We consider the scattering transform for Schrödinger operators with energy dependent potentials. We prove unique invertibility of the transform when there are no bound states and find a simplified recovery formula. We construct as a special case a one-soliton "breather". As an application we prove a global existence theorem for a class of non-linear partial differential equations.

A general existence and uniqueness theorem for degenerate elliptic Bellman equations in bounded domains is proved. Functional classes $C^{2+\alpha}(D)$ and $C^{1,1}(D)$ are the classes where solutions are looked for. This theorem has a very broad range of applicability. Equations $u_{t}=u_{xx}$ and $$ P_{m}(u_{xx})=\sum_{k=0}^{m-1}c_{k}(x)P_{k}(u_{xx}), $$ where $P_{k}(u_{xx})$ is the $k$th elementary symmetric polynomial of eigenvalues of the matrix $u_{xx}$ are particular cases of equations under consideration.

An eigenvalue problem for elliptic variational inequalities is considered. The existence of multiple solutions is proved, when the operator is a small (non-odd) perturbation of an odd operator. To this aim, techniques of nonsmooth critical point theory are employed

The linear integral equation of Abel-Volterra type $$ \varphi(x)={\frac{a(x)}{\Gamma(\alpha)}}\int^{x}_{0} {\frac{\varphi(t)}{(x-t)^{1-\alpha}}}\ dt+f(x) \quad (0<x<\infty,\ 0<\alpha <1)\tag{*} $$ is investigated. The asymptotic behavior of the solution $\varphi(x)$ as $x\to 0$ is studied, provided that the functions $a(x)$ and $f(x)$ have special power asymptotic expansions near zero. It is shown that in certain cases such an asymptotic solution $\varphi(x)$ of the equation $(*)$ coincides with its explicit solution. Examples are also given.

If the seasonal cycle as well as the long response times of the climate system are taken into account, one-layer energy balance climate models give rise to parameter-dependent functional reaction-diffusion equations with 1-periodic forcing and a time delay $T\gg 1$. We show that the principal branch of fixed points of the corresponding time-1-map is S-shaped in the sense that it is a simple curve with an even number of turning points. This curve connects $(0,\mathbf {0})$ and $(\infty , \infty )$ within $(0,\infty) \times C([-T,0]\times M,(0,\infty))$, $M$ a compact, oriented Riemannian surface. The paper is a continuation of [13], where a case without time-delay was considered.

This paper proves the global existence of small radially symmetric solutions to the nonlinear Schrödinger equations of the form $$ \begin{cases} i\partial_t u+\frac12\triangle u=F(u, \nabla u, \bar u, \nabla\bar u),& (t,x)\in\mathbb{R}\times\mathbb{R}^n,\\ u(0,x)=\varepsilon_0\phi(|x|),& x\in\mathbb{R}^n, \end{cases} $$ where $n\ge 3$, $\varepsilon_0$ is sufficiently small, $|x|=(\sum\limits_{1\le j\le n}x_j^2)^{1/2}$, $$ F =\sum_{\ell_0\le|\alpha|\le\ell_1}\lambda_\alpha u^{\alpha_1}\bar u^{\alpha_2} +\sum_{\substack {0\le|\alpha|\le\ell_2\\ 1\le |\beta|\le\ell_3}} \lambda_{\alpha\beta}u^{\alpha_1}\bar u^{\alpha_2}(\sum_{1\le j\le n}|\partial_ju|^2)^{\beta_1}(\sum_{1\le j\le n}(\partial_ju)^2)^{\beta_2} (\sum_{1\le j\le n}(\partial_j\bar u)^2)^{\beta_3} $$ with $\lambda_\alpha, \lambda_{\alpha\beta}\in\mathbb{C}$, $\ell_1, \ell_2, \ell_3\in\mathbb{N}$, $\ell_0=3$ for $n=3, 4$, and $\ell_0=2$ for $n\ge 5$. The method depends on the combination of a gauge transformation and generalized energy estimtes and does not require the condition such that $$ \partial_{\nabla u}F\ \text{ is pure imaginary} $$ which is needed for the classical energy method.

We study resonances and resonance states in the problem of the scattering of acoustic waves by a spherically symmetric inhomogeneity of the medium using the Lax-Phillips approach. The density of the medium is assumed to be constant outside the sphere $r=a$ and to have either (i) a finite jump on this sphere, or (ii) a singularity corresponding to infinite rarefaction, or (iii) a singularity corresponding to infinite condensation. We show that, for each value of the angular momentum $\ell$, there exists an infinite sequence of resonances accumulating to infinity. Explicit asymptotic formulae for resonances and resonance states are given. In case (i) the resonances are asymptotically close to a horizontal line in the upper half-plane, in case (ii) (or (iii)) they converge to the real axis and are asymptotically close to the eigenvalues of the Dirichlet (or Neumann) problem in the interior of the scatterer. We give an explicit description of all main elements of the Lax-Phillips scheme: incoming and outgoing subspaces, spectral representation and $S$-matrix. Using this description, the Foias-Sz.-Nagy theory of contraction operators and the above-mentioned asymptotic formulae for resonances, we show that in all three cases (i)--(iii) the resonance states form a Riesz basis in the orthogonal complement to the incoming and outgoing subspaces.

