We study the annihilation dynamics arising in the KPP-Fisher equation, proposed by Fisher in 1936 to model the propagation of a mutant gene and subsequently studied rigorously in the seminal work of Kolmogorov, Petrovskii and Piskunov. The approach is via a comparison theorem, where the comparison functions satisfy equations which are linearizable to the heat equation. In some sense, we have obtained a ‘linearization’ of the KPP-Fisher equation.

A nonlinear forward-backward heat equation with a regularization term was proposed by Barenblatt et al. [1, 2] to model the heat and mass exchange in stably stratified turbulent shear flow. It was proven to be well-posed in the case of given initial and Neumann boundary conditions. However, the solution was found to have an unphysical discontinuity with certain smooth initial functions. In this paper, a nonlinear heat equation with a time delay originally used by Barenblatt et al. [1, 2] to derive their model is investigated. The same type of initial-boundary value problem is shown to have a unique smooth global solution when the initial function is reasonably smooth. Numerical examples are used to demonstrate that its solution forms step-like profiles in finite times. A semi-discretization of the initial-boundary value problem is proved to have a unique asymptotically and globally stable equilibrium.

Dorsey et al. [8] have constructed formal solutions for the half-space Ginzburg–Landau model, when κ is small. Dorsey et al. deduce a formal expansion for the superheating field in powers of κ½ up to order 4. In this paper, we show how the formal construction gives a natural way for constructing a subsolution for the Ginzburg–Landau system. We improve the result obtained by Bolley & Helffer [2], and take a step in the proof of the Parr Formula [13], getting two terms in the lower bound for the superheating field as κ → 0.

The paper presents a method of computing periodic water waves based on solving an integral equation by means of discretization and automatically finding the mesh on which the functions to be found are approximated by the best way. The power of the method to describe ‘bad functions’ well makes it possible to reproduce all the main results of asymptotic theory for the almost-highest waves (Longuet-Higgins & Fox, 1977, 1978, 1996) by a direct numerical simulation. The method is able to compute two full periods of the oscillations of wave properties for all wave height-to-length ratios. The end of the second period corresponds to the wave steepness that achieves 99.99997% of the limiting value. So, the validity of the asymptotic formulae by Longuet-Higgins & Fox is proved for the steep waves of any finite depth. The refined value of the maximum slope of the free-surfaces is found to be 30.3787°.

Three-dimensional nonstationary flow of a viscous incompressible liquid is investigated in a layer, driven by a nonuniform distribution of temperature on its free boundaries. If the temperature given on the layer boundaries is quadratically dependent on horizontal coordinates, external mass forces are absent, and the motion starts from rest then the free boundary problem for the Navier–Stokes equations has an ‘exact’ solution in terms of two independent variables. Here the free boundaries of the layer remain parallel planes and the distance between them must be also determined. In present paper, we formulate conditions for both the unique solvability of the reduced problem globally in time and the collapse of the solution in finite time. We further study qualitative properties of the solution such as its behaviour for large time (in the case of global solvability of the problem), and the asymptotics of the solution near the collapse moment in the opposite case.

An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.

The paper deals with the probability density function (PDF) of the concentration of a scalar within a turbulent flow. Following some comments about the overall structure of the PDF, and its approach to a limit at large times, attention focusses on the so-called small scale mixing term in the evolution equation for the PDF. This represents the effect of molecular diffusion in reducing concentration uctuations, eventually to zero. Arguments are presented which suggest that this quantity could, in certain circumstances, depend inversely upon the PDF, and a particular example of this leads to a new closure hypothesis. Consequences of this, especially similarity solutions, are explored for the case when the concentration field is statistically homogeneous.

The existence of classical solutions with a free boundary changing its topology in time is proved for the multi-dimensional Hele-Shaw problem. We prove that, if some symmetry and monotonicity conditions on the problem data hold, then a cusp, arising at the contact between two free boundary components, disappears instantly and the free boundary becomes smooth and singly-connected.

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f (u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt ] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

We consider an eigenvalue problem of three-dimensional elasticity for a multi-structure consisting of a finite three-dimensional solid linked with a thin-walled elastic cylinder. An asymptotic method is used to derive the junction conditions and to obtain the skeleton model for the multi-structure. Explicit asymptotic formulae have been obtained for the first six eigen-frequencies.

This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.