Related papers: Dynamics of solutions in the generalized Benjamin-…

We consider the elliptic and parabolic superquadratic diffusive Hamilton-Jacobi equations with homogeneous Dirichlet conditions. For the elliptic problem in a half-space, we prove a Liouville-type classification, or symmetry result, which…

We consider the semilinear heat equation $$u_t-\Delta u=f(u) $$ for a large class of non scale invariant nonlinearities of the form $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function (which includes for…

We show that uniqueness results of the kind those obtained for KdV and Schr\"odinger equations ([7], [28]), are not valid for the dispersion generalized-Benjamin-Ono equation in the weighted Sobolev spaces $$H^s(\R)\cap L^2(x^{2r}dx),$$ for…

The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence,…

We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, a Schr\"odinger-type equation with a nonlocal, convolution-type nonlinearity: $iu_t+\Delta u +\left(|x|^{-(d-2)} \ast |u|^{p} \right) |u|^{p-2}u = 0,…

This paper studies the existence and uniqueness problem for the generalized Benjamin-Bona-Mahony (gBBM) equation with quasi-periodic initial data on the real line. We obtain an existence and uniqueness result in the classical sense with…

In this paper, we analyze finite difference schemes for Benjamin-Ono equation, u_t = uu_x + Hu_{xx}, where H denotes the Hilbert transform. Both the decaying case on the full line and the periodic case are considered. If the initial data…

We consider the blow up problem in the energy space for the critical (gKdV) equation in the continuation of part I and part II. We know from part I that the unique and stable blow up rate for solutions close to the solitons with strong…

In this work, we study the behavior of blow-up solutions to the multidimensional restricted Euler--Poisson equations which are the localized version of the full Euler--Poisson system. We provide necessary conditions for the existence of…

We consider the long time dynamics of large solutions to the Benjamin-Ono equation. Using virial techniques, we describe regions of space where every solution in a suitable Sobolev space must decay to zero along sequences of times.…

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…

We consider the $k$-dispersion generalized Benjamin-Ono ($k$-DGBO) equations. For nonlinearities with power $k \geq 4$, we establish local and global well-posedness results for the associated initial value problem (IVP) in both the critical…

This paper is devoted to the analysis of blow-up solutions for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities \[ iu_{t}+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u. \] When $p_1=\frac{4}{N}$ and…

We construct local solutions to the Benjamin-Ono equation for quasi-periodic initial data. The solution is unique among limits of smooth solutions and depends continuously on the data. Our result applies to a richer class of quasi-periodic…

We show that the initial-value problem for the Benjamin-Ono equation on $\mathbb{R}$ with $L^2(\mathbb{R})$ rational initial data with only simple poles can be solved in closed form via a determinant formula involving contour integrals. The…

Fourth-order semilinear parabolic equations of the Cahn--Hilliard-type (01) u_t + \D^2 u = \g u \pm \D (|u|^{p-1}u) in \Omega \times \re_+, are considered in a smooth bounded domain $\O \subset \ren$ with Navier-type boundary conditions on…

We prove that the modified Benjamin-Ono equation is globally wellposed in $H^s$ for $s\ge 1/2$.

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 establish the local well-posedness of the generalized Benjamin-Ono equation $\partial_tu+\mathcal{H}\partial_x^2u\pm u^k\partial_xu=0$ in $H^s(\R)$, $s>1/2-1/k$ for $k\geq 12$ and without smallness assumption on the initial data. The…

Blow-up rates are established for general solutions to the quasilinear diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ in the range of exponents $1<p<m$, $\sigma>0$. More precisely, if…