Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights

Abstract

We consider the wave equation with a weak internal damping with non-constant delay and nonlinear weights given by utt(x, t) − uxx(x, t) + µ 1 (t)ut(x, t) + µ 2 (t)ut(x, t − τ (t)) = 0 in a bounded domain. Under proper conditions on nonlinear weights µ 1 (t), µ 2 (t) and non-constant delay τ (t), we prove global existence and estimative the decay rate for the energy.

Dynamics of Timoshenko system with time-varying weight and time-varying delay

“…Here ϕ = ϕ(x, t), ψ = ψ(x, t) model the transverse displacement of the beam and the angular direction of the filament of the beam respectively and ρ 1 , ρ 2 , k, b are positive real numbers. The systems are subject to the Dirichlet boundary conditions ϕ(0, t) = ϕ(L, t) = ψ(0, t) = ψ(L, t) = 0, t > 0, (3) and the initial conditions We are interested in proving the exponential stability for such of each problem. From mathematical point of view, the problem (1) is very different of problem (2) because the partially damped Timoshenko system is exponentially stable if and only if the coefficients satisfy…”

“…For waves with time-varying delay and time-varying weights we cite the recent work of Barros et al [3] where was studied the equation given by…”

Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights

“…Since the weights are nonlinear, the operator is nonautonomous, we use the Kato variable norm technique [15] to show that the system is well-posed. We use the standard multiplicative method as in [2] to obtain the exponential stabilization. Finally, we give equivalence between exponential stabilization and observability inequality.…”

“…Preliminaries. In this section, we propose hypothesis for the time-varying delay and time-dependent weights as in [2,25]. Assumption 1.…”

“…Lemma 2.3. [2,27] Let (v, p, z) be a solution to system (17)- (25). Then the energy functional defined by (26…”