A potential well theory for the wave equation with nonlinear source and boundary damping terms

By Enzo Vitillaro. Glasgow Mathematical Journal, 2002

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Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions (2008, methodology): “In this work, we cannot use “directly” the existence result of Georgiev and Todorova [15] nor the results of Vitillaro [33, 34] because of the presence of the strong linear damping −∆ut and the dynamic boundary conditions on Γ1.”
On a wave equation with supercritical interior and boundary sources and damping terms (2011, methodology): “However, even without nonlinear boundary dynamics, an interior source of a super-critical order (3 < p ≤ 5) substantially perturbs the evolution generator so that the presence of damping becomes essential even for the local existence theory [6], [18], [32], [39]; the latter references along with [3]”
Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping (2008, background): “This has been recognized in the literature first in the case of ⁀interior sources [15, 28, 32, 3] and, more recently, in the case of boundary sources [20, 31, 5].”
Global existence for the wave equation with nonlinear boundary damping and source terms (2002, unknown): “In [27] a related problem was considered.”
A class of fourth order wave equations with dissipative and nonlinear strain terms (2008, background): “And there are only a few works on high order nonlinear evolution equations [3–31].”