Exponential Decay to Thermoelastic Systems over Noncylindrical Domains

Abstract

This paper is concerned with linear thermoelastic systems defined in domains with moving boundary. The uniform rate of decay of the energy associated is proved.

1. Introduction

In the study of asymptotic behavior for thermoelastic systems, a pioneering work is the one by Dafermos [1] concerned with the classical linear thermoelasticity for inhomogeneous and anisotropic materials, where the existence of a unique global solution and asymptotic stability of the system were proved. The existence of solution and asymptotic behavior to thermoelastic systems has been investigated extensively in the literature. For example, Muñoz Rivera [2] showed that the energy of the linear thermoelastic system (on cylindrical domain) decays to zero exponentially as 𝑡→∞. In [3], Burns et al. proved the energy decay for a linear thermoelastic bar. The asymptotic behaviour of a semigroup of the thermoelasticity was established in [4]. Concerning nonlinear thermoelasticity we can cite [5–7].

In the last two decades, several well-known evolution partial differential equations were extended to domains with moving boundary, which is also called noncylindrical problems. See, for instance, [8–10] and the references therein. In this work we studied the linear thermoelastic system in a noncylindrical domain with Dirichlet boundary conditions. This problem was early considered by Caldas et al. [11], which concluded that the energy associated to the system decreases inversely proportional to the growth of the functions that describes the noncylindrical domain. However they did not establish a rate of decay. The goal in the present work is to provide a uniform rate of decay for this noncylindrical problem.

Let us consider noncylindrical domains 𝑄⊂ℝ² of the form 𝑄=(𝑥,𝑡)∈ℝ²;𝑥=𝐾(𝑡)𝑦,𝑦∈(−1,1),𝑡∈(0,𝑇),(1.1) with lateral boundary Σ=_0<𝑡<𝑇{{−𝐾(𝑡)×{𝑡}}∪{𝐾(𝑡)×{𝑡}}},(1.2) where 𝐾∶[0,𝑇]→ℝ⁺ is a given 𝐶² function. Then our problem is𝑢𝜃_𝑡𝑡−𝑢_𝑥𝑥+𝛼𝜃_𝑥=0in𝑄,(1.3)_𝑡−𝑘𝜃_𝑥𝑥+𝛽𝑢_𝑥𝑡=0in𝑄,(1.4) with initial conditions𝑢(𝑥,0)=𝑢₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥),𝜃(𝑥,0)=𝜃₀(𝑥),−𝐾(0)<𝑥<𝐾(0),(1.5) and boundary conditions𝑢(−𝐾(𝑡),𝑡)=𝑢(𝐾(𝑡),𝑡)=0,𝜃_𝑥(−𝐾(𝑡),𝑡)=𝜃(𝐾(𝑡),𝑡)=0,0<𝑡<𝑇,(1.6) where 𝛼, 𝛽, and 𝑘 are positive real constants.

The function 𝐾(𝑡) and the constants 𝛼,𝛽, and 𝑘 satisfy the following conditions.

(H1) 𝐾∈𝐶²([0,𝑇],ℝ⁺) and 𝐾₀=min_{0≤𝑡≤𝑇}𝐾(𝑡)>0.(1.7)

(H2) There exists a positive constant 𝐾₁ such that 1−𝐾^(𝑡)𝑦²>𝐾₁.(1.8)

Problem (1.3)–(1.6) is slightly different from the one of [11] with respect to condition (1.6). Indeed, they assumed that 𝜃(−𝐾(𝑡),𝑡)=𝜃(𝐾(𝑡),𝑡)=0, for all 𝑡∈[0,𝑇]. Because of this mixed boundary condition in (1.6), we are able to construct a suitable Liapunov functional to derive decay rates of the energy. This is sufficient to provide a uniform rate of decay for this noncylindrical problem.

