Asymptotic Behavior of a Semilinear Wave Equation with Both Localized Memory and Delay Effects

In this paper, we investigate a semilinear wave equation that incorporates localized memory and delay effects. By imposing the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>δ</mml:mi> </mml:math> -condition on the memory kernel and including a semilinear source term, we ensure that the energy of the system decays exponentially. This approach effectively localizes these effects, allowing them to vanish on a set of positive measure. While the viscoelastic dissipation is distributed around the boundary in accordance with the Geometric Control Condition, the delay term is localized such that it does not satisfy this condition. Our findings demonstrate a flexible geometric interaction between memory and delay effects, ensuring stability even when the instability occurs in uncontrolled regions.