Anisotropic nonlinear diffusion with absorption: Existence and extinction (Q1914762)

Summary: The authors prove that the nonlinear parabolic partial differential equation \[ {\partial u\over \partial t}= \sum^n_{i, j= 1} {\partial^2\over \partial x_i \partial x_j} \varphi_{ij}(u)- f(u) \] 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 \(\varphi_{ij}\) and \(f\) which ensure the existence of a time \(t_0> 0\) for which \(u\) vanishes for all \(t\geq t_0\).