The existence and uniqueness of global solutions are derived by the arguments of [11] step by step, that is, to prove that the result of existence and uniqueness is based on transforming the system (1.3)–(1.6) into another initial boundary-value problem defined over a cylindrical domain whose sections are not time-dependent. This is done using a suitable change of variable. Then to show the existence and uniqueness for this equivalent system using Galerkin Methods and the existence result on noncylindrical domains will follows using the inverse of the transformation.

Therefore, we have the following result.

Theorem 1.1. Let Ω_𝑡 and Ω₀ be the intervals (−𝐾(𝑡),𝐾(𝑡)), 0<𝑡<𝑇, and (−𝐾(0),𝐾(0)), respectively. Then, given 𝑢₀,𝜃₀∈𝐻¹⁰(Ω₀)∩𝐻²(Ω₀) and 𝑢₁∈𝐻¹⁰(Ω₀), there exist unique functions 𝑢∶𝑄⟶ℝ,𝜃∶𝑄⟶ℝ(1.9) satisfying the following conditions: 𝑢𝑢∈𝐿^∞0,𝑇;𝐻¹⁰Ω_𝑡∩𝐻²Ω_𝑡,𝑢_𝑡∈𝐿^∞0,𝑇;𝐻¹⁰Ω_𝑡_𝑡𝑡,∈𝐿^∞0,𝑇;𝐿²Ω_𝑡,𝜃∈𝐿²0,𝑇,𝐻²Ω_𝑡,𝜃_𝑡∈𝐿²0,𝑇;𝐻¹⁰Ω_𝑡,(1.10) which are solutions of (1.3)–(1.6) in 𝑄.

2. Energy Decay

In [11] the authors proved that the energy associated with (1.3)–(1.6) decays at the rate 1/[𝐾(𝑡)]^𝛾1 with 𝛾₁>0; that is, the energy is decreasing inverserly proportional to the increase of sections of 𝑄. We make a slightly difference from the one of [11] with respect to the hypotheses about 𝐾; we are able to construct a suitable Liapunov functional to derive decay rates of the energy. This is done with the thermal dissipation only. More specifically, in this section we prove that the energy associated with (1.3)–(1.6) decays exponentially. Instead considering an auxiliary problem, we work directly on the original problem (1.3)-(1.4) in its noncylindrical domain.

In order to decay rates of the energy let us suppose the following hypotheses.

(H3) There exist positive constants 𝛿₀ and 𝛿₁ such that 0<𝛿₀≤𝐾^(𝑡)≤𝛿₁<1,𝑡≥0.(2.1)

(H4) There exists a positive constant 𝛿₂ such that 0<𝐾(𝑡)𝐾^(𝑡)≤𝛿₂,𝑡≥0.(2.2)

Let us introduce the energy functional 𝐸(𝑡)=𝐸(𝑡;𝑢,𝜃)=12^{𝐾(𝑡)−𝐾(𝑡)}||𝑢^𝑡||²+||𝑢_𝑥||²+𝛼𝛽||𝜃||²𝑑𝑥.(2.3)

Our main result is the following.

Theorem 2.1. Under the hypotheses (H1)–(H4), there exist positive constants 𝐶 and 𝛾 such that 𝐸(𝑡;𝑢,𝜃)≤𝐶𝐸(0;𝑢,𝜃)𝑒^−𝛾𝑡.(2.4)

The proof of Theorem 2.1 is given by using multipliers techniques. The notations and function spaces used here are standard and can be found, for instance, in the book by Lions [8].

Lemma 2.2. Let (𝑢,𝜃) be solution of (1.3)–(1.5) given by Theorem 1.1; then one obtains 𝑑𝑑𝑡𝐸(𝑡;𝑢,𝜃)≤−𝑘𝛽𝛼^{𝐾(𝑡)−𝐾(𝑡)}||𝜃_𝑥||²𝑑𝑥−𝐶₀||𝑢^𝑥(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||²,(2.5) where 𝐶₀=(𝛿₀/2)(1−𝛿²¹)>0.

