Instantaneous Shrinking in Nonlinear Diffusion-convection

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. In this article we shall consider the Cauchy problem (1) u, = («"•)„ +(11"), inRxR+, (2) u(x, 0) = u0(x) forxGR, where m and n are positive real constants and uQ is a given bounded continuous nonnegative function. Because, in general, problem (1), (2) does not admit a classical solution [4, 8, 10, 12], it is necessary to introduce the notion of a generalized solution. We shall say that a function u(x, t) is a generalized supersolution of equation (1) in a domain (3) D = (Vl,n2)x(0,T] with (4) -oo < nx < n2 < oo and 0 < T < oo Received by the editors June 15, 1988 and, in revised form, July 6, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 35K55, 35K65, 35B99.