Non Local Balance Laws in Traffic Models and Crystal Growth Rinaldo M. Colombo Dept. of Mathematics University of Brescia 25133 Brescia Italy Andrea Corli Dept. of Mathematics University of Ferrara 44100 Ferrara Italy Massimiliano D. Rosini Dept. of Mathematics University of Brescia 25133 Brescia Italy Abstract This note addresses the well posedness of Temple systems with non local sources. The resulting theorem holds globally in time and without requiring any smallness of the initial data. Its scope comprises models for traffic flow and for crystal growth. 2000 Mathematics Subject Classification: 35L50, 90B20, 82D25. Key words and phrases: Balance Laws, Traffic Flow Models, Crystal Growth Models. 1 Introduction This paper is concerned with nonlinear hyperbolic systems of balance laws in one space dimension, such as ∂ t u + ∂ x f (u)= G(t, u) . (1.1) Here, u denotes the unknown vector function, f defines a Temple system and G: [0, +∞[ × L 1 (R; R n ) → L 1 (R; R n ) is a (possibly) non local operator. The key feature of (1.1) is in the term G. Non local sources arise mainly as convolution operators in the theory of viscoelastic materials with memory [6], in models for radiating gases [12], in some regularizations of Chapman-Enskog expansions [12]. Below, we consider a model for traffic flows recently proposed in [11] and a further model concerning crystal growth presented in [22, 23]. In this paper we focus on the global existence and stability of solutions to (1.1) for the Cauchy problem, the non locality in the source term being referred to the space variable. For sources non local in time, i.e. systems with memory, see [7]. The well-posedness of the Cauchy problem for (1.1) with suitable non local source terms was recently proved in [12], in the case of initial data with small total variation. Here we relax this assumption requiring that the total variation of the data is merely bounded. On the other hand we require stronger assumptions on the convective part of (1.1), namely that it is of Temple type. For Temple systems and in the case of local source terms, the well-posedness of the initial value problem was studied in [8]. As in that paper, we do not assume here that the eigenvalues of Df are monotone along the Lax curves and no nonresonance condition is assumed. We refer to [13] for the related initial-boundary value problem. 1