Proof. From hypothesis 𝑢(𝐾(𝑡),𝑡)=0=𝑢(−𝐾(𝑡),𝑡) it follows that 𝑢_𝑡(𝐾(𝑡),𝑡)=−𝐾^(𝑡)𝑢_𝑥(𝐾(𝑡),𝑡)𝑒𝑢_𝑡(−𝐾(𝑡),𝑡)=𝐾^(𝑡)𝑢_𝑥(−𝐾(𝑡),𝑡).(2.6)
Multiplying (1.3) by 𝑢_𝑡, integrating in the variable 𝑥, and from (2.6) we obtain ^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑡𝑢_𝑡𝑑𝑥=12𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥−𝐾^(𝑡)||𝑢_𝑡(𝐾(𝑡),𝑡)||²−𝐾^(𝑡)||𝑢_𝑡(−𝐾(𝑡),𝑡)||²=12𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥−𝐾^(𝑡)³||𝑢_𝑥(𝐾(𝑡),𝑡)||²−𝐾^(𝑡)³||𝑢_𝑥(−𝐾(𝑡),𝑡)||².(2.7)
Now, applying integration by parts and using (2.6) it follows that ^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑥𝑥𝑢_𝑡𝑑𝑥==−𝐾𝑢_𝑥(𝐾(𝑡),𝑡)𝑢_𝑡(𝐾(𝑡),𝑡)+𝑢_𝑥(−𝐾(𝑡),𝑡)𝑢_𝑡(−𝐾(𝑡),𝑡)+^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑥𝑢_𝑡𝑥𝑑𝑥_(𝑡)2||𝑢^𝑥(𝐾(𝑡),𝑡)||²+𝐾^(𝑡)2||𝑢^𝑥(−𝐾(𝑡),𝑡)||²+12𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑥||²𝑑𝑥.(2.8) Thus, from inequalities (2.7) and (2.8) we have 12𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}||𝑢^𝑡||²+||𝑢_𝑥||²𝑑𝑥=−12𝐾𝐾^_(𝑡)³||𝑢^𝑥(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||²(𝑡)2||𝑢^𝑥−𝛼_{𝐾(𝑡)−𝐾(𝑡)}(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||²𝜃_𝑥𝑢_𝑡𝑑𝑥.(2.9)
Multiplying (1.4) by 𝜃 and integrating in the variable 𝑥 and using (2.6) we obtain 12𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}||𝜃||²𝑑𝑥=−𝑘^{𝐾(𝑡)−𝐾(𝑡)}||𝜃_𝑥||²𝑑𝑥+𝛽^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝜃_𝑥𝑑𝑥.(2.10)
Multiplying (2.10) by 𝛼/𝛽 and summing with (2.9) it follows that 12𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}||𝑢^𝑡||²+||𝑢_𝑥||²+𝛼𝛽||𝜃||²𝑑𝑥=−𝑘𝛽𝛼^{𝐾(𝑡)−𝐾(𝑡)}+𝐾||𝜃_𝑥||²𝑑𝑥_(𝑡)³2||𝑢^𝑥−𝐾_(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||²(𝑡)2||𝑢^𝑥(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||².(2.11)
Thus, following the hypothesis (H3), 12𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}||𝑢^𝑡||²+||𝑢_𝑥||²+𝛼𝛽||𝜃||²𝑑𝑥≤−𝑘𝛽𝛼^{𝐾(𝑡)−𝐾(𝑡)}−𝐾||𝜃_𝑥||²𝑑𝑥_(𝑡)21−𝛿²¹||𝑢_𝑥≤−𝑘𝛼(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||²𝛽^{𝐾(𝑡)−𝐾(𝑡)}−𝛿||𝜃_𝑥||²𝑑𝑥₀21−𝛿²¹||𝑢_𝑥(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||²,(2.12) which concludes the demonstration.

To estimate the term ∫^{𝐾(𝑡)−𝐾(𝑡)}|𝑢_𝑥|²𝑑𝑥 of the energy we use the following lemma.

Lemma 2.3. With the same hypothesis of Lemma 2.2, one gets 𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑢𝑑𝑥≤^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥−12^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑥||²𝑑𝑥+𝐶^p𝛼²2^{𝐾(𝑡)−𝐾(𝑡)}||𝜃_𝑥||²𝑑𝑥,(2.13) where 𝐶_𝑝 is Poincare's constant.

