Affiliations: Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA and 18-20 Avenue De La République, 92400, Courbevoie, France. E-mail: [email protected]
Abstract: In this article, a rigorous mathematical treatment of the dryland vegetation model introduced by Gilad et al. [Phys. Rev. Lett. 98(9) (2004), 098105-1–098105-4, J. Theoret. Biol. 244 (2007), 680–691] is presented. We prove the existence and uniqueness of solutions in (L1(Ω))3 and the existence of global attractors in L1(Ω;𝒟), where 𝒟 is an invariant region for the system. A key step is the regularization of the model by adding εΔ to the diffusion term and by approximating the initial data U0 by a sequence {U0,n} of smooth functions in (L1(Ω))3. The various a priori estimates and the maximum principle permit the passage to the limit as ε→0 and n→∞, proving the existence and uniqueness of solutions U in the specified space. Also, we deduce from the a priori estimates that the solution meets the necessary hypotheses (see Theorem 1.1 in Chapter 1 of Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1997) and hence, we obtain the existence of global attractors.
