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IMA Journal of Applied Mathematics Advance Access originally published online on December 17, 2004
IMA Journal of Applied Mathematics 2005 70(1):119-128; doi:10.1093/imamat/hxh053
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IMA Journal of Applied Mathematics Vol. 70 No. 1 © Institute of Mathematics and its Applications 2005; all rights reserved.

Spatial decay of transient end effects for nonstandard linear diffusion problems*

C. O. Horgan1 and R. Quintanilla2

1 Department of Civil Engineering, University of Virginia, Charlottesville, VA 22904, USA, 2 Matematica Aplicada 2, U.P.C., Barcelona, Spain

In this paper we investigate the spatial decay of transient end effects for some nonstandard linear diffusion problems. We consider classes of linear parabolic equations on three-dimensional semi-infinite cylinders with non-zero boundary conditions only on the near end. If zero initial conditions were imposed, it is well-known that solutions decay exponentially with distance from the end and many results on the rate of spatial decay have been established. Here we consider a different type of initial condition that arises in regularization of ill-posed problems: namely, that the field quantity at time t = T is assumed to be proportional to that at t = 0. A spatial decay estimate for solutions of such problems is obtained with optimal decay rate explicit in the proportionality parameter. Continuous dependence on this parameter is established. The results are extended to a pseudo-parabolic equation. Binary mixtures of rigid solids are then considered. The governing PDEs are a coupled system of two parabolic equations for the temperature in each phase. Differential inequality arguments are used to obtain decay estimates for cross-sectional norms of the solution pair.

Keywords: nonstandard linear diffusion problems; spatial decay of transient end effects.

* Dedicated to Ray W. Ogden on the occasion of his 60th birthday


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