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Existence, uniqueness, stability and asymptotic behaviour of solutions for a mathematical model of atherosclerosis

Silva, Telma; Sequeira, Adélia; Santos, Rafael F.; Tiago, Jorge

Discrete and Continuous Dynamical Systems, Series S 9, Issue1 (2016), 343 - 362

We study an atherosclerosis model described by a reaction-diffusion system of three equations, in one dimension, with homogeneous Neumann boundary conditions. The method of upper and lower solutions and its associated monotone iteration (the monotone iterative method) are used to establish existence, uniqueness and boundedness of global solutions for the problem. Upper and lower solutions are derived for the corresponding steady-state problem. Moreover, solutions of Cauchy problems defined for time-dependent system are presented as alternatives upper and lower solutions. The stability of constant steady-state solutions and the asymptotic behavior of the time-dependent solutions are studied.