By Yuming Qin, Zhiyong Ma
This booklet provides fresh findings at the international life, the individuality and the large-time habit of worldwide ideas of thermo(vis)coelastic platforms and comparable versions coming up in physics, mechanics and fabrics technological know-how corresponding to thermoviscoelastic platforms, thermoelastic platforms of sorts II and III, in addition to Timoshenko-type structures with earlier heritage. a part of the e-book is predicated at the study performed by way of the authors and their collaborators in recent times. The booklet will gain newcomers within the box and specialists alike.
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Extra info for Global Well-posedness and Asymptotic Behavior of the Solutions to Non-classical Thermo(visco)elastic Models
83), yields that if + δ is small enough, 1 d ϒ(t) ≤ −Nαβ −1 dt φx kˆ ∗ φx + 2φtx kˆ ∗ φtx + φttx kˆ ∗ φttx 0 + φxx kˆ ∗ φxx + φtxx kˆ ∗ φtxx dx C3 n(t, v, φ) + n(t, vt , φt ) 2 a2 + 2C4 + 1 + C16 δ + C14 β3 4 + kˆ ∗ φttx 2 + kˆ ∗ φxx 2 + − 2 ˆ + λ1 (kˆ (t))2 + λ2 (k(t)) kˆ ∗ φx 2 kˆ ∗ φtxx φ0xx 2 kˆ ∗ φtx + 2 2 +λ3 (kˆ (t))2 ( φ1x + λ4 (k2 (t))2 + ((k1 ∗ k2 )(t))2 + ((k1 ∗ k2 ) (t))2 p0x 2 + φ0x 2 . 59), taking δ and small enough, we deduce ϒ(t) + C3 2 t n(τ, v, φ) + n(τ, vt , φt ) dτ 0 2 t +C17 0 kˆ ∗ ∂ti φx 1 2 i=0 α1 α2 2 ≤ (β1 + β2 ) kˆ ∗ ∂ti φx + 2 dτ i=0 N 2 α 2 C4 + k˜ 1 (0) 2(β1 + β2 ) + α1 α2 β1 β2 2 2 + η0 = α3 2 .
11), we can easily obtain the result. Next we introduce the functional I3 (t) := ρ2 1 0 ψt (ϕx + ψ)d x + +∞ ρ1 b¯ 1 ρ1 1 ψ x ϕt d x + ϕt g(s)ηtx (x, s)dsd x.
20) and for all t ≥ 0, Let us denote v(x, t) = eδt u(x, t), φ(x, t) = eδt θ (x, t), p(x, t) = eδt q(x, t). 2 Global Existence and Exponential Stability where 33 F(t) = feδt + 2δvt − δ 2 v, G(t) = geδt + δφ + δβvx , 1 1 φ0 (x)dx = 0 θ0 (x)dx = 0. 28) that 1 1 φ(x, t)dx = 0 θ (x, t)dx = 0. 29) 0 To facilitate our analysis, let us introduce the linear problem ⎧ Vtt − Vxx + α x = F, ⎪ ⎪ ⎪ ⎪ ˆ ⎪ ⎨ t − k ∗ xx + βVxt = G, Pt + (1 − δ)P + k x = 0, ⎪ ⎪ ⎪ V (0, t) = V (1, t) = P(0, t) = P(1, t) = 0, ⎪ ⎪ ⎩ V (x, 0) = V0 , Vt (x, 0) = V1 , (x, 0) = 0 , P(x, 0) = P0 , with 1 0 (x)dx = 0.