EXISTENCE AND BLOW-UP OF SOLUTIONS FOR FRACTIONAL WAVE EQUATIONS OF KIRCHHOFF TYPE WITH VISCOELASTICITY

摘要

In this paper, we deal with the initial boundary value problem of the following fractional wave equation of Kirchhoff type utt + M([u]_(α,2)~2)(-Δ)~αu + (-Δ)~sut = ∫_0~tg(t - τ)(-Δ)~αu(τ)dτ + λ|u|~(q-2)u, where M : [0, ∞) → (0, ∞) is a nondecreasing and continuous function, [u]α,2 is the Gagliardo-seminorm of u, (-Δ)~α and (-Δ)~s are the fractional Laplace operators, g : R~+ → R~+ is a positive nonincreasing function and λ is a parameter. First, the local and global existence of solutions are obtained by using the Galerkin method. Then the global nonexistence of solutions is discussed via blow-up analysis. Our results generalize and improve the existing results in the literature.