Citation: Yaorong Su, Weiguang Xie, Jianbin Xu. Towards low-voltage organic thin film transistors (OTFTs) with solution-processed high-k dielectric and interface engineering[J]. AIMS Materials Science, 2015, 2(4): 510-529. doi: 10.3934/matersci.2015.4.510
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This paper is concerned with the initial boundary value problem
$ {utt(x,t)−uxx(x,t)+μ1(t)ut(x,t)+μ2(t)ut(x,t−τ(t))=0in Ω×]0,+∞[,u(0,t)=u(L,t)=0on ]0,+∞[,u(x,0)=u0(x),ut(x,0)=u1(x)on Ω,ut(x,t−τ(0))=f0(x,t−τ(0))in Ω×]0,τ(0)[, $
|
(1) |
where
This problem has been first proposed and studied in Nicaise and Pignotti [22] in case of constant coefficients
With a weight depending on time,
W. Liu in [19] studied the weak viscoelastic equation with an internal time varying delay term. By introducing suitable energy and Lyapunov functionals, he establishes a general decay rate estimate for the energy under suitable assumptions.
F. Tahamtani and A. Peyravi [29] investigated the nonlinear viscoelastic wave equation with source term. Using the Potential well theory they showed that the solutions blow up in finite time under some restrictions on initial data and for arbitrary initial energy.
Global existence and asymptotic behavior of solutions to the viscoelastic wave equation with a constant delay term was considered by M. Remil and A. Hakem in [28].
Global existence and asymptotic stability for a coupled viscoelastic wave equation with time-varying delay was studied in [3] by combining the energy method with the Faedo-Galerkin's procedure.
The stabilization problem by interior damping of the wave equation with boundary or internal time-varying delay was studied in [23] by introducing suitable Lyapunov functionals.
Energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks was considered in [11].
For problems with delay in different contexts we cite [9,10,30,32] with references therein. In absence of delay (
Time delay is the property of a physical system by which the response to an applied force is delayed in its effect, and the central question is that delays source can destabilize a system that is asymptotically stable in the absence of delays, see [7]. In fact, an arbitrarily small delay may destabilize a system that is uniformly asymptotically stable in the absence of delay unless additional control terms have been used, see for example [8,12,31]
By energy method in [24] was studied the stabilization of the wave equation with boundary or internal distributed delay. By semigroup approach in [27] was proved the well-posedness and exponential stability for a wave equation with frictional damping and nonlocal time-delayed condition. Transmission problem with distributed delay was studied in [18] where was established the exponential stability of the solution by introducing a suitable Lyapunov functional.
Here we consider a wave equation with non-constant delay and nonlinear weights, thus, the present paper is a generalization of the previous ones. The remaining part of this paper is organized as follows. In the section 2 we introduce some notations and prove the dissipative property of the full energy of the system. In the section 3, for an approach combining semigroup theory (see [21] and [4]) with the energy estimate method we prove the existence and uniqueness of solution. In section 4 we present the result of exponential stability.
We will need the following hypotheses:
(H1)
$ |μ′1(t)μ1(t)|≤M1,0<α0≤μ1(t),∀t≥0, $
|
(2) |
where
(H2)
$ |μ2(t)|≤βμ1(t), $
|
(3) |
$ |μ′2(t)|≤M2μ1(t), $
|
(4) |
for some
We now state a lemma needed later.
Lemma 2.1 (Sobolev-Poincare's inequality). Let
$ \left\| \Psi \right\|_q \leq c_* \left\| \Psi_x \right\|_2, \quad \mathit{\mbox{for}}\; \Psi \in H_0^1(]0,L[). $ |
Lemma 2.2 ([13][16]). Let
$ \int_{S}^{+\infty} E^{1+\sigma}(t)\,dt \leq \frac{1}{\omega}E^\sigma(0)E(S),\ 0\leq S < +\infty. $ |
Then
$ E(t)=0 ∀t≥Eσ(0)ω|σ|, if−1<σ<0,E(t)≤E(0)(1+σ1+ωσt)1σ ∀t≥0, ifσ>0,E(t)≤E(0)e1−ωt ∀t≥0, ifσ=0. $
|
As in [23], we assume that
$ τ(t)∈W2,+∞([0,T]), for T>0 $
|
(5) |
and there exist positive constants
$ 0<τ0≤τ(t)≤τ1, ∀t>0 $
|
(6) |
and
$ τ′(t)≤d<1, ∀t>0. $
|
(7) |
We introduce the new variable
$ z(x,ρ,t)=ut(x,t−τ(t)ρ), x∈Ω,ρ∈]0,1[,t>0. $
|
(8) |
Then
$ \tau(t)z_t(x,\rho,t) + (1-\tau'(t)\rho)z_\rho(x,\rho,t) = 0, \ x\in \Omega,\ \rho \in ]0,1[,\ t > 0 $ |
and problem (1) takes the form
$ {utt(x,t)−uxx(x,t)+μ1(t)ut(x,t)+μ2(t)z(x,1,t)=0inΩ×]0,+∞[,τ(t)zt(x,ρ,t)+(1−τ′(t)ρ)zρ(x,ρ,t)=0inΩ×]0,1[×]0,+∞[,u(0,t)=u(L,t)=0on]0,+∞[,u(x,0)=u0(x),ut(x,0)=u1(x)onΩ,z(x,ρ,0)=ut(x,−τ(0)ρ)=f0(x,−τ(0)ρ)inΩ×]0,1[. $
|
(9) |
We define the energy of the solution of problem (9) by
$ E(t)=12‖ut‖2L2(Ω)+12‖ux‖2L2(Ω)+ξ(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx, $
|
(10) |
where
$ ξ(t)=ˉξμ1(t) $
|
(11) |
is a non-increasing function of class
$ β√1−d<ˉξ<2−β√1−d. $
|
(12) |
Our first result states that the energy is a non-increasing function.
