Instantaneous shrinking in nonlinear diffusion-convection

Gilding, B. H.; Kersner, R.

doi:10.1090/S0002-9939-1990-1007496-9

Instantaneous shrinking in nonlinear diffusion-convection
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by B. H. Gilding and R. Kersner PDF

Proc. Amer. Math. Soc. 109 (1990), 385-394 Request permission

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.

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The existence uniqueness and properties of the solutions of a degenerate parabolic equation with diffusion-advection-absorption