The paper provides a complete description of the asymptotic behaviour of the positive solutions of the second-order parabolic equation \[ \partial u / \partial t - \triangle u = \lambda u - a(x) f(x,u) u \] with Dirichlet boundary conditions and non-negative initial data in a bounded domain \(\Omega\) in \(\mathbb R^n\), \(n \geq 1\), \(\partial \Omega\) of class \(C^3\), \(\lambda \in \mathbb R\), function \(a \geq 0\), \(a \neq 0\) of class \(C^\mu(\overline{\Omega})\) for some \(0<\mu\leq 1\), and \(a\) and \(f\) satisfying certain additional assumptions.
The author shows that depending on the value of \(\lambda\), the solutions fall within one of four classes of asymptotic behaviour, and uses the concept of metasolution as an extension of the so-called large solution to classify the results. The paper exends the results provided earlier by J. Lopez-Gomez [Electron. J. Differ. Equ. 2000, Conf. 05, 135–171, electronic only (2000; Zbl 1055.35049)].