Unboundedness phenomenon in a model of urban crime

Abstract

We show that spatial patterns (“hotspots”) may form in the crime model [Formula: see text] which we consider in [Formula: see text], [Formula: see text], [Formula: see text] with [Formula: see text], [Formula: see text] and initial data [Formula: see text], [Formula: see text] with sufficiently large initial mass [Formula: see text]. More precisely, for each [Formula: see text] and fixed [Formula: see text], [Formula: see text] and (large) [Formula: see text], we construct initial data [Formula: see text] exhibiting the following unboundedness phenomenon: Given any [Formula: see text], we can find [Formula: see text] such that the first component of the associated maximal solution becomes larger than [Formula: see text] at some point in [Formula: see text] before the time [Formula: see text]. Since the [Formula: see text] norm of [Formula: see text] is decreasing, this implies that some heterogeneous structure must form. We do this by first constructing classical solutions to the nonlocal scalar problem [Formula: see text] from the solutions to the crime model by taking the limit [Formula: see text] under the assumption that the unboundedness phenomenon explicitly does not occur on some interval [Formula: see text]. We then construct initial data for this scalar problem leading to blow-up before time [Formula: see text]. As solutions to the scalar problem are unique, this proves our central result by contradiction.