We analyze a model for non-isothermal superconductivity, derived independently by G. Maugin, and K. Miya and S. A. Zhou. The model is described by a parabolic system based on the Time-Dependent Ginzburg–Landau (TDGL) equation, the Maxwell equations, and an energy equation such that the Clausius–Duhem inequality holds. The principal unknown fields are the complex valued Ginzburg–Landau order parameter $\psi$, the magnetic vector potential $A$, and the temperature $T$. A significant feature for this model is that it accounts for the interchange of thermal and electro-magnetic energies through Joule heating. The sensitivity of superconducting materials to thermal variations is a major obstacle to their applications. In practice one sees that time varying currents and magnetic fields generate thermal energy, producing ‘hot spots’ and increasing the temperature. This can result in suppressing superconductivity, i.e. $\psi$ tending to $0$. Our principal result is that we exhibit this phenomenon. We prove that if the electro-magnetic energy of the superconductor is sufficiently large at time $t\,{=}\,0$ then $T$ will rise beyond the critical temperature, $T_c$, and $\psi\to 0$ as $t\to\infty$. Earlier analytic work investigating electro-dynamic effects in superconductivity centered on an isothermal model consisting of the TDGL equation and Maxwell equations. This setting takes into account just electro-magnetic effects and produces markedly different evolutions.

We study the dynamics of dissipation and blow-up in a critical-case unstable thin film equation. The governing equation is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for lubrication flows of thin viscous fluid layers on solid surfaces. %For a special balance between %destabilizing second-order terms and regularizing fourth-order terms, There is a critical mass for blow-up and a rich set of dynamics including families of similarity solutions for finite-time blow-up and infinite-time spreading. The structure and stability of the steady-states and the compactly-supported similarity solutions is studied.

The characterisation of those shapes that can be made by the gravity sag-bending manufacturing process used to produce car windscreens and lenses is modelled as an inverse problem in linear plate theory. The corresponding second-order partial differential equation for the Young's modulus is shown to change type (possibly several times) for certain target shapes. We consider the implications of this behaviour for the existence and uniqueness of solutions of the inverse problem for some frame geometries. In particular, we show that no general boundary conditions for the inverse problem can be prescribed if it is desired to achieve certain kinds of target shapes.

A model for the dynamic thermomechanical behavior of a viscoelastic beam which is in frictional contact with a rigid rotating wheel is presented. It describes a simple braking system in which the wheel comes to a stop as a result of the frictional traction generated by the beam. Friction is modelled with a temperature and slip rate dependent coefficient of friction. Frictional heat generation is taken into account as well as the wheel temperature evolution, and the wear of the beam's contacting end. The model is formulated as a variational inequality. A FEM numerical scheme for the model is described, implemented, and the results of numerical simulations are shown.

A variety of models on the interaction between glucose and insulin have been suggested over the last 50 years. One, developed by Sturis et al. [19], and consisting of six nonlinear ordinary differential equations, has been widely accepted. However, the model has the disadvantage of containing auxiliary variables which have no clinical interpretation. In this paper we study an alternative model which incorporates a time delay explicitly, negating the need for the auxiliary equations. A simplifying assumption of having just one insulin compartment reduces the number of equations still further. We then study the resulting system of two differential delay equations, establishing results on positivity, boundedness, persistence and global asymptotic stability. For the latter, two quite different approaches are employed: comparison principles and Lyapunov functionals. The two approaches provide different sets of sufficient conditions for global stability, so that we investigate different regions of parameter space.

This paper considers the nonlinear stability oftravelling wavefronts of a time-delayed diffusive Nicholson blowflies equation. We prove that, under a weighted L2 norm, ifa solution is sufficiently close to a travelling wave front initially, it converges exponentially to the wavefront as t → ∞. The rate ofconvergence is also estimated.