In this paper, we consider an inverse problem of finding a coefficient of right hand side of the following system of Kelvin–Voigt equations perturbed by an isotropic diffusion and damping terms vt+∇π−ϰΔvt−νdiv|D(v)|p−2D(v)=γ|v|m−2v+f(t)g(x,t), divv(x,t)=0. The damping term γ|v|m − 2v in the momentum equation realizes an absorbtion (sink) if γ ≤ 0, and a source if γ > 0. We show how the exponents p,  m, the coefficients ν,  ϰ,  γ, the dimension of the space d, and data of the problem should interact each other for the existence of weak solutions to the problem. We also establish the conditions for uniqueness of the solutions to this problem.