The authors investigate the following scalar PDE: \[ \partial_ t \beta (u) = \Delta u - g(x,u) \quad \text{on} \quad \mathbb{R}_ + \times \Omega \tag{1} \] with \(u=0\) on \(\mathbb{R}_ + \times \partial \Omega\) and \(\beta (u(0,x)= \beta (u_ 0(x))\) for \(x \in \Omega\) for some given \(u_ 0\). Here \(\Omega \subseteq \mathbb{R}^ d\) with \(d \leq 3\) is bounded. The aim of the authors is to prove the existence of so-called inertial sets (or exponential attractors) for the semigroup \(S(t)\), \(t \geq 0\) generated by (1). An inertial set \({\mathcal M}\) has all the properties of an inertial manifold except the manifold property, that is: (a) \({\mathcal M}\) is invariant under \(S(t)\), (b) it contains the global attractor \(A\) of \(S(t)\) (whose existence follows from the properties of \(S(t)\), (c) \({\mathcal M}\) attracts all trajectories of \(S(t)\), at exponential rate, (d) \({\mathcal M}\) has finite fractal dimension. There is a basic property, which when satisfied by \(S(t)\), guarantees the existence of an inertial set, namely the so-called discrete squeezing property. The authors now take as starting point a theorem proved in earlier papers (by Eden, Foias, Nicolaenko and Temam) which essentially states that if the semigroup \(S(t)\) is Lipschitz and has the squeezing property, then it admits an inertial set. The main task of the paper is to consider abstract evolution equations which include (1) as a special case (under some conditions on \(\beta\) and \(g)\) and to give sufficient conditions for these evolution equations in order that the generated semigroup \(S(t)\) has the squeezing property and thus admits an inertial set. These assumptions and also the related results are too technical to be discussed here. The paper under discussion relies heavily on a series of earlier papers by the authors.