Lemma 2.3. Let
$ E′(t)≤−μ1(t)(1−ˉξ2−β2√1−d)‖ut‖2L2(Ω)−μ1(t)(ˉξ(1−τ′(t))2−β√1−d2)‖z(x,1,t)‖2L2(Ω)≤0. $
|
(13) |
Proof. Multiplying the first equation (9) by
$ 12ddt(‖ut‖2L2(Ω)+‖ux‖2L2(Ω))+μ1(t)‖ut‖2L2(Ω)+μ2(t)∫Ωz(x,1,t)utdx. $
|
(14) |
Now multiplying the second equation (9) by
$ \tau(t)\xi(t)\int_{\Omega} \int_0^1\!\! z_t(x,\rho,t)z(x,\rho,t)\,d\rho\,dx = -\frac{\xi(t)}{2}\! \int_{\Omega} \int_0^1\!\!\! (1- \tau'(t)\rho)\frac{\partial}{\partial \rho}(z(x,\rho,t))^2\,d\rho\,dx. $ |
Consequently,
$ ddt(ξ(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx)=−ξ(t)2∫Ω∫10(1−τ′(t)ρ)∂∂ρ(z(x,ρ,t))2dρdx+ξ′(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx=ξ(t)2∫Ω(z2(x,0,t)−z2(x,1,t))dx+ξ(t)τ′(t)2∫Ω∫10z2(x,1,t)dρdx+ξ′(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx. $
|
(15) |
From (10), (14) and (15) we obtain
$ E′(t)=ξ(t)2‖ut‖2L2(Ω)−ξ(t)2‖z(x,1,t)‖2L2(Ω)+ξ(t)τ′(t)2‖z(x,1,t)‖2L2(Ω)+ξ′(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx−μ1(t)‖ut‖2L2(Ω)−μ2(t)∫Ωz(x,1,t)utdx. $
|
(16) |
Due to Young's inequality, we have
$ μ2(t)∫Ωz(x,1,t)utdx≤|μ2(t)|2√1−d‖ut‖2L2(Ω)+|μ2(t)|√1−d2‖z(x,1,t)‖2L2(Ω). $
|
(17) |
Inserting (17) into (16), we obtain
$ E′(t)≤−(μ1(t)−ξ(t)2−|μ2(t)|2√1−d)‖ut‖2L2(Ω)−(ξ(t)2−ξ(t)τ′(t)2−|μ2(t)|√1−d2)‖z(x,1,t)‖2L2(Ω)+ξ′(t)τ(t)2∫Ω∫10z2(x,ρ,t)dρdx≤−μ1(t)(1−ˉξ2−β2√1−d)‖ut‖2L2(Ω)−μ1(t)(ˉξ(1−τ′(t))2−β√1−d2)‖z(x,1,t)‖2L2(Ω)≤0. $
|
Lemma 2.4. Let
$ \|u_t(x,t)\|_{L^2(\Omega)}^{2} < -\frac{1}{\sigma}E'(t), $ |
where
Proof. From Lemma 2.3, we have that
$ −E′(t)≥μ1(t)(1−ˉξ2+β2√1−d)‖ut‖2L2(Ω)+μ1(t)(ˉξ(1−τ′(t))2+β√1−d2)‖z(x,1,t)‖2L2(Ω)≥0 $
|
and from (H1), we obtain
$ 0≤a0(1−ˉξ2+β2√1−d)‖ut‖2L2(Ω)≤μ1(t)(1−ˉξ2+β2√1−d)‖ut‖2L2(Ω)≤−E′(t) $
|
and the lemma is proved.
For the semigroup setup we
$ {Ut=A(t)U,U(0)=U0=(u0,u1,f0(⋅,−,τ(0)))T, $
|
(18) |
where the operator
$ AU=(v,uxx−μ1(t)v−μ2(t)z(x,1,t),−1−τ′(t)ρτ(t)zρ(x,ρ,t))T. $
|
(19) |
We introduce the phase space
$ \mathcal{H} = H_0^1(\Omega)\times L^2(\Omega)\times L^2(\Omega \times ]0,1[) $ |
and the domain of
$ D(A(t))={(u,v,z)T∈H/v=z(⋅,0) in Ω}, $
|
(20) |
where
$ H = H^2(\Omega)\cap H_0^1(\Omega)\times H_0^1(\Omega)\times L^2(\Omega; H_0^1(]0,1[)). $ |
Notice that the domain of the operator
$ D(A(t))=D(A(0)),∀t>0. $
|
(21) |
$ ⟨U,ˉU⟩H=∫Ωuxˉuxdx+∫Ωvˉvdx+ξ(t)τ(t)∫Ω∫10zˉzdρdx, $
|
(22) |
for
Using this time-dependent inner product and the next theorem we will get a result of existence and uniqueness.
Theorem 3.1. Assume that
(i)
(ii) (21) holds,
(iii) for all
(iv)
Then, problem (18) has a solution
Our goal is then to check the above assumptions for problem (18).
