The authors investigate nonnegative (weak) solutions of initial boundary value problems for a quasilinear parabolic equation of the following form: \[ (IBVP)\quad u_ t(x,t)=\Delta \phi (u(x,t))+a(x)f(u(x,t))\quad in\quad D\times (0,\infty), \]
\[ u=\chi \quad in\quad \partial D\times (0,\infty);\quad u=u_ 0\quad in\quad D\times \{0\}. \] Here, D is an open bounded domain in \(R^ N\) with smooth boundary \(\partial D\); \(\phi\) in \(C^ 1(0,\infty)\) is increasing, having Hölder continuous inverse, and \(\phi (0)=\phi '(0)=0\); \(f\in C^ 1(0,\infty)\) is increasing, \(f(0)=0\); a is a Hölder continuous function on D changing its sign. The data \(\chi\), \(u_ 0\) are nonnegative functions of class \(C^ 1(\partial D)\), \(L^{\infty}(D)\) respectively. The authors first provide a classification of associated (nonnegative) stationary solutions as follows: if \(D^+:=\{x\in D\); \(a(x)>0\}\), and \(D_ I\) is a specified union of connected components of \(D^+\), then \(S_ I\) is defined as the set of all stationary solutions which are positive on \(D_ I\). In this way, a maximal element with \(S_ I\) is naturally defined. Then the authors prove that any trajectory defined by (IBVP) has a (uniform) \(\omega\)-limit set consisting of precisely one element, which is the maximal solution within one of the \(S_ I's\) (the result applies in case \(D^+\) consists of a finite number of connected components: in case of infinite components a counterexample is provided). The authors’ approach is based on the maximum principle, and is carefully carried out in detail.