Partial differential equations and dynamical systems

1. Jorge A. Esquivel-Avila, “Blow-Up of Solutions with High Energies of a Coupled System of Hyperbolic Equations”, We consider an abstract coupled evolution system of second order in time. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We compare our results with those in the literature and show how we improve them., Abstract and Applied Analysis, 2019:Article ID 7405725 (2019), 1-11  
2. Jorge A. Esquivel-Avila, “Remarks on the qualitative behavior of the undamped Klein-Gordon equation”, We consider the undamped Klein‐Gordon equation in bounded domains with homogeneous Dirichlet boundary conditions. For any real value of the initial energy, particularly for supercritical values of the energy, we give sufficient conditions to conclude blow‐up in finite time of weak solutions. The success of the analysis is based on a detailed analysis of a differential inequality. Our results improve previous ones in the literature., Mathematical Methods in the Applied Sciences, 41:1 (2017), 103-111  
3. Jorge A. Esquivel-Avila, “Qualitative analysis of a nonlinear wave equation”, The paper concerns the qualitative behavior of solutions to the mixed problem in a cylinder with the Dirichlet boundary condition for the wave equation with a nonlinear dissipative term and nonlinear source term. Using the concepts of a stable set (potential well) and an unstable set, necessary and sufficient conditions for blow-up of solutions and necessary and sufficient conditions for convergence of all bounded solutions as t→∞ are given., Discrete Contin. Dyn. Syst., 10:3 (2004), 787-804  
4. Jorge A. Esquivel-Avila, “The dynamics of a nonlinear wave equation”, We consider a wave equation in a bounded domain with linear dissipation and with a nonlinear source term. We give characterizations of all the solutions with respect to their qualitative properties: global existence and nonexistence, boundedness, blow-up, and convergence to equilibria, J. Math. Anal. Appl., 279:1 (2003), 135-150  
5. Jorge A. Esquivel-Avila, “A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations”, We give necessary and sufficient conditions for existence of global and nonglobal solutions of a nonlinear wave equation in a bounded domain. We consider nonlinear dissipation and a nonlinear source term. We also analyze the qualitative behavior of solutions forwards and backwards for the wave equation without dissipation. In this case we present characterizations of blow-up and asymptotic behavior. Finally, we extend some of our results to a nonlinear Kirchhoff equation. We use the concepts of stable and unstable sets introduced by Payne and Sattinger in 1975., Nonlinear Anal., 52:4 (2003), 1111-1127    

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