Nonlinear Analysis: Real World Applications 52 (2020) 103026 Contents lists available at ScienceDirect Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa Blow-up result and energy decay rates for binary mixtures of solids with nonlinear damping and source terms M.L. Santos a,* , M.M. Freitas b , A.J.A. Ramos b a Institute of Exact and Natural Sciences, Doctoral Program in Mathematics, Federal University of Pará, Augusto Corrêa Street, Number 01, 66075-110, Belém PA, Brazil b Federal University of Pará, Raimundo Santana Street s/n Salinópolis PA, 68721-000, Brazil article info Article history: Received 27 April 2019 Received in revised form 28 August 2019 Accepted 29 August 2019 Available online 27 September 2019 Dedicated to Prof. Jaime E. Muñoz Rivera on the occasion of his 60th Birthday Keywords: Nonlinear damping and sources Well-posedness Energy identity Blow-up Nonlinear semigroups Decay energy abstract In this paper we study the long-time behavior of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good" part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. Published by Elsevier Ltd. 1. Introduction This article is concerned with a special case of a theory of binary mixture of solids with nonlinear damping and source terms. The theory of mixtures of solids has been widely investigated in the last decades, see the references [1–4] for a detailed presentation. Qualitative properties of solutions to the problem defining this kind of material have been the scope of many investigations. Several results concerning existence, uniqueness, continuous dependence and asymptotic stability can be found in the literature see the references [5–8]. Despite the ample work available on the subject, there has been less focus on the interaction of nonlinear sources and damping on the one-dimensional binary mixture of solids framework. The problem we want to study can be stated as follows: Given L> 0 and T> 0, we consider a rod composed by a mixture of two interacting continua with reference configuration [0,L]. Denoting by u(x,t),w(x,t): [0,L] × [0,T ] → R ∗ Corresponding author. E-mail addresses: ls@ufpa.br (M.L. Santos), mirelson@ufpa.br (M.M. Freitas), ramos@ufpa.br (A.J.A. Ramos). https://doi.org/10.1016/j.nonrwa.2019.103026 1468-1218/Published by Elsevier Ltd.