On considère une solution $u$, assez régulière, d’une équation aux dérivées partielles non linéaire. Si $u$ est conormale par rapport a une hypersurface simplement caractéristique pour l’équation linéarisée, on étudie l’équation de transport satisfaite par son symbole principal, et on en déduit la propagation de la propriété “$u$ est conormale classique”.

Global solvability and asymptotics of semilinear parabolic Cauchy problems in ${ℝ}^n$ are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over ${ℝ}^n$, $n\in \mathbb{N}$. In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.

Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is $L^2$-critical. On the other...

The existence, uniqueness and regularities of the generalized global solutions and classical global solutions to the equation $$u_t=-A\left(t\right)u_{x^4}+B\left(t\right)u_{x^2}+g{\left(u\right)}_{x^2}+f{\left(u\right)}_x+h{\left(u_x\right)}_x+G\left(u\right)$$ with the initial boundary value conditions $$u(-\ell ,t)=u(\ell ,t)=0,u_{x^2}(-\ell ,t)=u_{x^2}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ or with the initial boundary value conditions $$u_x(-\ell ,t)=u_x(\ell ,t)=0,u_{x^3}(-\ell ,t)=u_{x^3}(\ell ,t)=0,u(x,0)=\varphi \left(x\right),$$ are proved. Moreover, the asymptotic behavior of these solutions is considered under some conditions.

This paper is devoted to the study of cloaking via anomalous localized resonance (CALR) in the two- and three-dimensional quasistatic regimes. CALR associated with negative index materials was discovered by Milton and Nicorovici [21] for constant plasmonic structures in the two-dimensional quasistatic regime. Two key features of this phenomenon are the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others, and the connection between the localized resonance...

We study oscillatory solutions of semilinear first order symmetric hyperbolic system $Lu=f(t,x,u,\bar{u})$, with real analytic $f$.The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in $T,X$ with only the natural hypothesis of coherence.In the special case where $L$ has constant coefficients and the phases are linear, the solutions have asymptotic description$$u^{ϵ}=U(t,x,t/ϵ,x/ϵ)+o\left(1\right)$$where the profile $U(t,x,T,X)$ is almost periodic in $(T,X)$.The main novelty in the analysis is the space of profiles which...

We investigate the long-time behaviour of solutions to the Korteweg-de Vries equation with a zero order dissipation and an additional forcing term, when the space variable varies over $R$, and prove that it is described by a maximal compact attractor in $H^2\left(R\right)$.

Le but de cet article est l’étude de la compétition Réaction-Diffusion pour un problème de type ${\displaystyle \beta {\left(w\right)}_t-d_{\epsilon }diva(x,Dw)+r_{\epsilon }g\left(x,\beta \left(w\right)\right)=f,}$ où $\mathbf{a}$ est un opérateur de Lerray-Lions, $\beta$ est une fonction continue croissante et la réaction $g$ est une fonction croissante qui dépend de l’espace $x$. On suppose que les coefficients de diffusion $d_{\epsilon }$ et de Réaction $r_{\epsilon }$ dépendent du paramètre $\epsilon$ avec $d_{\epsilon }$ et/ou $r_{\epsilon }$ tends vers $+\infty$ lorsque ${\displaystyle \epsilon \to 0}$. Dans le cas où, le coefficient de réaction est très rapide, nous étudions le comportement asymptotique lorsque ${\displaystyle t\to \infty }$ de la solution...

Les données, i.e. l’ouvert $\Omega$ et la force appliquée $f$, sont supposées de classe $𝒞$. Il est montré que toute solution des équations de Navier-Stokes dans l’ouvert $\Omega$, bornée dans $H^1(\Omega {)}^N$ ($N=2$ ou $3$) sur un intervalle de temps semi-infini $(t_0+\infty )$, est aussi bornée, pour $t\to +\infty$, dans tous les espaces $H^m(\Omega {)}^N$. Il en résulte que tout ensemble fonctionnel invariant ou attracteur borné dans $H^1\left(\Omega \right)(N$ (ou même $H^{1/2+ϵ}(\Omega {)}^N$, $ϵ\>0$) est porté par ${𝒞}^{\infty }(\bar{\Omega })$. Le cas où les forces appliquées dérivent d’un potentiel (i.e. $f=0$) est abordé : il est montré que toute solution...