Proof. From the outline condition 𝑢(−𝐾(𝑡),𝑡)=𝑢(𝐾(𝑡),𝑡)=0 follows that 𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑢𝑑𝑥=^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥+^{𝐾(𝑡)−𝐾(𝑡)}𝑢𝑢_𝑡𝑡𝑑𝑥.(2.14)
Replacing 𝑢_𝑡𝑡=𝑢_𝑥𝑥−𝛼𝜃_𝑥 in the derivative above we get 𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑢𝑑𝑥=^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥−^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑥||²𝑑𝑥+𝛼^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑥𝜃𝑑𝑥.(2.15)
Applying Cauchy-Schwartz's inequality, Young's inequality, and Poincare's inequality in (2.15) we have 𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑢𝑑𝑥≤^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥−12^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑥||²𝑑𝑥+𝐶^𝑝𝛼²2^{𝐾(𝑡)−𝐾(𝑡)}||𝜃_𝑥||²𝑑𝑥.(2.16) Therefore our conclusion follows.

To estimate the term ∫^{𝐾(𝑡)−𝐾(𝑡)}|𝑢_𝑡|²𝑑𝑥 of the energy we introduce the function 𝑞=∫^{𝑥−𝐾(𝑡)}𝜃𝑑𝑠. By these conditions we have the following lemma.

Lemma 2.4. With the same hypothesis of Lemma 2.2, there are positive constants 𝐶₁ and 𝐶₂ such that 𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑞𝑑𝑥≤𝐶₁^{𝐾(𝑡)−𝐾(𝑡)}||𝜃_𝑥||²𝑑𝑥+𝛽32^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑥||²𝑑𝑥−𝛽4^{𝐾(𝑡)−𝐾(𝑡)}+𝜀𝐾(𝑡)||𝑢_𝑥||𝑢_𝑡||²𝑑𝑥(𝐾(𝑡),𝑡)||²+𝐶₂||𝑢^𝑥(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||²,(2.17) where 𝐶₁=((𝐶_𝑝/2𝛿₀)+𝛼𝐶_𝑝+(𝑘²/2𝛽)+(𝐶_𝑝/2)+(8/𝛽)) and 𝐶₂=𝛿₂(1+2𝛽𝛿₁+𝛿³¹).

