In this article we study and derive explicit formulas for the boundary layers occurring in the linearized channel flows in the limit of small viscosity. Our study is based on classical boundary layer techniques combined with a new global treatment of the pressure term.

We study the initial-value problems for the Davey-Stewartson systems and the Ishimori equations. Elliptic-hyperbolic and hyperbolic-elliptic cases were treated by the inverse scattering techniques ([2--4, 10, 13--15, 32] for the Davey-Stewartson systems and [28, 29, 33] for the Ishimori equations). Elliptic-elliptic and hyperbolic-elliptic cases were studied (in [16, 17] for the Davey-Stewartson systems and [31] for the Ishimori equations) without the use of the inverse scattering techniques. Existence of a weak solution to the Davey-Stewartson systems for the elliptic-hyperbolic case is also obtained in [16] with a smallness condition on the data in $L^2$ and a blow-up result was also obtained for the elliptic-elliptic case. By using the sharp smoothing property of solutions to the linear Schrödinger equations the local existence of a unique solution to the Davey-Stewartson systems for the elliptic-hyperbolic and hyperbolic-hyperbolic cases was established in [30] in the usual Sobolev spaces with a smallness condition on the data. We prove the local existence of a unique solution to the Davey-Stewartson systems for the elliptic-hyperbolic and hyperbolic-hyperbolic cases in some analytic function spaces without a smallness condition on the data. Furthermore we prove existence of global small solutions of these equations for the elliptic-hyperbolic and hyperbolic-hyperbolic cases in some analytic function spaces.

We consider the equation of turbulent burst propagation $$ u_t=l(t)\Delta u^m-k\frac{u^m}{l(t)} $$ with $m>1$ and $k \ge 0$. Here $l(t)$ denotes the radius of the support of the turbulent distribution. The equation possesses special self-similar solutions of the form $$ u(x,t)= t^{-\mu/(m-1)}f(\zeta),\ \ \zeta=|x|t^{\mu-1}, $$ which are non-negative and have compact support in the space variables. A most interesting feature of such solutions is that the decay exponent is anomalous. It has been proved that general solutions approach for large $t$ these self-similar profiles (intermediate asymptotics). We prove here the analytic dependence with respect to the relevant parameter $k$, of the anomalous exponent appearing in the special solution. The method used is in itself a complete proof of the existence of the self-similar solution with anomalous exponent, different from that recently given in [11]. This new proof also gives us the analyticity of $f^{m-1}$, as a function of $k$ and $\z$. The main technical contribution is a shooting method from the free boundary, which uses a fixed-point theorem for complex analytic functions, and produces analytic regularity. A major novelty of the paper consists in showing that the family of self-similar solutions can be continued for $k<0$ (representing a reaction term) down to a critical value $-k_*<0$ at which $\mu$ has an asymptote.

A bifurcation problem or a nonlinear eigenvalue problem for degenerate quasilinear elliptic equations with Dirichlet boundary condition is studied. A bifurcation branch of radially symmetric solutions for the problem is obtained. The asymptotic behavior of the energy along to this branch is also calculated.

We consider quasiperiodic positive definite Lagrangian systems and establish the existence of one or more solutions homoclinic (namely asymptotic as $t\to\pm\infty$) to a quasiperiodic solution. These results are obtained by means of a variational approach. An application is carried out to quasiperiodically perturbed Lagrangian systems.

This paper is concerned with the asymptotic behavior of gradients of solutions of the minimal surface equation in two dimensions, for solutions which vanish on the long sides of a semi-infinite strip. Using arguments based on maximum principles, an exponential decay estimate for the gradient is obtained with a decay rate which coincides with that for Laplace's equation. The estimate is fully explicit in terms of the boundary data on the near end of the strip. The techniques are first illustrated for harmonic functions on semi-infinite strips or cylinders. The results are relevant to principles of Saint-Venant and Phragmén-Lindelöf type.

