We consider a boundary value problem in a strip, which is obtained by linearization of the two-dimensional problem of a stream about a slender cylinder, semisubmerged in a heavy fluid of finite depth; it is assumed that the cylinder has uniform, subcritical speed in the direction orthogonal to its generators. We discuss in particular the waveless statement of the problem, characterized by the asymptotic condition of vanishing flow oscillations at infinity. By suitable variational formulations of the problem, we find two classes of solutions, differing by their regularity properties. The most regular solutions exist for particular shapes of the cylinder's section and provide a velocity field which is everywhere continuous and bounded in the strip. The solutions of the other class exist for every (reasonably smooth) symmetric cylinder's profile and have finite energy, but are singular at two points on the strip boundary, representing the points where the free surface meets the cylinder's hull. The relevance of these results for the solvability of the nonlinear problem is discussed.

This paper proves the estimate $\Vert{{u_\varepsilon}} - u \Vert_{L^1(Q_T)} \le C \varepsilon^{1/5}$, where, for all $\varepsilon>0$, ${{u_\varepsilon}}$ is the weak solution of $({{u_\varepsilon}})_t + {\hbox{div}} ({{\bf q}} \ f({{u_\varepsilon}}))- {\Delta}(\varphi({{u_\varepsilon}})+\varepsilon {{u_\varepsilon}}) = 0$ with initial and boundary conditions, $u$ is the entropy weak solution of $u_t + {\hbox{div}} ({{\bf q}} f(u))- {\Delta}(\varphi(u)) = 0$ with the same initial and boundary conditions, and $C>0$ does not depend on $\varepsilon$. The domain $\Omega$ is assumed to be regular and $T$ is a given positive value.

We give an extension of the a priori estimate, obtained in [8], for a solution of the inhomogeneous wave equation in ${\bf R}^n\times{\bf R}$, where $n=2$ or $n=3$. As an application, we study the asymptotic behavior as $t \to \pm \infty$ of solutions to systems of semilinear wave equations. The discrepancy of the speeds of propagation may make a significant difference from the case of common propagation speeds. (See also Theorem 3.3 and 3.4). Whether such a phenomenon occurs or not depends on the type of the interaction determined by the nonlinearities.

We study the elliptic-hyperbolic Davey-Stewartson system $$ \left\{ \begin{array}{l} i\partial_t u + (\partial_{x_1}^2 + \partial_{x_2}^2) u = c_1 |u|^2 u + c_2 u \partial_{x_1} {\varphi}, \qquad (t,x) \in {\bf R} \times {\bf R}^2, \\ (\partial_{x_1}^2 - \partial_{x_2}^2) {\varphi} = \partial_{x_1} |u|^2, \\ u(0,x) = u_0 (x), \qquad x \in {\bf R}^2, \end{array} \right. $$ where $c_1$, $c_2 \in {\bf R}$, $u$ is a complex valued function and ${\varphi}$ is a real valued function. The system can be translated to nonlocal nonlinear Schr{\"o}dinger equations after a rotation and a rescaling $$ {\mbox{(NLS)}} \qquad\qquad\qquad \left\{ \begin{array}{l} i\partial_t u + (\partial_{x_1}^2 + \partial_{x_2}^2) u \\ = c_0 |u|^2u + c_1 u \displaystyle{\int_{x_2}^\infty} \partial_{x_1} |u|^2 ({x_1},{x_2}')d{x_2}' + c_2 u \displaystyle{\int_{x_1}^\infty} \partial_{x_2} |u|^2 ({x_1}',{x_2})d{x_1}',\\ u(0,x) = u_0(x), \end{array} \right. $$ where $c_0,c_1,c_2 \in {\bf C}$. Our purpose in this paper is to prove that if the norm $ \| ( \prod_{j=1}^2 \cosh \theta x_j )u_0 \|_{3,0} + \| ( \prod_{j=1}^2 \cosh \theta x_j )u_0 \|_{0,3} $ for $\theta \neq 0$ is sufficiently small, then small solutions of (NLS) exist and become analytic with respect to $x$ for any $t \ne 0$ . Here we denote the weighted Sobolev space by ${\bf H}^{m,s} = \bigm\{ \phi \in {\bf L}^2 ; \| \phi \|_{m,s} = \bigm\| (1 + x_1^2 + x_2^2)^{s/2} (1 - \partial_{x_1}^2 - \partial_{x_2}^2)^{m/2} \phi \bigm\| < \infty \bigm\}$, $m,s \in {\bf R}^+$.

We study the comparison principle and interior Hölder continuity of viscosity solutions of $$ F(x,u(x),Du(x),D^2u(x))+H(x,Du(x))-f(x)=0\quad\mbox{in }\Omega , $$ where $F$ satisfies the standard "structure condition'' and $H$ has superlinear growth with respect to $Du$. Following Caffarelli, Crandall, Kocan and Święch [3], we first present the comparison principle between $L^p$-viscosity subsolution and $L^p$-strong supersolutions. We next show the interior Hölder continuity for $L^p$-viscosity solutions of the above equation. For this purpose, modifying some arguments in [1] by Caffarelli, we obtain the Harnack inequality for them when the growth order of $H$ with respect to $Du$ is less than $2$.