We consider the Cauchy problem for the nonlinear Schrödinger equation in one space dimension with interaction satisfying null gauge condition. We prove the local well-posedness of the problem in the Sobolev space $H^{1/2}$. The method depends on the nonlinear gauge transformation and on sharp smoothing estimates for the null gauge form.

This paper deals with decay properties for the solutions of second-order damped abstract evolution equations. A large number of applications is given, concerning damped wave systems, more general systems of evolution type, and Lagrangian quasi-variational systems. We consider the case in which the time-dependence of the damping term is bounded.

In this paper we give some new necessary and sufficient conditions under which the zero solution of the linear delay-differential equations with variable coefficients $$ x'(t)=A(t)x(t-\tau) \tag L $$ is exponentially asymptotically stable. For example, in the case $$ A(t) = -\rho(t)\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}, $$ where $\rho(t) > 0,$ $\lim_{t \to \infty} \int_{t-\tau}^t \rho(s)\,ds = q>0$ and $|\theta| < \frac{\pi}{2},$ the zero solution of (L) is exponentially asymptotically stable if and only if $ q <\frac{\pi}{2}-|\theta|$.

We study the problem of minimal regularity required to ensure local well-posedness for systems of nonlinear wave equations with different propagation speeds in three space dimensions $$ \begin{align} (\partial_t^2 -C_1^2\Delta)u &= F(u,v,\partial u,\partial v), \\ (\partial_t^2 -C_2^2\Delta)v &= G(u,v,\partial u,\partial v). \end{align} $$ We prove that if $C_2 > C_1$ and $F$, $G$ have the form $\p u \cdot v$, then the problem is well-posed in $H^1$. Our proof is based on the same type of space-time estimates as those of Klainerman and Machedon.

In this paper, nonautonomous Lotka--Volterra systems of "pure-delay type" are considered and some sufficient conditions on the global asymptotic stability are obtained. As a corollary, we show that, under the conditions of Theorem 2.1 in Kuang [11], the system remains globally asymptotically stable provided the delays are sufficiently small. Both finite and infinite delays are allowed in the systems. Our results give an affirmative answer to the two open problems due to Kuang. The results are established by constructing suitable Lyapunov functionals.

Equations involving the $p$-Laplacian with a term in the critical growth range are considered. Existence results are obtained under minimal assumptions on the lower order perturbation. The problem is studied by means of variational methods; in particular, a problem with linking geometry is treated thanks to the orthogonalization technique introduced in [13].

The linear transport equation is supplemented with a hysteresis operator to obtain a local model of adsorption--desorption. The resulting nonlinear initial-boundary-value problem is shown to have a unique differentiable global solution when the hysteresis functional has a symmetric convex graph. An explicit solution is given for an elementary example, and a formula is given for the representation of smooth cases of such functionals.

An inverse problem for a nonlinear ordinary differential equation is discussed. We prove an existence theorem of a nonlinear term with which a boundary value problem admits a solution. This is an improvement of earlier work by A. Lorenzi. We also prove a uniqueness theorem of the nonlinear term.

We show that every weak solution $\mathbf{u}$ of the steady Navier--Stokes equations in a bounded domain $\Omega \subseteq \mathbb{R}^N$, $N\ge 5$, satisfying additionally $\mathbf{u}\in L^q(\Omega )$, where $ q\ge 4$ and $q > N/2$ (for the Dirichlet problem) or $ q\ge 4$ and $q > N/4$ (for the space periodic problem), is regular.

We propose a new method to prove the strong asymptotic stability of certain distributed evolutionary systems. It can be applied in some cases where LaSalle's invariance principle cannot be applied due to the lack of compactness.