The existence of a global time is often taken for granted but should instead be considered as a matter of investigation. By using the tools of global Lorentzian geometry I prove that, under physically reasonable conditions, the impossibility of finding a global time implies the singularity of spacetime.

We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a 3D setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate...

Let S be a complete, open simply connected surface. Suppose that the integral of the Gauss curvature over arbitrary measurable sets is less than tt/2 in magnitude. We show that the surface admits a global Tchebychev net.

The behavior of elastic waves in a three-dimensional isotropic incompressible material is studied. Unlike compressible elastodynamics, where there are nonlinear interactions of shear and pressure waves, with incompressible elastodynamics the only waves present are shear waves. In an isotropic system, shear waves are linearly degenerate, and therefore global solutions to the perturbative incompr...

We treat the Cauchy problem for nonlinear system of viscoelasticity with memory term. We study the existence and time decay of the solution to this nonlinear problem. The kernel of the memory term includes integrable singularity at zero and polynomial decay at infinity. We prove the existence of a global solution for space dimensions n ≥ 3 and arbitrary quadratic nonlinearities.

is invertible. Here we treat the case of inputs coming from free waves located far away from the origin. In order to simplify the presentation, we will only consider inputs that would have come from our first guess (that is |x|f is bounded in L). Hence, we now consider the case when at least one function f or g is supported away from the origin and assume that for all y1 in the support of f and...

In this paper we prove a global existence result for pair diffusion models in two dimensions. Such models describe the transport of charged particles in semiconductor heterostructures. The underlying model equations are continuity equations for mobile and immobile species coupled with a nonlinear Poisson equation. The continuity equations for the mobile species are nonlinear parabolic PDEs invo...

Wave maps (i.e. nonlinear sigma models) with torsion are considered in 2+1 dimensions. Global existence of smooth solutions to the Cauchy problem is proven for certain reductions under a translation group action: invariant wave maps into general targets, and equivariant wave maps into Lie group targets. In the case of Lie group targets (i.e. chiral models), a geometrical characterization of inv...

In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for u ≥ 0 and p ∈ (0,∞) we obtain ∫ R3 dx u(x) ≤ ( p+ 1 p )2 ∫ R3 dx {(−4)−1u(x)}|∇u p 2 (x)|. We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically,...