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).