We study the stability and instability properties of standing waves for the equation $iu_t+\Delta u+a\vert u\vert^{p-1}u+E_1(\vert u\vert^2)u=0$ in $\Bbb R^2$ or $\Bbb R^3$, which derives from the generalized Davey-Stewartson system in the elliptic-elliptic case. We show that if $n=2$ and $a(p-3)<0$, then the standing waves generated by the set of minimizers for the associated variational problem are stable. We also show that if $n=3$, $a>0$ and $1+4/3<p<5$ or $a<0$ and $1<p<3$, then the standing waves are strongly unstable. We employ the concentration compactness principle due to Lions and the compactness lemma due to Lieb to solve the associated minimization problem.

In this paper, we establish some existence results for boundary and periodic value problems for systems of nonlinear differential equations with right-hand side satisfying a Berntein-Nagumo growth condition. Hartman's condition ($|f| \le 2k(\langle x,f\rangle + |x'|^2) + K$) is not assumed. This assumption is replaced by one which is automatically satisfied in the scalar case.

The existence of multiple periodic solutions of a nonlinear Lienard system is treated. The proof is based on the theory of topological degree and monotone operators.

In this paper we are concerned with the uniqueness of positive solutions of boundary value problems for quasilinear differential equations of the type $(|u'|^{m-2}u')' + p(t)f(u) = 0, $ $ m>1$. The key ingredient of the method is the generalized Prüfer transformation. These problems arise, for example, in the study of the $m$-Laplace equation in annular regions.

This paper is devoted to two problems for the nonautonomous integrodifferential equation in a Banach space $X$ of hyperbolic type $$ \text{$u'(t) = A(t)u(t)+ \int_0^t B(t,s)u(s) ds + f(t)$ for $t\in[0,T]$, and $u(0) = u_0$.} $$ One is the problem of existence and uniqueness of classical solutions without assuming that the common domain of $A(t)$ is dense in $X$, and the other is the regularity problem in the case where the common domain of $A(t)$ is dense in $X$. The regularity result will play a role in developing an abstract theory which can be applied to the second-order integrodifferential equations with the third kind boundary conditions.

We consider the three-dimensional exterior problem for steady Navier-Stokes equations. We prove, under an assumption on the smallness of external data, existence (and uniqueness) of solutions with the same spatial decay at infinity as that of the fundamental solution of the Stokes operator. In this sense the presented result is optimal and naturally completes classical results of Finn [11].

We introduce necessary and sufficient conditions for the existence of periodic solutions of the equation $$ u'' + f(u)u' + g(u) = h(t), $$ where $g$ has a singularity of repulsive type at the origin and $f$ is arbitrary.

We study the Cauchy problem for nonsymmetric hyperbolic differential operators with different time-scales $P=\frac{1}{\epsilon} P_0(\frac{\partial}{\partial x}) + P_1(x,t,\frac{\partial}{\partial x}),$ $0<\epsilon \ll1$. Sufficient conditions for well-posedness independently of $\epsilon$ are derived. The bounded derivative principle is also shown to be valid, i.e., there exists smooth initial data such that a number of time derivatives are uniformly bounded initially. This gives an existence theory for the limiting equations when $\epsilon \rightarrow 0$. We apply our theory to the slightly compressible upper convected Maxwell model describing viscoelastic fluid flow.

In this paper we establish an asymptotic formula for "small" solutions of the delay equation $\dot x(t) = a x(t-b|x(t)|)$, where $a$ and $b$ are positive constants.

We consider a system of the form $$ \begin{align} & u''+au'=H_v(u,v)-h(t) \\ & v''+ bv'=H_u(u,v) - k(t), \end{align} $$ where $h,k $ are locally integrable and $T$-periodic, and $H$ is a $C^1$ function defined on $(0,\infty)\times (0,\infty)$, for which a good model is given by $$ H(u,v) = -( {1\over u^\alpha } + {1\over v^\beta } ),\quad \alpha ,\beta > 0 . $$ We state conditions under which existence of positive, $T$-periodic solutions for this system is ensured. We also study the problems of uniqueness and existence of multiple solutions in some special cases.