Related papers: Long time behavior of solutions for a damped Benja…

This paper studies the parabolic $p$-Laplace equation with $p>2$ in a moving domain under a Neumann type boundary condition corresponding to the total mass conservation. We establish the existence and uniqueness of a weak solution by the…

We consider the asymptotic behavior of the soltion to the wave equation with time-dependent damping and analytic nonlinearity. Our main goal is to prove the convergence of a global solution to an equilibrium as time goes to infinity by…

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi Equation with Neumann boundary condition and initial data a continious function. Then, we study the large time behavior of the solutions.

We study linear time fractional diffusion equations in divergence form of time order less than one. It is merely assumed that the coefficients are measurable and bounded, and that they satisfy a uniform parabolicity condition. As the main…

We study the long-time behavior of the solution of a damped BBM equation $u_t + u_x - u_{xxt} + uu_x + \mathscr{L}_{\gamma}(u) = 0$. The proposed dampings $\mathscr{L}_{\gamma}$ generalize standards ones, as parabolic…

We prove local well-posedness of the Benjamin-Ono equation for a class of bounded initial data including periodic and bore-like functions. As a consequence, we obtain local well-posedness in $H^s(\mathbb{R})+H^\sigma(\mathbb{T})$ for…

In this paper we study the inviscid limit of the Benjamin-Ono-Burgers equation in the energy space $ H^{1/2} (\R) $ or $ H^{1/2}(\T) $. We prove the strong convergence in the energy space of the solution to this equation toward the solution…

We study the long time behaviour of solutions for the weakly damped forced Kawahara equation on the torus. More precisely, we prove the existence of a global attractor in $L^2$, to which as time passes all solutions draw closer. In fact, we…

We consider the Benjamin-Ono equation on the real line for initial data in weighted Sobolev spaces. After the application of the gauge transform, the flow is shown to be Lipschitz continuous and to present a nonlinear smoothing effect. As a…

We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation where $0<\alpha \leq 1$ \begin{eqnarray*} \left\{ \begin{array}{l} \partial_t u+|\partial_x|^{1+\alpha}\partial_x u+uu_x=0,\\ u(x,0)=u_0(x), \end{array}…

We consider the quantum hydrodynamic system on a $d$-dimensional irrational torus with $d=2,3$. We discuss the behaviour, over a "non trivial" time interval, of the $H^s$-Sobolev norms of solutions. More precisely we prove that, for generic…

In this paper, we consider global strong solutions and uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model in R^d, where d=2 and 3. The well-recognized problem of the global existence of smooth solutions for the 2D…

We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent…

We study the large time behavior of Fujita-Kato type solutions to the Vlasov-Navier-Stokes system set on $\mathbb{T}^3 \times \mathbb{R}^3$. Under the assumption that the initial so-called modulated energy is small enough, we prove that the…

We will consider the damped Newton method for strongly monotone and Lipschitz continuous operator equations in a variational setting. We will provide a very accessible justification why the undamped Newton method performs better than its…

In this paper we prove long time existence for a large class of fully nonlinear, reversible and parity preserving Schr\"odinger equations on the one dimensional torus. We show that for any initial condition even in $x$, regular enough and…

Regularity and uniqueness of weak solutions of the compressible barotropic Navier-Stokes equations with constant viscosity coefficients is proven for small time in dimension $N=2,3$ under periodic boundary conditions. In this paper, the…

The large time $t$ asymptotics for scalar, constant coefficient,linear, third order, dispersive equations are obtained for asymptotically time-periodic Dirichlet boundary data and zero initial data on the half-line modeling a wavemaker…

We prove that the Benjamin-Ono equation is globally well-posed in $ H^s(\T) $ for $ s\ge 0 $. Moreover we show that the associated flow-map is Lipschitz on every bounded set of $ {\dot H}^s(\T) $, $s\ge 0$, and even real-analytic in this…

We consider the long-time behavior of the entropy solution of a first-order scalar conservation law on a Riemannian manifold. In the case of the Torus, we show that, under a weak property of genuine non-linearity of the flux, the solution…