The authors prove that the nonlinear parabolic partial differential equation ∂ u ∂ t = ∑ i , j = 1 n ∂ 2 ∂ x i ∂ x j φ i j ( 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 φ i j and f which ensure the existence of a time t 0 > 0 for which u vanishes for all t ≥ t 0 .
International Journal of Mathematics and Mathematical Sciences
Lair, A. V., & Oxley, M. E. (1996). Anisotropic nonlinear diffusion with absorption: existence and extinction. International Journal of Mathematics and Mathematical Sciences, 19(3), 427–434.