First, we prove
The proof is the same as the one Lemma
Let
$ 0 = \langle (u,v,z)^T,(f,g,h)^T \rangle_{\mathcal{H}} = \int_{\Omega} u_x f_x\,dx + \int_{\Omega} v g\,dx + \xi(t)\tau(t) \int_{\Omega} \int_0^1 z h\,d\rho\,dx, $ |
for all
We first take
$ \int_{\Omega} \int_0^1 zh\,d\rho\,dx = 0. $ |
Since
The above orthogonality condition is then reduced to
$ 0 = \int_{\Omega} u_xf_x\,dx, \quad \forall (u,v,z)^T \in D(\mathcal{A}(0)). $ |
By restricting ourselves to
$ 0 = \int_{\Omega} u_xf_x\,dx, \quad \forall (u,0,0)^T \in D(\mathcal{A}(0)). $ |
Since
We consequently
$ D(A(0) is dense in H. $
|
(23) |
Secondly, we notice that
$ ‖Φ‖t‖Φ‖s≤ec2τ0|t−s|,∀t,s∈[0,T], $
|
(24) |
where
$ ‖Φ‖2t−‖Φ‖2secτ0|t−s|=(1−ec2τ0|t−s|)(‖ux‖2L2(Ω)+‖v‖2L2(Ω))+(ξ(t)τ(t)−ξ(s)τ(s)ecτ0|t−s|)∫Ω∫10z2(x,ρ,t)dρdx. $
|
It is clear that
$ \tau(t) = \tau(s) + \tau'(r)(t-s), $ |
where
Hence
$ \xi(t)\tau(t) \leq \xi(s)\tau(s) + \xi(s)\tau'(r)(t-s), $ |
which implies
$ \frac{\xi(t)\tau(t)}{\xi(s)\tau(s)} \leq 1 + \frac{|\tau'(r)|}{\tau(s)}|t-s|. $ |
Using (5) and
$ \frac{\xi(t)\tau(t)}{\xi(s)\tau(s)} \leq 1 + \frac{c}{\tau_0}|t-s| \leq e^{\frac{c}{\tau_0}|t-s|}, $ |
which proves (24) and therefore
Now we calculate
$ ⟨A(t)U,U⟩t=∫Ωvxuxdx+∫Ω(uxx−μ1(t)v−μ2(t)z(⋅,1))vdx−ξ(t)∫Ω∫10(1−τ′(t)ρ)zρ(x,ρ)z(x,ρ)dρdx. $
|
Integrating by parts, we obtain
$ ⟨A(t)U,U⟩t=−μ1(t)‖v‖2L2(Ω)−μ2(t)∫Ωz(⋅,1)vdx−∫Ω∫10(1−τ′(t)ρ)∂∂ρz2(x,ρ)dρdx. $
|
Since
$ \left(1- \tau'(t)\rho \right)\frac{\partial}{\partial \rho} z^2(x,\rho) = \frac{\partial}{\partial \rho} \left( \left(1- \tau'(t)\rho \right) z^2(x,\rho) \right) + \tau'(t)z^2(x,\rho), $ |
we have
$ ∫10(1−τ′(t)ρ)∂∂ρz2(x,ρ)dρ=(1−τ′(t))z2(x,1)−z2(x,0)+τ′(t)∫10z2(x,ρ)dρ. $
|
So we get
$ ⟨A(t)U,U⟩t=−μ1(t)‖v‖2L2(Ω)−μ2(t)∫Ωz(x,1)vdx+ξ(t)2‖z(x,0)‖2L2(Ω)−ξ(t)(1−τ′(t))2‖z(x,1)‖2L2(Ω)−ξ(t)τ′(t)2∫Ω∫10z2(x,ρ)dρdx. $
|
Therefore, from (16) and (17), we deduce
$ ⟨A(t)U,U⟩t≤−μ1(t)(1−ˉξ2−β2√1−d)‖v‖2L2(Ω)−μ1(t)(ˉξ(1−τ′(t))2−β√1−d2)‖z(x,1,t)‖2L2(Ω)+ξ(t)|τ′(t)|2τ(t)τ(t)∫Ω∫10z2(x,ρ)dρdx. $
|
Then, we have
$ ⟨A(t)U,U⟩t≤−μ1(t)(1−ˉξ2−β2√1−d)‖v‖2L2(Ω)−μ1(t)(ˉξ(1−τ′(t))2−β√1−d2)‖z(x,1,t)‖2L2(Ω)+κ(t)⟨U,U⟩t, $
|
where
$ \kappa(t) = \frac{\sqrt{1+ \tau'(t)^2}}{2\tau(t)}. $ |
From the (13), we obtain
$ ⟨A(t)U,U⟩t−κ(t)⟨U,U⟩t≤0, $
|
(25) |
which means that the operator
Moreover,
$ \frac{d}{dt}\mathcal{A}(t)U = \left(0,0,\frac{\tau''(t)\tau(t)\rho-\tau'(t)(\tau'(t)\rho-1)}{\tau(t)^2}z_{\rho} \right)^T, $ |
with
$ ddt˜A(t)∈L∞∗([0,T],B(D(A(0)),H)), $
|
(26) |
the space of equivalence classes of essentially bounded, strongly measurable functions from
Now, we will show that
$ \left( \lambda I - \mathcal{A}(t) \right)U = F, $ |
that is verifying following system of equations
$ {λu−v=f1,λv−uxx+μ1(t)v−μ2(t)z(⋅,1)=f2,λz+1−τ′(t)ρτ(t)zρ=f3. $
|
(27) |
Suppose that we have found
$ v=λu−f1. $
|
(28) |
It is clear that
$ z(x,0)=v(x), for x∈Ω. $
|
(29) |
Following the same approach as in [22], we obtain, by using equation for
$ z(x,\rho) = v(x)e^{-\vartheta(\rho,t)} + \tau(t)e^{-\vartheta(\rho,t)} \int_{0}^{\rho} f_3(x,s)e^{\vartheta(s,t)}\,ds, $ |
if
$ z(x,\rho) = v(x)e^{\zeta(\rho,t)} + e^{\zeta(\rho,t)} \int_{0}^{\rho} \frac{\tau(t)f_3(x,s)}{1-s\tau'(s)} e^{-\zeta(s,t)}\,ds, $ |
otherwise, where
From (28), we obtain
$ z(x,ρ)=λu(x)e−ϑ(ρ,t)−f1(x,ρ)e−ϑ(ρ,t)+τ(t)e−ϑ(ρ,t)∫ρ0f3(x,s)eϑ(s,t)ds, $
|
(30) |
if
$ z(x,ρ)=λu(x)eζ(ρ,t)−f1(x,ρ)eζ(ρ,t)+eζ(ρ,t)∫ρ0τ(t)f3(x,s)1−sτ′(s)e−ζ(s,t)ds, $
|
(31) |
otherwise.
