Pécsi Tudományegyetem
Műszaki és Informatikai Kar Pécsi Tudományegyetem
Műszaki és Informatikai Kar

This note concerns reaction–diffusion processes which display remarkable behaviour. Everywhere the concentration, density or temperature exceeds some critical level until at some moment in time it decreases to the critical level at one point in space. At this instant, the complete profile immediately drops to the critical level at every point in space, and then remains there.

This paper concerns processes described by a nonlinear partial differential equation that is an extension of the Fisher and KPP equations including densitydependent diffusion and nonlinear convection. The set of wave speeds for which the equation admits a wavefront connecting its stable and unstable equilibrium states is characterized. There is a minimal wave speed. For this wave speed there is a unique wavefront which can be found explicitly. It displays a sharp propagation front. For all greater wave speeds there is a unique wavefront which does not possess this property. For such waves, the asymptotic behaviour as the equilibrium states are approached is determined.

The main subject of this paper is the model (| f ""| n−1 f "")" + 1 n+1 f f "" = 0, f (0) = f " (0) = 0, f " (∞) = 1, arising in the study of a 2D laminar boundary-layer with power-law viscosity, f = f (η) is the non-dimensional stream function and n > 0. Besides proving the existence and uniqueness results, we investigate the precise behavior of f for small and large η. In particular, for n > 1 we show that f (η) is linear for η ≥ η0 > 0. This means that in original (x, y) coordinates, the domain y ≥ a0x 1 n+1 , a0 = const., the velocities u = const., v = 0 and the boundary-layer is the domain y < a0x 1 n+1 (“spatial localization of the layer”).

Abstract. The Cauchy problem for a nonlinear diffusion-convection equation is studied. The equation may be classified as being of degenerate parabolic type with one spatial derivative and a time derivative. It is shown that under certain conditions solutions of the initial-value problem exhibit instantaneous shrinking. This is to say, at any positive time the spatial support of the solution is bounded above, although the support of the initial data function is not. This is a phenomenon which is normally only associated with nonlinear diffusion with strong absorption. In conjunction, a previously unreported phenomenon is revealed. It is shown that for a certain class of initial data functions there is a critical positive time such that the support of the solution is unbounded above at any earlier time, whilst the opposite is the case at any later time.