The authors study the parabolic problem (I) \[ u_ t=\Delta \phi (u)+uf(x,u),\quad in\quad (0,\infty)\times \Omega,\quad \Omega \subset R^ d,\quad bounded, \]
\[ u=0,\quad on\quad (0,\infty)\times \partial \Omega,\quad u=u_ 0,\quad on\quad \{0\}\times \Omega \] and the stationary problem associated to it: (II) \[ \Delta \phi (u)+uf(x,u)=0,\quad in\quad \Omega;\quad u=0\quad on\quad \partial \Omega \] under suitable assumptions on \(\phi\) and f: in particular, there exists \(k>0\) such that f(x,u)\(\leq 0\) for any \(x\in \Omega\) and \(u\geq k\), and \(0\leq u_ 0\in L^{\infty}(\Omega)\). The authors discuss attractivity properties of the stationary solutions of (II) with respect to solutions to (I). They show that for any connected component of \(P:=\{x\in \Omega | f(x,0)>0\}\) there is a solution of (II) strictly positive on that component. As the authors state, this situation is consistent with the possibility that either (i) every solution of (II) be strictly positive in \(\Omega\) or (ii) certain solutions of (II) exhibit a free boundary. Accordingly, the authors informally present the following as a typical result:
“Let P be disconnected; suppose f(x,.) nonincreasing (x\(\in \Omega)\). If the connected components of P are far from each other and f(.,0) is ‘negative enough’ in the whole intermediate region, every solution of (II) vanishes at some distance from them”.
The authors state in section 2 a series of theorems on support properties of stationary solutions, existence of free boundaries, and attractivity properties of stationary solutions.
Section 3 is devoted to the proofs, and section 4 includes examples and numerically calculated graphs. The bibliography includes 20 items.