In particular, if
$ z(x,1)=λu(x)e−ϑ(1,t)−f1(x,1)e−ϑ(1,t)+τ(t)e−ϑ(1,t)∫10f3(x,s)eϑ(s,t)ds, $
|
(32) |
and if
$ z(x,1)=λu(x)eζ(1,t)−f1(x,1)eζ(1,t)+eζ(1,t)∫10τ(t)f3(x,s)1−sτ′(s)e−ζ(s,t)ds. $
|
(33) |
By using (27) and (28), the function
$ λ2u−uxx+μ1(t)v+μ2(t)z(⋅,1)=f2+λf1. $
|
(34) |
Solving the equation (34) is equivalent to finding
$ ∫Ω(λ2uη+uxηx+μ1(t)vη+μ2(t)z(⋅,1)η)dx=∫Ω(f2+λf1)ηdx, $
|
(35) |
for all
Consequently, the equation (35) is equivalent to the problem
$ Υ(u,η)=L(η), $
|
(36) |
where the bilinear form
$ \Upsilon: H_0^1(\Omega) \times H_0^1(\Omega) \rightarrow \mathbb{R} $ |
and the linear form
$ L: H_0^1(\Omega) \rightarrow \mathbb{R} $ |
are defined by
$ \Upsilon(u, \eta) = \int_{\Omega} \left( \lambda^2 u\eta + u_x \eta_x \right)\,dx + \int_{\Omega} \lambda u \left( \mu_1(t) + \mu_2(t)N_1 \right)\eta\,dx $ |
and
$ L(\eta) = \int_{\Omega} \left( \mu_1(t)f_1 \eta + \mu_2(t)N_2 \right)\eta\,dx + \int_{\Omega} (f_2 + \lambda f_1)\eta\,dx, $ |
where
$ N_1 = \left\{ e−ϑ(1,t),ifτ′(t)=0,eζ(1,t),ifτ′(t)≠0 \right. $
|
and
$ N_2 = \left\{ −f1(x,1)e−ϑ(1,t)+τ(t)e−ϑ(1,t)∫10f3(x,s)eϑ(s,t)ds,ifτ′(t)=0,−f1(x,1)ezeta(1,t)+ezeta(1,t)∫10τ(t)f3(x,s)1−sτ′(t)e−ζ(s,t)ds,ifτ′(t)≠0. \right. $
|
It is easy to verify that
$ u \in H_0^1(\Omega). $ |
Applying the classical elliptic regularity, it follows from (35) that
$ u \in H^2(\Omega). $ |
Therefore, the operator
$ λI−˜A(t)=(λ+κ(t))I−A(t) is surjective, $
|
(37) |
for any
Then, (24), (25) and (37) imply that the family
$ {˜Ut=˜A(t)˜U,˜U(0)=U0=(u0,u1,f0(⋅,−,τ(0)))T $
|
(38) |
has a unique solution
$ U(t) = e^{\int_0^t \kappa(s)\,ds}\tilde{U}(t) $ |
because
$ Ut(t)=κ(t)e∫t0κ(s)ds˜U(t)+e∫t0κ(s)ds˜Ut(t)=e∫t0κ(s)ds(κ(t)+˜A(t))˜U(t)=A(t)e∫t0κ(s)ds˜U(t)=A(t)U(t), $
|
which concludes the proof.
The existence and uniqueness are obtained by the following result.
Theorem 3.2 (Global solution). For any initial datum
$ U \in C([0,+\infty[, \mathcal{H}) $ |
for problem (18).
Moreover, if
$ U \in C([0,+\infty[, D(\mathcal{A}(0))) \cap C^1([0,+\infty[, \mathcal{H}). $ |
Proof. A general theory for equations of type (18) has been developed using semigroup theory [14], [15] and [26]. The simplest way to prove existence and uniqueness results in to show that the triplet
In this section we shall investigate the asymptotic behavior of problem (1). The stability result will be obtained using Lemma 2.2.
Theorem 4.1 (Stability Result). Let
$ u∈C([0,+∞[,H10(Ω))∩C1([0,+∞[,L2(Ω)), $
|
$ z∈C([0,+∞[,L2(Ω)×]0,1[). $
|
Proof. From now on, we denote by
Given
$ ∫TSEq∫Ωu(utt−uxx+μ1(t)ut+μ2(t)z(x,1,t))dxdt=0. $
|
Notice that
$ u_{tt}u = \left( u_t u \right)_t - u_t^2, $ |
using integration by parts and the boundary conditions we know that
