Exponential Stability for the 2D Wave Model with Localized Memory in a Past History Framework and Nonlinearity of Arbitrary Growth

In this paper, we discuss the asymptotic stability as well as the wellposedness of the viscoelastic damped wave equation posed on a bounded domain Ω of R 2 , ∂ t 2 u - Δ u + ∫ 0 ∞ g ( s ) div [ a ( x ) ∇ u ( · , t - s ) ] d s + b ( x ) ∂ t u + f ( u ) = 0 , in Ω × R + , subject to a locally distributed viscoelastic effect driven by a nonnegative function a ( x ) which is positive around the entire neighborhood of ∂ Ω and supplemented with a frictional damping b ( x ) ≥ 0 acting effectively on ∂ A where A = { x ∈ Ω / a ( x ) = 0 } . Assuming that well-known geometric control condition ( ω ′ , T 0 ) holds, supposing that the relaxation function g is bounded by a function that decays exponentially to zero and the function f possesses an arbitrary growth, we show that the solutions to the corresponding partial viscoelastic model decay exponentially to zero. We can also treat the focusing case for those solutions with energy less than d of the ground state, where d is the level of the Mountain Pass Theorem.