Existence and Stability Estimate for the Solution of the Ageing Hereditary Linear Viscoelasticity Problem

and Applied Analysis 3 holding for any t ∈ 0, T . Here ( a ij x e j k ) t : ∫ t 0 a ij x, t, τ · e j k x, τ dτ 2.4 are Volterra integral operators with kernels a ij x, t, τ ; a hk ij 0 x, t are instantaneous elastic coefficients out-of-integral terms and a ij x : a hk ij 0 x, t a ij x ; fi0 are components of a vector of external forces; φi x, t are components of a vector of boundary traction on the part ∂Ωσ of the external boundary; ψ x, t are components of the displacement vector on the rest part ∂Ωu of the boundary. All functions are supposed to be continuous w.r.t. t ∈ 0, T and sufficiently smooth w.r.t. x in domain Ω for performing a partial integration . The whole viscoelastic operator tensor a ij x n×n n×n is assumed to be symmetric at each point x ∈ Ω: a ij x a kh ji x a ik hj x a hj ik x . 2.5 The tensor a ij 0 x, t n×n n×n is additionally positive definite, with elements bounded at each point x ∈ Ω c0η j k η j k ≤ a ij 0 x, t η i hη j k ≤ C0ηj kη j k , 2.6 for all η k η k j ∈ R and t ∈ 0, T where the constants 0 < c0 ≤ C0 < ∞ are independent of x and t. For isotropic materials a ij λδhiδkj μδijδhk μδikδhj . Example 2.1. i Often, the kernels a ij x, t, τ are of the convolution type and are taken in the exponential form a ij x, t, τ ⎧⎪⎨ ⎪⎩ m ∑ p 1 α ij p x e−βp x t−τ , if t ≥ τ, 0, if t < τ, 2.7 where the βp are piece-wise continuous functions, often just constants, and the α ij p x are piece-wise continuous functions for x ∈ Ω. ii The a ij x, t, τ may also be kernels of the Abel type typical example for the relaxation kernels of concrete and cement : a ij x, t, τ ⎪⎪⎨ ⎪⎪⎩ A ij 1 x, t, τ t − τ −α A ij 2 x, t, τ τ −β A ij 3 x, t, τ t −γ , if t ≥ τ, 0, otherwise, 2.8 with 0 ≤ α, β, γ < 1. TheA ij p, p 1, 2, 3, are continuous in t and τ , and piece-wise continuous in x ∈ Ω. 4 Abstract and Applied Analysis 3. Weak Problem Formulation and Main Results In order to obtain the variational formulation, we multiply 2.1 by test functions vi x ∈ H1 0 Ω, ∂Ωu , i 1, . . . , n, where H 1 0 Ω, ∂Ωu : {v ∈ H1 Ω : v x 0, x ∈ ∂Ωu}, and integrate over the whole domain Ω. Integrating by parts and taking into account boundary condition 2.2 , we obtain the following variational problem. Find uj ∈ H1 Ω , j 1, . . . , n, satisfying 2.2 and ∫ Ω a ij ∂uj ∂xk ∂vi ∂xh dx l v , l v : ∫