$ 0=[Eq(t)∫Ωuutdx]TS−∫TSqEq−1(t)E′(t)∫Ωuutdxdt−∫TSEq(t)‖ut‖2L2(Ω)dt+∫TSEq(t)‖ux‖2L2(Ω)dt+∫TSEq(t)∫Ωμ1(t)uutdxdt+∫TSEq(t)∫Ωμ2(t)uz(x,1,t)dxdt. $
|
(39) |
Similarly, we multiply the second equation of (9) by
$ 0=∫TS∫Ω∫10Eq(t)ξ(t)e−2ρτ(t)z(τ(t)zt+(1−ρτ′(t))zρ)dρdxdt $
|
$ =12∫Ω∫10∫TSEq(t)ξ(t)e−2ρτ(t)∂∂tz2dtdρdx+12∫TSEq(t)ξ(t)∫Ω∫10e−2ρτ(t)(1−ρτ′(t))∂∂ρz2dρdxdt. $
|
Using integration by parts and the boundary conditions we know that
$ 0=[ξ(t)τ(t)2Eq(t)∫Ω∫10e−2ρτ(t)z2dρdx]TS−12∫TSqEq−1(t)E′(t)ξ(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt−12∫TSqEq(t)ξ′(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt+12∫TSEq(t)ξ(t)∫Ω[e−2ρτ(t)(1−τ′(t))z2(x,1,t)−z2(x,0,t)]dxdt+∫TSEq(t)ξ(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt. $
|
(40) |
Since
$ ∫TSqEq(t)ξ′(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt≤0. $
|
(41) |
Moreover, as
$ −12∫TSEq(t)ξ(t)∫Ωe−2ρτ(t)(1−τ′(t))z2(x,1,t)dxdt≤0, $
|
(42) |
then, from (40), (41) and (42), we have that
$ ∫TSEq(t)ξ(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt≤−[ξ(t)τ(t)2Eq(t)∫Ω∫10e−2ρτ(t)z2dρdx]TS+12∫TSqEq−1(t)E′(t)ξ(t)τ(t)∫Ω∫10e−2ρτ(t)z2dρdxdt−12∫TSEq(t)ξ(t)∫Ωz2(x,0,t)dxdt. $
|
(43) |
Using the definition of
$ γ0∫TSEq+1dt≤−[Eq(t)∫Ωuutdx]TS−[ξ(t)τ(t)2Eq(t)∫Ω∫10e−2ρτ(t)z2dρdx]TS+q∫TSEq−1(t)E′(t)∫Ωuutdxdt+q∫TSξ(t)τ(t)2Eq−1(t)E′(t)∫Ω∫10e−2ρτ(t)z2dρdxdt $
|
$ +2∫TSEq(t)‖ut‖2L2(Ω)dt−∫TSEq(t)∫Ωμ1(t)uutdxdt−∫TSEq(t)∫Ωμ2(t)uz(x,1,t)dxdt+12∫TSξ(t)Eq(t)e−2ρτ(t)∫Ωz2(x,0,t)dxdt, $
|
(44) |
where
Using the Young and Sobolev-Poincaré inequalities and Lemma 2.3, we find that
$ −[Eq(t)∫Ωuutdx]TS≤Eq(S)∫Ωu(x,S)ut(x,S)dx−Eq(T)∫Ωu(x,T)ut(x,T)dx≤cEq+1(S). $
|
Now, we known that
$ −[ξ(t)τ(t)2Eq(t)∫Ω∫10e−2ρτ(t)z2dρdx]TS≤ξ(S)τ(S)2Eq(S)∫Ω∫10e−2ρτ(S)z2(x,ρ,S)dρdx≤cEq(S)ξ(S)τ(S)∫Ω∫10z2(x,ρ,S)dρdx≤cEq+1(S). $
|
By (13), we have
$ ∫TSEq−1(t)E′(t)∫Ωuutdxdt≤c∫TS(−E′(t))Eq(t)dt≤cEq+1(S). $
|
Similarly,
$ ∫TSEq−1(t)E′(t)ξ(t)τ(t)2∫Ω∫10e−2ρτ(t)z2dρdxdt≤cEq+1(S). $
|
From Lemma 2.4, we deduce that
$ ∫TSEq(t)‖ut‖2L2(Ω)dt≤−c∫TSEq(t)E′(t)dt≤cEq+1(S). $
|
Now, we get that
$ |∫TSEq(t)∫Ωμ1(t)uutdxdt|≤μ1(0)|∫TSEq(t)∫Ωuutdxdt|≤c(ε1)∫TSEq(t)∫Ωu2tdxdt+ε1∫TSEq(t)∫Ωu2xdxdt≤c(ε1)∫TSEq(t)(−E′(t))dt+ε1∫TSEq(t)E(t)dt≤c(ε1)Eq+1(S)+ε1∫TSEq+1(t)dt $
|
(45) |
and from (H2) we obtain that
$ |∫TSEq(t)∫Ωμ2(t)uz(x,1,t)dxdt|≤βμ1(0)|∫TSEq(t)∫Ωφz(x,1,t)dxdt|≤c(ε2)Eq+1(S)+ε2∫TSEq+1(t)dt. $
|
(46) |
Finally,
$ 12∫TSEq(t)ξ(t)∫Ωz2(x,0,t)dxdt≤ˉξμ1(0)2∫TSEq(t)‖ut‖2L2(Ω)dt≤c∫TSEq(t)(−E′(t))dt≤cEq+1(S). $
|
Choosing
$ \int_S^T E^{q+1}\,dt \leq \frac{1}{\gamma}E^{q+1}(S). $ |
Since
$ \int_S^T E^{q+1}\,dt \leq \frac{1}{\gamma}E(0)E^{q}(S). $ |
We choose
$ E(t) \leq E(0)e^{1-\gamma t}. $ |
This ends the proof of Theorem 4.1.
