In this paper we obtain a bifurcation result for parameterized families of $k$ set-contractive perturbations of closed Fredholm operators of index zero by means of an elementary degree theory and its relationship to a homotopy invariant for curves of closed Fredholm operators called the parity. This theorem is applied to the study of bifurcation of nonlinear Sturm Liouville problems on an unbounded interval.

Let $\Omega$ be a $C^{1,1}$ and bounded domain in $R^{n},$ with $n\geq2.$ Let $K$ and $f$ be two nonnegative functions on $\Omega$ belonging to $L^{p} ( \Omega ) $ for some $p>n/2$ and let $\alpha>0.$ In this paper we prove existence and uniqueness of a strong solution $u\in W_{loc}^{2,p} ( \Omega ) \cap C ( \overline{\Omega} ) $ for the elliptic Dirichlet problem $-\Delta u+\lambda u=Ku^{-\alpha}+f$ in $\Omega,$ $u>0$ in $\Omega,$ $u=0$ in $\partial\Omega.$ Moreover, for the case $\lambda=0,$ $f=0,$ we prove, under the weaker hypothesis $p>\frac{ ( \alpha^{2}+1 ) n}{2\alpha^{2}+n},$ the existence of a solution $u\in W_{loc}^{2,p} ( \Omega ) $ for the above problem that satisfies, in a suitable extended sense, the boundary condition $u=0$ on $\partial\Omega.$

Local well-posedness for the generalized KdV equation is obtained in a class of functions analytic on a strip around the real axis without shrinking the width of the strip in time. The proof relies on space-time estimates that previously have been used mainly for low-regularity spaces.

It is established, via variational methods, the existence and multiplicity of solutions for a class of semilinear elliptic equations in ${{\mathbb R}^N}$. The condition on the potential implies that the associated linear problem possesses a sequence of positive eigenvalues. This fact is used to study double resonant problems under a local nonquadraticity condition at infinity and pointwise limits. For the existence of solution the nonlinearity may satisfy a critical growth condition.

The long time behavior of a class of autonomous semilinear parabolic equations is investigated. We employ the recently introduced concept of Inertial Manifold with Delay to construct a new family of approximate inertial manifolds (AIM) of an exponential order. These AIMs contain all steady states of the system under consideration.

In this paper we study the system of elastic waves in a bounded domain in ${{\mathbb R}}^3$ with a localized dissipation given by a function $q(x)$ in $L^r(\Omega )$. The dissipation is effective only on a neighborhood of part of the boundary of the domain. We prove algebraic decay rate of energy. In order to obtain this result we develop new energy identities for this system. Our result extend previous results for the system of elasticity and wave equation.

Given an interval matrix system with discrete- or continuous-time dynamics, the flow-invariance of a rectangular set with arbitrary time-dependence is introduced as a concept of geometric nature. A theorem for its characterization provides a link between the vector function describing the time-dependence of the rectangular set and a constant matrix, which, in a certain sense, dominates the interval matrix. For rectangular sets with exponential time-dependence, this link becomes a system of algebraic inequalities. If there exist flow-invariant rectangular sets approaching the state space origin for an infinite time horizon, then the interval matrix system exhibits two special types of asymptotic stability (called componentwise asymptotic stability - for arbitrarily time-dependent rectangular sets, and componentwise exponential asymptotic stability - for exponentially time-dependent rectangular sets), whose equivalence is proved. A necessary and sufficient condition for componentwise asymptotic stability is derived, which adequately exploits Schur or Hurwitz stability of the constant matrix dominating the interval matrix.

If the Navier-Stokes equations are averaged by a local, spacial filter (denoted by an over-bar) the following system results: $$\nabla \cdot {\overline{u}} = 0 \ \text{and}\ {\overline{u}}_t + {\overline{u}} \cdot \nabla {\overline{u}} - Re^{-1} \ \Delta {\overline{u}} + \nabla \cdot \mathbb R(u) + \nabla {\overline{p}} = {\overline{f}},$$ where $\mathbb R(u): = {\overline{u \ u}} - {\overline{u}} \ {\overline{u}}$ denotes the Reynold's stresses. To close this system, various models for the Reynold's stresses of the form $\mathbb R(u) \cong \tilde {\mathbb R}({\overline{u}})$ are used. The desire is that the solution to the resulting system (call it $w$) be close to ${\overline{u}}$. However, models are often validated instead by calculating $u$ and explicitly checking $||\mathbb R(u) - \tilde {\mathbb R}({\overline{u}})||$. This report studies conditions under which the latter being small implies $w$ is close to ${\overline{u}}$. (we call this {\it verifiability} of a model.) Since ${\overline{u}} \rightarrow u$ as $\delta \rightarrow 0$, we also study when it can be proven that $w\rightarrow u$ as $\delta \rightarrow 0$.