Download Global Well-posedness and Asymptotic Behavior of the by Yuming Qin, Zhiyong Ma PDF

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

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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.

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