Some asymptotic properties for solutions of one-dimensional advection–diffusion equations with Cauchy data in Lp(R)

We state and discuss a number of fundamental asymptotic properties of solutions $u(?,t)$ to one-dimensional advection–diffusion equations of the form $u_t+f{(u)}_x=(a{(u)u_x)}_x$, $x\in \mathbb{R}$, $t>0$, assuming initial values $u(?,0)=u_0\in L^p(\mathbb{R})$ for some $1?p<\infty$. To cite this article: P. Braz e Silva, P.R. Zingano, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On sub-harmonic bifurcations

Under fairly general hypotheses, we investigate by elementary methods the structure of the p-periodic orbits of a family $h_u$ of transformations near $(u_0,x_0)$ when $h_{u_0}(x_0)=x_0$ and $\mathrm{d}{h}_{u_0}(x_0)$ has a simple eigenvalue which is a primitive p-th root of unity. To cite this article: M. Chaperon et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Un théorème de Liouville pour l'opérateur de Schrödinger avec dérive

Let $(M,g)$ be a complete Riemannian manifold without boundary of dimension n and V be a $C^2$ vector field on M such that $r(x)|V(x)|$ is bounded. Suppose that ${\mathrm{Ric}}_g(x)??\min \{\lambda (r(x))?{\mu }_{\mathrm{?}V}(x),\beta (r(x))\}$ outside a compact set of M, where ${\mu }_{\mathrm{?}V}$ denotes the upper eigenvalue of ?V and $\lambda ,\beta$ are non-negative decreasing functions such that ${\lim }_{t\to +\infty }{t}^2\lambda (t)=0$. There exists positive numbers $b_n$ and $c_n$ which depend only on n and $6V{6}_{\infty }$ such that if h is a $C^2$ function defined on M with $\mathrm{\Delta }h??c_n{a}^2$ and ${\mathrm{lim?sup}}_{R\to \infty }{R}^{\mathrm{?2}}{\min }_{x\in B_p(3R)?B_p(R)}h(x)??b_n{a}^2$, where $0?a<{\mathrm{lim?inf}}_{j\to \infty }h(z_j)$, where $(z_j)$ is a sequence of M such that $r(z_j)\to \infty$, then the equation $\mathrm{\Delta }u(x)+V(x)?\mathrm{?}u(x)+h(x)u(x)=0$ has no positive $C^2$ solution on M. To cite this article: S. Asserda, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Life-span of classical solutions to one dimensional initial–boundary value problems for general quasilinear wave equations

This paper is devoted to studying the initial–boundary value problem for one dimensional general quasilinear wave equations $u_{tt}?u_{xx}=b(u,Du)u_{xx}+2a_0(u,Du)u_{tx}+F(u,Du)$ on exterior domain. We obtain the sharp lower bound of the life-span of classical solutions to the initial–boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations.     

On the radial distribution of Julia sets of entire solutions of f(n)+A(z)f=0

This paper is devoted to studying the dynamical properties of solutions of $f^{(n)}+A(z)f=0$, where $n(?2)$ is an integer, and $A(z)$ is a transcendental entire function of finite order. We find the lower bound on the radial distribution of Julia sets of $E(z)$ provided that $E=f_1{f}_2?f_n$ and $\{f_1,f_2,\dots ,f_n\}$ is a solution base of such equations.     

Invariant measures of stochastic partial differential equations and conditioned diffusions

This work establishes and exploits a connection between the invariant measure of stochastic partial differential equations (SPDEs) and the law of bridge processes. Namely, it is shown that the invariant measure of $u_t=u_{xx}+f(u)+\sqrt{2?}\eta (x\text{,}t)$, where $\eta (x\text{,}t)$ is a space–time white-noise, is identical to the law of the bridge process associated to $\mathrm{d}U=a(U)\mathrm{d}x+\sqrt{?}\mathrm{d}W(x)$, provided that a and f are related by $?a^{″}(u)+2a^{\prime }(u)a(u)=?2f(u)$, $u\in \mathbb{R}$. Some consequences of this connection are investigated, including the existence and properties of the invariant measure for the SPDE on the line, $x\in \mathbb{R}$. To cite this article: M.G. Reznikoff, E. Vanden-Eijnden, C. R. Acad. Sci. Paris, Ser. I 340 (2005).