[1] | Halik M, Klauk H, Zschieschang U, et al. (2004) Low-voltage organic transistors with an amorphous molecular gate dielectric. Nature 431: 963-966. |
[2] |
Moon H, Zeis R, Borkent E, et al. (2004) Synthesis, crystal structure, and transistor performance of tetracene derivatives. J Am Chem Soc 126: 15322-15223. doi: 10.1021/ja045208p
![]() |
[3] |
Jung B., Lee K, Sun J, et al. (2010) Air-operable, high-mobility organic transistors with semifluorinated side chains and unsubstituted naphthalenetetracarboxylic diimide cores: high mobility and environmental and bias stress stability from the perfluorooctylpropyl side chain. Adv Funct Mater 20: 2930-2944. doi: 10.1002/adfm.201000655
![]() |
[4] |
Dickey K, Anthony J, Loo Y, (2006) Improving organic thin-Film transistor performance through solvent-vapor annealing of solution-processable triethylsilylethynyl anthradithiophene. Adv Mater 18: 1721-1726. doi: 10.1002/adma.200600188
![]() |
[5] |
Coropceanu V, Cornil J, Demetrio A, et al. (2007) Charge transport in organic semiconductors. Chem Rev 107: 926-952. doi: 10.1021/cr050140x
![]() |
[6] |
Li W, Auciello O, Premnath R, et al. (2010) Giant dielectric constant dominated by Maxwell-Wagner relaxation in Al2O3/TiO2 nanolaminates synthesized by atomic layer deposition. Appl Phys Lett 96: 162907. doi: 10.1063/1.3413961
![]() |
[7] |
Lee J, Kim J, Im S, et al. (2003) Pentacene thin-film transistors with Al2O3+x gate dielectric films deposited on indium-tin-oxide glass. Appl Phys Lett 83: 2689. doi: 10.1063/1.1613997
![]() |
[8] |
Maunoury C, Dabertrand K, Martinez E, et al. (2007) Chemical interface analysis of as grown HfO2 ultrathin films on SiO2. J Appl Phys 101: 034112. doi: 10.1063/1.2435061
![]() |
[9] |
Di C, Yu G, Liu Y, et al. (2006) High-performance low-cost organic field-effect transistors with chemically modified bottom electrodes. J Am Chem Soc 128: 16418-16419. doi: 10.1021/ja066092v
![]() |
[10] |
Di C, Yu G, Liu Y, et al. (2008) High-performance organic field-effect transistors with low-cost copper electrodes. Adv Mater 20: 1286-1290. doi: 10.1002/adma.200701812
![]() |
[11] |
Di C, Liu Y, Yu G, et al. (2009) Interface engineering: an effective approach toward high-performance organic field-effect transistors. Acc Chem Res 42: 1573-1583. doi: 10.1021/ar9000873
![]() |
[12] |
Ma H, Yip H, Huang F, et al. (2010) Interface engineering for organic electronics. Adv Funct Mater 20: 1371-1388. doi: 10.1002/adfm.200902236
![]() |
[13] |
Chua L, Zaumseil J, Chang J, et al. (2005) General observation of n-type field-effect behaviour in organic semiconductors. Nature 434: 194-199. doi: 10.1038/nature03376
![]() |
[14] |
Frank M, Sayan S, Dörmann S, et al. (2004) Hafnium oxide gate dielectrics grown from an alkoxide precursor: structure and defects. J Mater Sci Eng B 109: 6-10. doi: 10.1016/j.mseb.2003.10.020
![]() |
[15] |
Lu Y, Lee W, Lee H, et al. (2009) Low-voltage organic transistors with titanium oxide/polystyrene bilayer dielectrics. Appl Phys Lett 94: 113303. doi: 10.1063/1.3097010
![]() |
[16] |
Fleischli F, Suarez S, Schaer M, et al. (2010) Organic thin-film transistors: the passivation of the dielectric-pentacene interface by dipolar self-assembled monolayers. Langmuir 26: 15044-15049. doi: 10.1021/la102060u
![]() |
[17] |
Wu W, Liu Y, Wang Y, et al. (2008) High-performance, low-operating-voltage organic field-effect transistors with low pinch-off voltages. Adv Funct Mater 18: 810-815. doi: 10.1002/adfm.200701125
![]() |
[18] | Tang M, Okamot T, Bao Z, (2006) High-performance organic semiconductors: asymmetric linear acenes containing sulphur. J Am Chem Soc 128: 16002-16003. |
[19] |
Acton O, Osaka I, Ting G, et al. (2009) Phosphonic acid self-assembled monolayer and amorphous hafnium oxide hybrid dielectric for high performance polymer thin film transistors on plastic substrates. Appl Phys Lett 95: 113305. doi: 10.1063/1.3231445
![]() |
[20] | Gao W, Dickinson L, Grozinger C, et al. (2009) Self-assembled monolayers of alkylphosphonic acids on metal oxides. Langmuir 12: 6429-6435. |
[21] |
Ma H, Acton O, Ting G, et al. (2008) Low-voltage organic thin-film transistors with π-σ-phosphonic acid molecular dielectric monolayers. Appl Phys Lett 92: 113303. doi: 10.1063/1.2857502
![]() |
[22] |
McElwee J, Helmy R, Fadeev A, et al. (2005) Thermal stability of organic monolayers chemically grafted to minerals. J Colloid Interface Sci 285: 551-556. doi: 10.1016/j.jcis.2004.12.006
![]() |
[23] |
Acton O, Ting G, Ma H, et al. (2008) π‐σ‐Phosphonic acid organic monolayer/sol-gel hafnium oxide hybrid dielectrics for low‐voltage organic transistors. Adv Mater 20: 3697-3701. doi: 10.1002/adma.200800810
![]() |
[24] |
Lu X, Minari T, Kumatani A. et al. (2011) Effect of air exposure on metal/organic interface in organic field-effect transistors. Appl Phys Lett 98: 243301. doi: 10.1063/1.3599056
![]() |
[25] |
Di C, Yu G, Liu Y, et al. (2008) Efficient modification of Cu electrode with nanometer-sized copper tetracyanoquinodimethane for high performance organic field-effect transistors. Phys Chem Chem Phys 10: 2302-2307. doi: 10.1039/b718935d
![]() |
[26] |
Gu W, Jin W, Wei B, at al. (2010) High-performance organic field-effect transistors based on copper/copper sulphide bilayer source-drain electrodes. Appl Phys Lett 97: 243303. doi: 10.1063/1.3526737
![]() |
[27] |
