On spatial decay in a nonlinear equation with viscoelastic-type dissipation

We study spatial decay estimates for a nonlinear wave equation with viscoelastic-type dissipation, where the nonlinear dissipative function satisfies suitable growth conditions. For solutions defined on semi-infinite cylinders and cone-like regions, we establish Phragmén–Lindelöf alternatives: either the energy grows exponentially or polynomially with distance, or it decays exponentially or polynomially to zero. The results extend previous contributions on spatial decay in nonlinear dissipative systems.