We consider an a priori global strong solution to the Navier-Stokes equations. We prove it behaves like a small solution for large time. Combining this asymptotics with uniqueness and averaging in time properties, we obtain the stability of such a global solution.

Nous discutons l’asymptotique des noyaux de Bergman pour des puissances élevées de fibrés de droites, d’après deux travaux récents avec B.Berndtsson et R. Berman.

Let -ℒ be the generator of a Lévy semigroup on L¹(ℝⁿ) and f: ℝ → ℝⁿ be a nonlinearity. We study the large time asymptotic behavior of solutions of the nonlocal and nonlinear equations uₜ + ℒu + ∇·f(u) = 0, analyzing their $L^p$-decay and two terms of their asymptotics. These equations appear as models of physical phenomena that involve anomalous diffusions such as Lévy flights.

The nonlinear integro-differential system associated with the penetration of a magnetic field into a substance is considered. The asymptotic behavior as $t\to \infty$ of solutions for two initial-boundary value problems are studied. The problem with non-zero conditions on one side of the lateral boundary is discussed. The problem with homogeneous boundary conditions is studied too. The rates of convergence are given. Results presented show the difference between stabilization characters of solutions of these...

We study asymptotic behavior of solutions to multifractal Burgers-type equation $u_t+f{\left(u\right)}_x=Au$, where the operator A is a linear combination of fractional powers of the second derivative $-{\partial }^2/\partial x^2$ and f is a polynomial nonlinearity. Such equations appear in continuum mechanics as models with fractal diffusion. The results include decay rates of the $L^p$-norms, 1 ≤ p ≤ ∞, of solutions as time tends to infinity, as well as determination of two successive terms of the asymptotic expansion of solutions.

In the recent years, many results have been established on positive solutions for boundary value problems of the form $-div\left(\right|\nabla u\left(x\right){|}^{p-2}\nabla u\left(x\right))=\lambda f\left(u\left(x\right)\right)$ in Ω, u(x)=0 on ∂Ω, where λ > 0, Ω is a bounded smooth domain and f(s) ≥ 0 for s ≥ 0. In this paper, a priori estimates of positive radial solutions are presented when N > p > 1, Ω is an N-ball or an annulus and f ∈ C¹(0,∞) ∪ C⁰([0,∞)) with f(0) < 0 (non-positone).

We prove that minimizers $u\in W^{1,n}$ of the functional $E_{}\left(u\right)=1/n{\int }_{|}{\nabla u|}^{n}dx+1/\left(4^n\right){\int }_{(}1-{\left|u\right|}^2{)}^{2}dx$, ⊂ ${ℝ}^n$, n ≥ 3, which satisfy the Dirichlet boundary condition $u_{}=g$ on for g: → $S^{n-1}$ with zero topological degree, converge in $W^{1,n}$ and $C_{loc}^{\alpha }$ for any α<1 - upon passing to a subsequence ${}_k\to 0$ - to some minimizing n-harmonic map. This is a generalization of an earlier result obtained for n=2 by Bethuel, Brezis, and Hélein. An example of nonunique asymptotic behaviour (which cannot occur in two dimensions if deg g = 0) is presented.

This work is concerned with the asymptotic analysis of a time-splitting scheme for the Schrödinger equation with a random potential having weak amplitude, fast oscillations in time and space, and long-range correlations. Such a problem arises for instance in the simulation of waves propagating in random media in the paraxial approximation. The high-frequency limit of the Schrödinger equation leads to different regimes depending on the distance of propagation, the oscillation pattern of the initial...

We describe the long-time behaviour of solutions of parabolic equations in the case when some solutions may blow up in a finite or infinite time. This is done by providing a maximal compact invariant set attracting any initial data for which the corresponding solution does not blow up. The abstract result is applied to the Frank-Kamenetskii equation and the N-dimensional Navier-Stokes system with small external force.