A new framework for modelling the evolution of the thermal bar system in a lake is presented. The model assumes that the thermal bar is located between two regions: the deeper region, where spring warming leads to overturning of the entire water column, and the near shore shallower region, where a stable surface layer is established. In this model the thermal bar moves out slightly more quickly than predicted by a simple thermal balance. Also, the horizontal extent of the thermal bar region increases as it moves out from the shore.

In this paper, we analytically investigate multi-component Cahn–Hilliard and Allen–Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form ∫Ω½Γ∇c : ∇c + ½|∇z|2+Wch(c)+Wel(e,c,z)dx with a polynomial or logarithmic chemical energy density Wch, an inhomogeneous elastic energy density Wel and a quadratic structure of the gradient of damage variable z. For the corresponding elastic Cahn–Hilliard and Allen–Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for strain e.

The subcritical flow of a stream over a bottom obstruction or depression is considered with particular interest in obtaining solutions with no downstream waves. In the linearised problem this can always be achieved by superposition of multiple obstructions, but it is not clear whether this is possible in a full nonlinear problem. Solutions computed here indicate that there is an effective nonlinear superposition principle at work as no special shape modifications were required to obtain wave-cancelling solutions. Waveless solutions corresponding to one or more trapped waves are computed at a range of different Froude numbers and are shown to provide a rather elaborate mosaic of solution curves in parameter space when both negative and positive obstruction heights are included.

A free boundary problem, which comes from the model of the perpetual American call options with utility functions in financial market, is investigated. It is a degenerative parabolic free boundary problem and is studied by the line method. The existence, regularity and uniqueness of the solution as well as some properties of the free boundary are established.

We consider a nonlinear, strongly coupled, parabolic system arising in the modelling of burglary in residential areas. This model appeared in Pitcher (Eur. J. Appl. Math., 2010, Vol. 21, pp. 401–419), as a more realistic version of the Short et al. (Math. Models Methods Appl. Sci., 2008, Vol. 18, pp. 1249–1267) model. The system under consideration is of chemotaxis-type and involves a logarithmic sensitivity function and specific interaction and relaxation terms. Under suitable assumptions on the data of the problem, we give a rigorous proof of the existence of a global and bounded, classical solution, thereby solving a problem left open in previous work on this model. Our proofs are based on the construction of approximate entropies and on the use of various functional inequalities. We also provide explicit numerical conditions for global existence when the domain is a square, including concrete cases involving values of the parameters which are expected to be physically relevant.

For the parabolic–elliptic Keller–Segel system in 2 it has been proved that if the initial mass is less than 8π/χ, a global solution exists, and in case the initial mass is larger than 8π/χ, blow-up happens. The case of several chemotactic species introduces an additional question: What is the analog for the critical mass obtained for the single species system? We find a threshold curve in the two species case that allows us to determine if the system is a blow-up or a global in time solution. No radial symmetry is assumed.