Longtime dynamics for a nonlinear viscoelastic equation with time-dependent memory kernel

Abstract

This paper investigates the well-posedness, the existence and the regularity of the time-dependent global attractor for a viscoelastic equation in Ω⊂R3: |∂tu|ρ∂ttu−∂ttΔu−ht(0)Δu−∫0∞∂sht(s)Δu(t−s)ds+f(u)=hwith time-dependent memory kernel which is used to model aging phenomena of the material. By using the novel theory framework recently developed in literature (Conti et al., 2018 [12,13]) and establishing some delicate integration estimates along the trajectory of the solutions in the time-dependent phase space, we show that when ρ∈(1,4], the growth exponent p of f(u) is up to the critical range 1≤p≤5, and the time-dependent memory kernel satisfies the same conditions as in Conti et al., (2018) [12,13], the model is well-posed. Especially, when ρ∈(1,4) and 1≤p<5, the related process has an invariant time-dependent global attractor which has optimal regularity.