### A free boundary problem describing the saturated-unsaturated flow in a porous medium.

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Given a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This...

We study in the space of continuous functions defined on [0,T] with values in a real Banach space E the periodic boundary value problem for abstract inclusions of the form ⎧ $x\in S(x\left(0\right),se{l}_{F}\left(x\right))$ ⎨ ⎩ x (T) = x(0), where, $F:[0,T]\times \to {2}^{E}$ is a multivalued map with convex compact values, ⊂ E, $se{l}_{F}$ is the superposition operator generated by F, and S: × L¹([0,T];E) → C([0,T]; ) an abstract operator. As an application, some results are given to the periodic boundary value problem for nonlinear differential inclusions governed by m-accretive...

This is an expanded version, enriched by references, of my inaugural speech held on November 7, 2001 at the Real Academia de Ciencas Exactas, Físicas y Naturales in Madrid. It explains in a nontechnical way, accessible to a general scientific community, some of the motivation and basic ideas of my research of the last twenty years on a functional-analytical approach to nonlinear parabolic problems.

A class of infinite-dimensional dissipative dynamical systems is defined for which there exists a unique equilibrium point, and the rate of convergence to this point of the trajectories of a dynamical system from the above class is exponential. All the trajectories of the system converge to this point as t → +∞, no matter what the initial conditions are. This class consists of strongly dissipative systems. An example of such systems is provided by passive systems in network theory (see, e.g., MR0601947...

In a recent paper [9] we presented a Galerkin-type Conley index theory for certain classes of infinite-dimensional ODEs without the uniqueness property of the Cauchy problem. In this paper we show how to apply this theory to strongly indefinite elliptic systems. More specifically, we study the elliptic system $-\Delta u={\partial}_{v}H(u,v,x)$ in Ω, $-\Delta v={\partial}_{u}H(u,v,x)$ in Ω, u = 0, v = 0 in ∂Ω, (A1) on a smooth bounded domain Ω in ${\mathbb{R}}^{N}$ for "-"-type Hamiltonians H of class C² satisfying subcritical growth assumptions on their first order derivatives....

We study the Cauchy problem in ℝ³ for the modified Davey-Stewartson system $i\partial \u209cu+\Delta u=\lambda \u2081\left|u\right|\u2074u+\lambda \u2082b\u2081u{v}_{x\u2081}$, $-\Delta v=b\u2082{\left(\right|u\left|\xb2\right)}_{x\u2081}$. Under certain conditions on λ₁ and λ₂, we provide a complete picture of the local and global well-posedness, scattering and blow-up of the solutions in the energy space. Methods used in the paper are based upon the perturbation theory from [Tao et al., Comm. Partial Differential Equations 32 (2007), 1281-1343] and the convexity method from [Glassey, J. Math. Phys. 18 (1977), 1794-1797].