On the positive solutions of some nonlinear diffusion problems (Q1065274)

The author considers the nonlinear parabolic diffusion equation, \[ (*)\quad u_ t-a(x,u_ x)_ x+b(x,u)=\lambda g(x,u) \] together with initial boundary conditions of Dirichlet type, where a, b and g are monotone increasing functions, with respect to the second variable. He proves that (*) has a positive steady state solution if and only if, \(\lambda \in (0,\lambda^*)\) or \(\lambda \in (0,\lambda^*]\) where \(\lambda^*\leq \sup_{u} \mu_ 1\{u\}\) where \(\mu_ 1\{u\}\) denotes the first eigenvalue of the stationary problem linearized at the ''point'' u. The minimal positive steady state solutions are shown to be stable with respect to (*).