## 1 Introduction  
Catalytic (i.e., catalyst-coated) diesel particulate filters (DPFs) have been proposed to allow regeneration (i.e., oxidation of soot) at low temperatures [1], thus overcoming the drawbacks of the thermal process, including the formation of excessively hot regions that may cause irreversible damage to the filter [2]. The regeneration performance of catalytic DPFs is dependent not only on catalyst activity [3; 4], but also on the quality of the contact established between soot and catalyst particles [3; 4; 5; 6]. As observed for DPFs wash-coated with nano-metric ceria particles, the soot trapped inside the walls of the filter can come into intimate contact with a well-dispersed catalyst [5; 7], whereas this contact remains rather poor for the soot cake layer accumulated on top of the walls [5], which represents most of the soot retained in the filter. Although some soot particles in the cake layer may relocate during the regeneration process, possibly coming into contact with the catalyst, most of the soot accumulated as cake remains far from the catalytic walls, thus burning via the thermal path [5]. This may lead to severe temperature excursions also when regenerating catalytic filters. Spatio-temporal infrared measurements of temperature taken during the combustion process of the cake layer accumulated on planar catalytic single-layer filters have shown that, depending on the operating conditions, two different regimes of regeneration may be established, slow regeneration and fast regeneration [8; 9; 10]. In the regime of slow regeneration, combustion of soot occurs uniformly all over the filter, leading to moderate temperature rises. Conversely, in the regime of fast regeneration, combustion of soot occurs via propagation of sharp reaction fronts, resulting in high peak temperatures. Regeneration dynamics evolves from the slow regime to the fast regime as a function of increasing oxygen concentration [8], exhaust gas temperature [9], and soot load [10]. The transition from slow to fast regeneration is rather abrupt [8; 10]. Studies on filter regeneration have also been performed by means of models of different complexity. [PERSON] et al. [11] used a one-dimensional model of a single-channel filter to develop simple criteria for predicting the maximum temperature under both the regimes of slow and fast regeneration. Results of two-dimensional simulations by [PERSON] et al. [12] have highlighted the sensitivity of the filter temperature profile to changes in soot load, inlet gas temperature, and, at low flow rate, oxygen concentration. In work by [PERSON] et al. [13], the transition from the slow regime to the fast regime with increasing inlet gas temperature was simulated through a three-dimensional CFD-based model of thermal regeneration. More recently, we proposed a two-dimensional CFD-based model of regeneration of a single-channel catalytic DPF [14], and performed simulations varying the catalyst activity, the inlet gas velocity, and the amount of soot loaded as cake [14; 15; 16]. Our attention was mainly focused on the interaction between combustion of the soot trapped inside the catalytic wall of the filter and combustion of the cake. Preliminary results were also obtained in the absence of cake [16]. An abrupt transition from slow to fast regeneration was predicted to occur with increasing catalyst activity [14; 15] and soot load [16]. This transition progressively shifts towards higher catalyst activity with increasing inlet gas velocity [15]. In the slow regime, combustion of the cake proceeds independently of combustion of the soot trapped inside the catalytic wall. Regeneration is a \"safe\" process, but it takes a long time to complete. In the fast regime, the catalytic wall behaves as a \"pilot\" for the cake, causing its violent ignition. Under such conditions, the time for regeneration is suitable for practical applications. However, the cake burns via a propagating front in an uncontrolled fashion, leading to \"runway\" of the catalytic filter.