptions we know that _τ1${\tau }_1$ and _τ2${\tau }_2$ possess acims preserving the distribution functions F⁽¹⁾$F^{(1)}$ and F⁽²⁾$F^{(2)}$. The main result shows that for any convex combination F   of F⁽¹⁾$F^{(1)}$ and F⁽²⁾$F^{(2)}$ we can find a map η   with values between the graphs of _τ1${\tau }_1$ and _τ2${\tau }_2$ (that is, a selection) such that F is the η-invariant distribution function. Examples are presented. We also study the relationship of the dynamics of our multivalued maps to random maps.     

Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (−Δ)^su=λe^u${(-\mathrm{\Delta })}^{s}u=\lambda e^u$ in a bounded domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7$n\le 7$ for all s∈(0,1)$s\in (0,1)$ whenever Ω   is, for every i=1,...,n$i=1,...,n$, convex in the _xi$x_i$-direction and symmetric with respect to {_xi=0}$\{x_i=0\}$. The same holds if n=8$n=8$ and s?0.28206...$s?0.28206...$, or if n=9$n=9$ and s?0.63237...$s?0.63237...$. These results are new even in the unit ball Ω=_B1$\Omega =B_1$.     

Applications of equivariant degree for gradient maps to symmetric Newtonian systems

We consider G=Γ×S¹$G=\Gamma \times S^1$ with Γ$\Gamma$ being a finite group, for which the complete Euler ring structure in U(G)$U\left(G\right)$ is described. The multiplication tables for Γ=_D6$\Gamma =D_6$, _S4$S_4$ and _A5$A_5$ are provided in the Appendix. The equivariant degree for G$G$-orthogonal maps is constructed using the primary equivariant degree with one free parameter. We show that the G$G$-orthogonal degree extends the degree for G$G$-gradient maps (in the case of G=Γ×S¹$G=\Gamma \times S^1$) introduced by G?ba in [K. G?ba, W. Krawcewicz, J. Wu, An equivariant degree with applications to symmetric bifurcation problems I: Construction of the degree, Bull. London. Math. Soc. 69 (1994) 377–398]. The computational results obtained are applied to a Γ$\Gamma$-symmetric autonomous Newtonian system for which we study the existence of 2π$2\pi$-periodic solutions. For some concrete cases, we present the symmetric classification of the solution set for the systems considered.