This paper constructs a family of discrete games, whose value functions converge to the unique viscosity solution of the Neumann boundary problem of the curve shortening flow equation. To derive the boundary condition, a billiard semiflow is introduced and its basic properties near the boundary are studied for convex and more general domains. It turns out that Neumann boundary problems of mean curvature flow equations can be intimately connected with purely deterministic game theory.

This paper addresses the question of change of decay rate from exponential to algebraic for diffusive evolution equations. We show how the behaviour of the spectrum of the Dirichlet Laplacian yields the passage from exponential decay in bounded domains to algebraic decay or no decay at all in the case of unbounded domains. It is well known that such rates of decay exist. The purpose of this paper is to explain what makes the change in decay happen. We also discuss what kind of data is needed to obtain various decay rates.

Local and global well posedness results are established for the initial-value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving ill-posedness results otherwise. The global results allow us to study the norm growth of solutions corresponding to the Schrödinger equation term. We use ideas recently introduced to study the classical Zakharov systems.

The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear non-autonomous evolution equations in Hilbert spaces where the assumption is made that the domains of the quadratic forms associated with the generators are independent of time. Finally, these results are applied to time-dependent Schrödinger operators with moving point interactions in 1D.

The problem $-\epsilon^2\nabla\cdot(P( x)\nabla u)+F(V( x),u)=0$ is studied in the whole of ${\mathbb R}^n$, where $V(x)$ is a multidimensional potential and $P(x)$ a matrix function. Under general conditions solutions are constructed for small positive $\epsilon$ having simple concentration properties. The asymptotic form of the solution is studied as $\epsilon\to 0$ as well as its positivity.

In this paper we consider a one-dimensional linear thermoelastic system of Timoshenko type with past history acting only in one equation. We consider the model where the heat conduction is given by Green and Naghdi's theories and prove exponential and polynomial stability results for the equal and nonequal wave-speed propagation. Our results are established under conditions on the relaxation function weaker than those in [9].