This paper is devoted to the stability of a viscoelastic problem with Balakrishn\^{a}n-Taylor damping and time delay involving variable-exponent nonlinearity. Under some assumptions on the relaxation function, we establish the general decay estimate for the energy via suitable Lyapunov functionals. The problem considered is novel and meaningful because the presence of the flutter panel equation and the spillover problem with memory and variable exponents time delay control. Our result generalizes and improves previous conclusion in the literature.