This paper discusses the existence and the convergence of inertial manifolds for approximations to semi-linear evolution equations in Banach spaces. Our approximation considered here is closely related to Chernoff's product formulas. It is shown that the approximation possesses an inertial manifold and this manifold converges to the inertial manifold for the evolution equation. A "parabolic" version of Chernoff's lemma is established and used to prove the convergence theorems. As an application the schemes of "Crank-Nicholson type" are considered. Finally, the existence of inertial manifolds for the evolution equation is discussed under the condition that the approximations possess inertial manifolds.

We consider an autonomous perturbation of an autonomous system of ordinary differential equations, when a periodic solution of the autonomous system has a nontrivial multiplier equal to $1$ or $-1$. We derive criteria for the perturbed system to have an orbitally asymptotically stable periodic solution using a technique of stable fixed points of monotone operators together with a bifurcation technique due to M.A. Krasnoselskii.

We state a global existence and decay result for a class of semilinear evolution equations in a Banach space. The result is applied to semilinear parabolic equations of second and fourth order in $[0, \infty)\times \Bbb R^n.$

A local existence and uniqueness result for the functional differential equation in a Banach space $X$ $$ \begin{equation} x'(t)=f(t)x(t)+g(t)x_t, \quad x_0=\phi, \ x(0)=h, \ \{\phi,h\}\in L^1(-R,0;X)\times X \tag{FDE} \end{equation} $$ is obtained, for the case where the operators $f(t)$ satisfy only a local dissipativity condition and the operators $g(t)$ are only locally Lipschitz continuous. This is done by relating (FDE) to the evolution equation in ${L^1(-R,0;X)\times X}$ $$ \begin{equation} u'(t)=A(t)u(t),\quad u(0)=\{\phi , h\}, \tag{E} \end{equation} $$ where $$ \begin{align} &D(A(t)) = \{\{\phi,h\}\in L^1(-R,0;X)\times X ; \ \phi\in W^{1,1}(-R,0;X), h\in D(f(t)),\phi (0)=h\} \\ &(t)\{\phi ,h\} =\{\phi ',\, f(t)h+g(t)\phi\}. \end{align} $$ It is shown that if $u(t)$ is the limit solution of (E), then $u(t)=\{x_t,x(t)\}$, where $x(t)$ is the integral solution

We consider the propagation of electromagnetic waves in a non-homogeneous medium. The related constitutive relations contain time and space dependent convolution kernels. Since they are a priori unknown, a basic question concerns their identification. In the present paper, this is obtained by reducing the problem to a system of nonlinear integral equations of the second kind. Via a Contraction Theorem, we prove local (in time) existence and uniqueness results. Lipschitz continuous dependence upon the data is also proved.

The Cauchy problem of the Korteweg-de Vries equation on a periodic domain $S$, a unit circle in the complex plane establishes a nonlinear map $K$ from the initial data $\phi \in H^s(S) $ to the solution $u(x,t) \in C([-T,T];H^s(S) )$ for $s\geq 0$ (cf. [4] and [10]). Based on Bourgain's new approach [4] to periodic solutions of the KdV equation, it is shown that the nonlinear map $K$ is analytic from $H^s(S) $ to $C([-T,T];H^s(S) ) $ in the following sense. For any $\phi \in H^s(S) $, there exists a $\delta > 0 $ such that if $h\in H^s(S) $ with $\|h\| _{H^s(S) } \leq \delta $ and $\int _S h(x)dx = 0$, then $K (\phi + h) $ has the following Taylor series expansion: $$ K(\phi +h ) =\sum ^{\infty }_{n=0} \frac{K^{(n)} (\phi ) [h^n ] }{n!}, $$ where the series converges in the space $C([-T,T];H^s(S) ) $ uniformly for $\|h\| _{H^s (S)} \leq \delta $ and $K^{(n)} (\phi ) $ is the n-th order Fr\'{e}chet derivatives of $K$ at $\phi $. As a consequence, the periodic solution $u(x,t)$ of the KdV equation can be obtained by solving a series of linear problems since each term in the above Taylor series is a solution of a linearized KdV equation.

We develop a perturbation theory for strongly continuous backward evolutionary systems and their adjoint systems. The theory is not based on generating families but on certain operator families called step responses and cumulative outputs. The perturbation problem is reduced to solving an abstract Stieltjes integral equation of nonconvolution type. The theory is well suited for treating structured population models.

We study the existence of positive solutions for elliptic systems of the form (1) below, when $f$ has asymptotic behavior at $+\infty$ as $u^\sigma$ and $g$ satisfies some subcritical growth. Our result guarantees a nontrivial solution $(u,v)$ to the system $$ -\Delta u = u^\sigma + v^q,\;\;\; -\Delta v = u^p \quad \text {in} \quad \Omega; \quad u=v=0 \quad \text {on} \quad \partial \Omega, $$ where $1 <pq <\sigma < (N+2)/(N-2).$

Let $ \Omega$ be a bounded, smooth domain in ${\Bbb R}^N$, $N\ge 2$. We consider the elliptic boundary value problem $$ \begin{align} \Delta u + u_+^p - u_-^q & =0 \quad\hbox{ in } \ \Omega , \\ u & =0 \quad\hbox{ on } \ \partial \Omega , \end{align} $$ where $1< p < (N+2)/(N-2)$, $0<q<1$, $u_+ = \max\{ u,0\}$, $u_- = -\min\{ u,0\}$. We prove that for certain small domains $\Omega$, no solution to this problem which changes sign in $\Omega$ exists. This answers affirmatively a conjecture raised in [3] where existence of at least one signed solution is established under a largeness condition for the domain.