In this article, we consider the Cauchy problem for sixth-order damped Boussinesq equation in Rn. The well-posedness of global solutions and blow-up of solutions are obtained. The asymptotic behavior of the solution is established by the multiplier method.

This talk describes some applications of two kinds of observation estimate for the wave equation and for the damped wave equation in a bounded domain where the geometric control condition of C. Bardos, G. Lebeau and J. Rauch may failed. 1 The wave equation and observation We consider the wave equation in the solution u = u(x, t)    ∂ t u−∆u = 0 in Ω× R , u = 0 on ∂Ω× R , (u, ∂tu) (·, 0) = (u...

In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for th...

This paper is concerned with the Cauchy problem of the modified Kawahara equation. By using the Fourier restriction norm method introduced by Bourgain, and using the I-method as well as the L 2 conservation law, we prove that the modified Kawahara equation is globally well-posed for the initial data in the Sobolev space H s (R) with s > − 3 22 .

Considering the Cauchy problem for the Korteweg-de Vries-Burgers equation ut + uxxx + ǫ|∂x| u+ (u)x = 0, u(0) = φ, where 0 < ǫ, α ≤ 1 and u is a real-valued function, we show that it is globally well-posed in Hs (s > sα), and uniformly globally well-posed in H s (s > −3/4) for all ǫ ∈ (0, 1). Moreover, we prove that for any T > 0, its solution converges in C([0, T ]; Hs) to that of the KdV equa...

We prove local well-posedness results for the semi-linear wave equation for data in H γ , 0 < γ < n−3 2(n−1) , extending the previously known results for this problem. The improvement comes from an introduction of a two-scale Lebesgue space X r,p k .

For xed = (x; t), we consider the solution u(f) to u (x; t) + Au(x; t) = f(x) (x; t); x 2 ; t > 0 u(x; 0) = u(x; 0) = 0; x 2 ; Bju(x; t) = 0; x 2 @ ; t > 0; 1 j m; where u = @u @t , u = @ u @t , R, r 1 is a bounded domain with smooth boundary, A is a uniformly symmetric elliptic di erential operator of order 2m with t-independent smooth coe cients, Bj , 1 j m, are t-independent boundary di eren...

Abstract. It is shown how Andrews’ multidimensional extension of Watson’s transformation between a very-well-poised 8φ7-series and a balanced 4φ3-series can be used to give a straightforward proof of a conjecture of Zudilin and the second author on the arithmetic behaviour of the coefficients of certain linear forms of 1 and Catalan’s constant. This proof is considerably simpler and more stream...

The quasigeostrophic model is a simplified geophysical fluid model at asymptotically high rotation rate or at small Rossby number. We consider the quasigeostrophic equation with dissipation under random forcing in bounded domains. We show that global unique solutions exist for appropriate initial data. Unlike the deterministic quasigeostrophic equation whose well-posedness is well-known, there ...

where := −∂2 t +∆ and u[0] := (u, ut)|t=0. The equation is semi-linear if F is a function only of u, (i.e. F = F (u)), and quasi-linear if F is also a function of the derivatives of u (i.e. F = F (u,Du), where D := (∂t,∇)). The goal is to use energy methods to prove local well-posedness for quasilinear equations with data (f, g) ∈ Hs × Hs−1 for large enough s, and then to derive Strichartz esti...