A Damped Nonlinear Hyperbolic Equation with Nonlinear Strain Term - Investigaci'on, normalsize Instituto de; Matem'aticas-UNMSM, Facultad de Ciencias; Per'u, Lima (Journal of Applied Nonlinear Dynamics, 2026)
In this work, we investigate an initial boundary value problem related to the nonlinear hyperbolic equation u_tt + u_xxxx + α u_xxxxt = f(u_x)_x, for f(s) = |s|^ρ + |s|^σ, 1 < ρ, σ, α > 0. Under suitable conditions, we prove the existence of global solutions and the exponential decay of energy. The nonlinearity f(s) introduces some obstacles in the process of obtaining a priori estimates and we overcome this difficulty by employing an argument due to Tartar (1978). The exponential decay is obtained via an integral inequality introduced by Komornik (1994).