The nonlinear diffusion equation with general diffusivity $D(c)$, namely \begin{equation} \tag{1} \frac{\partial c}{\partial t} = \frac{\partial}{\partial x} (D(c)\frac{\partial c}{\partial x}), \end{equation} arises in many important areas of science and technology and considerable attention has been given to the power law diffusivities $D(c) = \alpha(c + \beta)^m$ where $\alpha, \beta$ and $m$ denote constants or with appropriate change of variables simply $D(c) = c^m$. It happens that the few known exact solutions of \begin{equation} \tag{2} \frac{\partial c}{\partial t} = \frac{\partial}{\partial x} (D^m\frac{\partial c}{\partial x}), \end{equation} turn out to be similarity solutions, even though originally they were not derived as such. In this talk we present an overview of recent results relating to similarity solutions for nonlinear diffusion and related phenomena.

In particular we consider the general nonlinear diffusion equation \begin{equation} \tag{3} \frac{\partial c}{\partial t} = x^l \frac{\partial}{\partial x}\{c^m x^n | \frac{\partial c}{\partial x}|^f \frac{\partial c}{\partial x}\}, \end{equation} which includes a wide variety of different physical processes, where $l$, $m$, $n$ and $p$ denote arbitrary constants. We also consider high order nonlinear evolution equations of the form \begin{equation} \tag{4} \frac{\partial c}{\partial t} = \sum_{j=1}^n \alpha_j \frac{\partial^j}{\partial x^j} (c^{mj}), \end{equation} for certain real constants $m_j$ and $\alpha_j$ denote any real constants. Equations of this general form include a number of standard equations such as the nonlinear diffusion equation, Burger's equation, the Kortewegde Vries equations and other equations which have been proposed for the flow of a thin viscous fluid sheet on an inclined bed and fluid flow in porous media.

All of the above equations admit similarity solutions of the general form \begin{equation} \tag{5} c(s,t) = x^\alpha \phi(x/t^\beta), \end{equation} for certain constants $a$ and $\beta$ and the given partial differential equation may be reduced to an ordinary differential equation. For particular values of $a$ and $\beta$ this ordinary differential equation admits a first integral and the question arises as to why special integrals exist for these particular parameter values. We show that these first integrals can be alternatively derived making use of the first order partial differential equation for $c(x, t)$ rather than the explicit functional form (5). This approach presents similarity solutions in an entirely new light and by equal utilization of both the partial differential equation to be solved and the first order partial differential equation we are able to deduce previous integrals virtually immediately without 'reference to the underlying ordinary differential equation.

Supercooled Stefan problems describe the freezing of a liquid initially cooled below its freezing point. The liquid and solid phases are separated by a sharp interface that constitutes a moving boundary. Such problems are notoriously ill-posed and not only is a planar moving boundary unstable to small perturbations, but the problem is prone to so called finite time blow up if the degree of initial undercooling is too great. A number of modifications to the the isothermal phase change condition have been proposed in order to prevent finite time blow up. These include applying a Gibbs-Thomson condition and/or a kinetic undercooling condition at the free boundary. A survey of recent results for supercooled Stefan problems, including effects of these modifications, is presented.

The problem of dendritic growth in solidification is formulated as a free boundary problem with an emphasis on the role of surface tension and curvature. The solvability theory of steady-state selection is described together with an analysis of time-dependent perturbations of the steady state that suggests a mechanism for sidebranching due to the selective amplification of noise.

Sufficient conditions are obtained for the oscillation of all solutions of the homogeneous Neumann and Dirichlet boundary value problems associated with the neutral parabolic system \[ \frac{\partial}{\partial t} [u_i(x,t) - c_iu_i(x,t - \tau)] - D_i \nabla^2 u_i(x,t) + \sum_{j=1}^m a_{ij}u_j(x,t, - \sigma_j) = 0 \] for $i = 1, 2, ..., m; x \in \Omega \subset R^n, t \gt 0$ where $\nabla^2$ denotes the Laplacian in $R^n$.

A problem of widespread industrial and theoretical importance is the separation of fine solids from liquids. The suspended solids are consolidated under the influence of a body force applied to the particles, for example, gravitational force in gravity thickening or an applied pressure in a pressure filter. In this paper, we outine the dynamics of such problems using the settling of a suspension in a closed bottom container under the force of gravity - that is batch settling.

The fourth order nonlinear diffusion equation $u_t + (u^nu_{xxx})_x = 0 (n \gt 0)$ governs a number of important physical processes, such as the flow of a surface tension dominated thin liquid film and the diffusion of dopant in semiconductors. This equation will be analysed using a perturbation scheme in the limit of small $n (ie 0 \lt n \ll 1)$. In this limit, the solution is determined by a system of nonlinear hyperbolic equations. An analysis of the solution shows that if the initial condition is of compact support, the solution does not move outside of its initial domain. Shocks, corresponding to jumps in $u_x$, can form in the solution. An examination of the shock jump condition shows that a shock cannot propagate outside of the domain of the initial condition. It is concluded that all solutions of $u_t + (u^nu_{xxx})_x = 0 (n \gt 0)$ for $0 \lt n \ll 1$ are waiting-time solutions.