For $s > 1$ two sequences of CH solutions living in a bounded subset of the Sobolev space $H^s(\mathbb{R})$ are constructed, whose distance at the initial time is converging to zero while at any later time it is bounded below by a positive constant. This implies that the solution map of the CH equation is not uniformly continuous in $H^s(\mathbb{R})$.

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 converges to its average value in $L^{p}$, $1\leq p < +\infty$. We give a partial result in the general case.

In this paper, an elliptic system with sign-changing weights is considered. By variational methods, we show that the problem has at least three nontrivial solutions.

In this paper, we establish the existence of global weak solutions to the stochastic Landau-Lifshitz equation in $\mathbb R^d$ for any $d > 0$. When restricted to the real line $\mathbb R^1$, we show that the Cauchy problem of this equation admits a unique global smooth solution with the difference method.

There are many papers devoted to the study of shallow water flows especially by numerical methods. In this note, we give a mathematical analysis, based on the theory of semigroups of operators on Hilbert space, of two linearized problems involving the Saint-Venant equations governing shallow water flows. The first one concerns the classical linearized Saint-Venant problem, while the second one deals with the water quality problem: the evolution of a concentration of products in a flow solution of the first problem.

In this paper we propose a variational model for image restoration based on the minimization of a convex functional with a gradient constraint. This approach has a smoothing effect on the degraded image, as it does not allow rapid changes in the image content. We also consider the evolutionary model of the resulting nonlinear differential equation. Furthermore, we discuss the dual problem associated with the model and propose a numerical algorithm for approximating its solution.

This paper is concerned with nonlocal generalizations of the inviscid Burgers equation arising as amplitude equations for weakly nonlinear surface waves. Under homogeneity and stability assumptions on the involved kernel it is shown that the Cauchy problem is locally well posed in $H^2(\mathbb R)$, and a blow-up criterion is derived. The proof is based on a priori estimates without loss of derivatives, and on a regularization of both the equation and the initial data.

In this paper, we present four models for distributed delay logistic equations with harvesting. We study the behavior of the solutions and the stability of the equilibria in each models. Further, we consider the existence of bifurcation for a distributed delay logistic equation with harvesting and a discounted past. We show that the long-term outcomes are affected by the discounting of the past history, and the nature of the harvesting strategy (constant or proportional harvesting are considered). Finally, a numerical simulation is performed to illustrate the obtained results.

We investigate the Navier-Stokes equations in a suitable functional setting, in a three-dimensional bounded Lipschitz domain $\Omega$, equipped with ``free boundary'' conditions. In this context, we employ the Fujita-Kato method and prove the existence of a local mild solution. Our approach makes essential use of the properties of the Hodge-Laplacian in Lipschitz domains.

We establish local and global well posedness for the initial-value problem associated to the one-dimensional Schröodinger-Debye (SD) system for data in Sobolev spaces with low regularity. To obtain local results we prove two new sharp bilinear estimates for the coupling terms of this system in the continuous and periodic cases. Concerning global results, in the continuous case, the system is shown to be globally well posed in $H^s\times H^s, -3/14 < s < 0$. For initial data in Sobolev spaces with high regularity ($H^s\times H^s,\; s > 5/2$), Bidégaray [4] proved that there are one-parameter families of solutions of the SD system converging to certain solutions of the cubic nonlinear Schröodinger equation (NLS). Our results below $L^2\times L^2$ say that the SD system is not a good approach to the cubic NLS in Sobolev spaces with low regularity, since the cubic NLS is known to be ill posed below $L^2$. The proof of our global result uses the \textbf{I}-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.

We prove, by using a method introduced by Constantin [6], that if the solution of the Cauchy problem associated with the KP-BBM-II equation has compact support for all times, then this solution vanishes identically. The only restriction is that the support in the $y$-direction has to be small.