International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 3, Pages 427-434
doi:10.1155/S0161171296000610
Anisotropic nonlinear diffusion with absorption: existence and extinction
Department of Mathematics and Statistics, Air Force Institute of Technology/ENC, 2950 P Street, Wright-Patterson Air Force Base, 45133-7765, Ohio, USA
Received 14 April 1994; Revised 19 October 1994
Copyright © 1996 Alan V. Lair and Mark E. Oxley. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The authors prove that the nonlinear parabolic partial differential equation
∂u∂t=∑i,j=1n∂2∂xi∂xjφij(u)−f(u) with homogeneous Dirichlet boundary conditions and a nonnegative initial condition has a nonnegative generalized solution u. They also give necessary and sufficient conditions on the constitutive
functions φij and f which ensure the existence of a time t0>0 for which u vanishes for all t≥t0.