Su Y, Wang C, Xie W, et al. (2011) Low-voltage organic field-effect transistors (OFETs) with solution-processed metal-oxide as gate dielectric. ACS Appl Mater Interfaces 3: 4662-4667. doi: 10.1021/am201078v
![]() |
[28] |
Su Y, Xie W, Li Y, et al. (2013) A low-temperature, solution-processed high-k dielectric for low-voltage, high-performance organic field-effect transistors (OFETs). J Phys D Appl Phys 46: 095105. doi: 10.1088/0022-3727/46/9/095105
![]() |
[29] |
Su Y, Ouyang M, Liu P, et al. (2013) Insights into the interfacial properties of low-voltage CuPc field-effect transistor. ACS Appl Mater Interfaces 5: 4960-4965. doi: 10.1021/am4006447
![]() |
[30] |
Su Y, Wang M, Xie F, et al. (2013) In situ modification of low-cost Cu electrodes for high-performance low-voltage pentacene thin film transistors (TFTs). Org Electron 14: 775-781. doi: 10.1016/j.orgel.2012.12.025
![]() |
[31] | Su Y, Xie W, Xu J, (2014) Facile modification of Cu source-drain (S/D) electrodes for high-performance, low-voltage n-channel organic thin film transistors (OTFTs) based on C60. Org Electron 15: 3259-3267. |
[32] |
Su Y, Jiang J, Ke N, et al. (2013) Low-voltage flexible pentacene thin film transistors with solution-processed dielectric and low-cost source-drain (S/D) electrodes. J Mater Chem C 1: 2585-2592. doi: 10.1039/c3tc00577a
![]() |
[33] |
Forrest S, (1997) Ultrathin organic films grown by organic molecular beam deposition and related techniques. Chem Rev 97: 1793-1896. doi: 10.1021/cr941014o
![]() |
[34] | Yang S, Shin K, Par C, (2015) The effect of gate-dielectric surface energy on pentacene morphology and organic field-effect transistor characteristics. Adv Funct Mater 15: 1806-1814. |
[35] |
Gao J, Xu J, Zhu M, et al. (2007) Thickness dependence of mobility in CuPc thin film on amorphous SiO2 substrate. J Phys D Appl Phys 40: 5666-5669. doi: 10.1088/0022-3727/40/18/022
![]() |
[36] |
Qi Q, Yu A, Wang L, et al. (2010) Behavior of pentacene initial nucleation on various dielectrics and its effect on carrier transport in organic field-effect transistor. J Nanosci Nanotechnol 10: 7103-7107. doi: 10.1166/jnn.2010.2802
![]() |
[37] |
Chung Y, Verploegen E, Vailionis A, et al. (2011) Controlling electric dipoles in nanodielectrics and its applications for enabling air-stable n-channel organic transistors. Nano Lett 11: 1161-1165. doi: 10.1021/nl104087u
![]() |
[38] |
Nakamura M, Goto N, Ohashi N, et al. (2005) Potential mapping of pentacene thin-film transistors using purely electric atomic-force-microscope potentiometry. Appl Phys Lett 86: 122112. doi: 10.1063/1.1891306
![]() |
[39] |
Wang S, Minari T, Miyadera T, et al. (2007) Bias stress instability in pentacene thin film transistors: contact resistance change and channel threshold voltage shift. Appl Phys Lett 91: 203508. doi: 10.1063/1.2813640
![]() |
[40] |
Kim Y, Jeon D, (2010) Effect of deposition temperature on the morphology and contact resistance of Au on pentacene. J Appl Phys 108: 016101. doi: 10.1063/1.3445268
![]() |
[41] |
Diao L, Frisbie C, Schroepfer D, et al. (2007) Electrical characterization of metal/pentacene contacts. J Appl Phys 101: 014510. doi: 10.1063/1.2424396
![]() |
[42] |
Zaumseil J, Baldwin K, Rogers J, (2003) Contact resistance in organic transistors that use source and drain electrodes formed by soft contact lamination. J Appl Phys 93: 6117-6124. doi: 10.1063/1.1568157
![]() |
[43] |
Watkins N, Yan L, Gao Y, (2002) Electronic structure symmetry of interfaces between pentacene and metals. Appl Phys Lett 80: 4384. doi: 10.1063/1.1485129
![]() |
[44] |
Kang S, Yi Y, Kim C, et al. (2006) Energy level diagrams of C60/pentacene/Au and pentacene/C60/Au. Synth Met 156: 32-37. doi: 10.1016/j.synthmet.2005.10.001
![]() |
[45] |
Zhang D, Liu Y, Liu Y, et al. (2004) The electrical properties and the interfaces of Cu2O/ZnO/ITO p-i-n heterojunction. Physica B: Cond Mat 351: 178-183. doi: 10.1016/j.physb.2004.06.003
![]() |
[46] |
Tseng C, Cheng Y, Lee M, et al. (2007) Study of anode work function modified by self-assembled monolayers on pentacene/fullerene organic solar cells. Appl Phys Lett 91: 233510. doi: 10.1063/1.2823579
![]() |
[47] |
Zhou Y, Fuentes-Hernandez C, Shim J, et al. (2012) A universal method to produce low-work function rlectrodes for organic electronics. Science 336: 327-332. doi: 10.1126/science.1218829
![]() |
[48] | Yu Y, Zhao Y, Ryu S, et al. (2009) Tuning the graphene work function by electric field effect. Nano Lett 9: 3430-3434. |
[49] |
Rentenberger S, Vollmer A, Zojer E, (2006) UV∕ozone treated Au for air-stable, low hole injection barrier electrodes in organic electronics. J Appl Phys 100: 053701. doi: 10.1063/1.2336345
![]() |
[50] |
Scheinert S, Grobosch M, Paasch G, et al. (2012) Contact characterization by photoemission and device performance in P3HT based organic transistors. J Appl Phys 111: 064502. doi: 10.1063/1.3693541
![]() |
[51] |
Shibata K, Ishikawa K, Takezoe H, et al. (2008) Contact characterization by photoemission and device performance in P3HT based organic transistors. Appl Phys Lett 92: 023305. doi: 10.1063/1.2834374
![]() |
[52] | Kumatani A, Li Y, Darmawan P, et al. (2013) On practical charge injection at the metal/organic semiconductor interface. Sci Rep 3: 1026. |
[53] |
Heimel G, Romaner L, Bredas J, et al. (2006) Interface energetics and level alignment at covalent metal-molecule Junctions: π-conjugated thiols on gold. Phys Rev Lett 96: 196806. doi: 10.1103/PhysRevLett.96.196806
![]() |
[54] |
Heimel G, Romaner L, Zojer E, et al. (2008) The interface energetics of self-assembled monolayers on metals. Acc Chem Res 41: 721-729. doi: 10.1021/ar700284q
![]() |
[55] |
Li H, Paramonov P, Bredas J, (2010) Theoretical study of the surface modification of indium tin oxide with trifluorophenyl phosphonic acid molecules: impact of coverage density and binding geometry. J Mater Chem 20: 2630-2637. doi: 10.1039/b921768a
![]() |
[56] |
Yang R, Park J, Colesniuc C, et al. (2009) Analyte chemisorption and sensing on n- and p-channel copper phthalocyanine thin-film transistors. J Chem Phys 130: 164703. doi: 10.1063/1.3078036
![]() |
[57] | Ma H, Acton O, Hutchins D, et al. (2012) Multifunctional phosphonic acid self-assembled monolayers on metal oxides as dielectrics, interface modification layers and semiconductors for low-voltage high-performance organic field-effect transistors. Phys Chem Chem Phys 14: 14110-14126. |
[58] |
Acton O, Ting G, Ma H, et al. (2008) Low-voltage high-performance C60 thin film transistors via low-surface-energy phosphonic acid monolayer/hafnium oxide hybrid dielectric. Appl Phys Lett 93: 083302. doi: 10.1063/1.2975175
![]() |
[59] |
Colleaux F, Ball J, Wobkenberg P, et al. (2011) Bias-stress effects in organic field-effect transistors based on self-assembled monolayer nanodielectrics. Phys Chem Chem Phys 13: 14387-14393. doi: 10.1039/c1cp20769e
![]() |
[60] |
Nanditha D, Dissanayake M, Hatton R, (2007) Operation of a reversed pentacene-fullerene discrete heterojunction photovoltaic device. Appl Phys Lett 90: 113505. doi: 10.1063/1.2713345
![]() |
[61] |
Chu C, Shrotriya V, Li G, et al. (2006) Tuning acceptor energy level for efficient charge collection in copper-phthalocyanine-based organic solar cells. Appl Phys Lett 88: 153504. doi: 10.1063/1.2194207
![]() |
[62] |
Chu C, Sung C, Kekuda D, et al. (2009) Flexible fullerene field-effect transistors fabricated through solution processing. Adv Mater 21: 4845-4849. doi: 10.1002/adma.200901215
![]() |
[63] |
Ishii H, Hayashi N, Ito E, et. (2004) Kelvin probe study of band bending at organic semiconductor/metal interfaces: examination of Fermi level alignment. Phys Stat Sol A 201: 1075-1094. doi: 10.1002/pssa.200404346
![]() |
[64] |
Kuribara K, Wang H, Uchiyama N, et al. (2012) Organic transistors with high thermal stability for medical applications. Nat Commun 3: 723. doi: 10.1038/ncomms1721
![]() |
[65] |
Kraus M, Richler S, Opitz A, et al. (2010) High-mobility copper-phthalocyanine field-effect transistors with tetratetracontane passivation layer and organic metal contacts. J Appl Phys 107: 094503. doi: 10.1063/1.3354086
![]() |
[66] |
Ramanathan S, Park C, McIntyre P, (2002) Electrical properties of ultrathin zirconia films grown by UV ozone oxidation. J Appl Phys 91: 4521-4527. doi: 10.1063/1.1459103
![]() |
[67] |
Gonon P, El Kamel F, (2007) Dielectric response of Cu/amorphous BaTiO3/Cu capacitors. J Appl Phys 101: 073901. doi: 10.1063/1.2716871
![]() |
[68] |
Klauk H, Zschieschang U, Pflaum J, et al. (2007) Ultralow-power organic complementary circuits. Nature 445: 745-748. doi: 10.1038/nature05533
![]() |
[69] |
Kim S, Yoon W, Jang M, et al. (2012) Damage-free hybrid encapsulation of organic field-effect transistors to reduce environmental instability. J Mater Chem 22: 7731-7738. doi: 10.1039/c2jm13329f
![]() |
[70] |
Yang H, Kim S, Yang L, et al. (2007) Pentacene nanostructures on surface-hydrophobicity-controlled polymer/SiO2 bilayer gate-dielectrics. Adv Mater 19: 2868-2872. doi: 10.1002/adma.200700560
![]() |
[71] |
Yang H, Yang L, Ling M, et al. (2008) Aging susceptibility of terrace-like pentacene films. J Phys Chem C 112: 16161-16165. doi: 10.1021/jp8055224
![]() |
[72] |
Nabok D, Puschnig P, Ambrosch-Draxl C, et al. (2007) Crystal and electronic structures of pentacene thin films from grazing-incidence x-ray diffraction and first-principles calculations. Phys Rev B 76: 235322. doi: 10.1103/PhysRevB.76.235322
![]() |
[73] | Yang C, Yoon J, Kim S, et al. Bending-stress-driven phase transitions in pentacene thin films for flexible organic field-effect transistors. Appl Phys Lett 92: 243305. |
[74] |
Kang S, Noh Y, Baeg K, et al. (2008) Effect of rubbed polyimide layer on the field-effect mobility in pentacene thin-film transistors. Appl Phys Lett 92: 052107. doi: 10.1063/1.2830694
![]() |
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15. | Meng Hu, Xin-Guang Yang, Jinyun Yuan, Stability and Dynamics for Lamé System with Degenerate Memory and Time-Varying Delay, 2024, 89, 0095-4616, 10.1007/s00245-023-10080-8 | |
16. | Waled Al-Khulaifi, Manal Alotibi, Nasser-Eddine Tatar, Exponential decay in a delayed wave equation with variable coefficients, 2024, 9, 2473-6988, 27770, 10.3934/math.20241348 | |
17. | Yan-Fang Li, Zhong-Jie Han, Gen-Qi Xu, Stabilization of nonlinear non-uniform piezoelectric beam with time-varying delay in distributed control input, 2023, 377, 00220396, 38, 10.1016/j.jde.2023.08.031 | |
18. | Luqman Bashir, Jianghao Hao, Salah Boulaaras, Muhammad Fahim Aslam, Tong Zhang, Exponential decay and blow-up results of a logarithmic nonlinear wave equation having infinite memory and strong time-varying delay, 2025, 2025, 1687-2770, 10.1186/s13661-025-02079-7 |