Proof. Calculate the derivative 𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑞𝑑𝑥=^{𝐾(𝑡)−𝐾(𝑡)}𝜕𝜕𝑡𝑢^𝑡+𝐾_𝑞𝑑𝑥+𝐾^(𝑡)𝑢_𝑡(𝐾(𝑡),𝑡)𝑞(𝐾(𝑡),𝑡)=(𝑡)𝑢_𝑡(−𝐾(𝑡),𝑡)𝑞(−𝐾(𝑡),𝑡)_{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑡𝑞𝑑𝑥+^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑞_𝑡𝑑𝑥+𝐾^(𝑡)𝑢_𝑡(𝐾(𝑡),𝑡)𝑞(𝐾(𝑡),𝑡).(2.18)
From (1.3) and recording that 𝑞=∫^{𝑥−𝐾(𝑡)}𝜃𝑑𝑠, we get 𝐼₁=∶𝑢_𝑥(𝐾(𝑡),𝑡)𝑞(𝐾(𝑡),𝑡)−^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑥𝑞_𝑥𝑑𝑥+^{𝐾(𝑡)−𝐾(𝑡)}𝛼𝜃𝑑𝑑𝑥^{𝑥−𝐾(𝑡)}+𝜃𝑑𝑠𝑑𝑥_{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑑𝑥^{𝑥−𝐾(𝑡)}𝜃_𝑡𝑑𝑠+𝐾^(𝑡)𝑢_𝑡(𝐾(𝑡),𝑡)𝑞(𝐾(𝑡),𝑡).(2.19)
As 𝑞_𝑥=𝜃 and 𝑞_𝑡=∫^{𝑥−𝐾(𝑡)}𝜃_𝑡𝑑𝑠 we obtain 𝐼₁=+𝑢_𝑥(𝐾(𝑡),𝑡)^{𝐾(𝑡)−𝐾(𝑡)}𝜃𝑑𝑥−^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑥𝜃𝑑𝑥+𝛼^{𝐾(𝑡)−𝐾(𝑡)}||𝜃||²𝑑𝑥_{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑞_𝑡𝑑𝑥+𝐾^(𝑡)𝑢_𝑡(𝐾(𝑡),𝑡)^{𝐾(𝑡)−𝐾(𝑡)}𝜃𝑑𝑥.(2.20)
Now, integrating (1.4) from −𝐾(𝑡) to 𝑥, multiplying by 𝑢_𝑡, and after integrating from −𝐾(𝑡) to 𝐾(𝑡), it follows that 𝐼₂==∶𝑘^{𝐾(𝑡)−𝐾(𝑡)}_{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑞_𝑡𝑑𝑥𝜃_𝑥𝑢_𝑡𝑑𝑥−𝛽^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥+𝛽𝑢_𝑡(−𝐾(𝑡),𝑡)^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑑𝑥.(2.21)
Replacing (𝐼₂) in (𝐼₁) and from (2.6) we get 𝐼₁=𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}=𝑢_𝑥𝑢_𝑡𝑞𝑑𝑥+𝛼(𝐾(𝑡)_{𝐾(𝑡)−𝐾(𝑡)},𝑡)^{𝐾(𝑡)−𝐾(𝑡)}𝜃𝑑𝑥−^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑥𝜃𝑑𝑥+𝛽𝐾_||𝜃||²𝑑𝑥+𝑘^{𝐾(𝑡)−𝐾(𝑡)}𝜃_𝑥𝑢_𝑡𝑑𝑥−𝛽^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥(𝑡)𝑢_𝑥(−𝐾(𝑡),𝑡)^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑑𝑥−𝐾^(𝑡)²𝑢_𝑥(𝐾(𝑡),𝑡)^{𝐾(𝑡)−𝐾(𝑡)}𝜃𝑑𝑥.(2.22)
Estimating some terms of (2.22) we obtain 𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑞𝑑𝑥≤𝛼^{𝐾(𝑡)−𝐾(𝑡)}||𝜃||²𝑑𝑥+𝐾^(𝑡)𝐾(𝑡)||𝑢_𝑥(𝐾(𝑡),𝑡)||²+𝐶^𝑝2𝐾^(𝑡)^{𝐾(𝑡)−𝐾(𝑡)}−||𝜃_𝑥||²𝑑𝑥_{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑥𝜃𝑑𝑥−𝛽4^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥+𝑘²2𝛽+𝐶2^𝑝^{𝐾(𝑡)−𝐾(𝑡)}+2𝛽𝐾(𝑡)𝐾_||𝜃_𝑥||²𝑑𝑥(𝑡)²||𝑢_𝑥(−𝐾(𝑡),𝑡)||²+𝐾(𝑡)𝐾^(𝑡)⁴||𝑢_𝑥(𝐾(𝑡),𝑡)||².(2.23)
Applying Poincare's inequality in the first term of the previous inequality, using the hypothesis (H3), and grouping the common terms, we obtain 𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑞𝑑𝑥≤𝛼𝐶_𝑝+𝐶^𝑝2𝛿₀+𝑘²2𝛽+𝐶2^𝑝^{𝐾(𝑡)−𝐾(𝑡)}||𝜃_𝑥||²𝑑𝑥−^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑥𝜃𝑑𝑥−𝛽4^{𝐾(𝑡)−𝐾(𝑡)}+𝐾(𝑡)𝐾_||𝑢_𝑡||²𝑑𝑥(𝑡)1+2𝛽𝛿₁+𝛿³¹||𝑢_𝑥(−𝐾(𝑡),𝑡)||²+||𝑢_𝑥(𝐾(𝑡),𝑡)||².(2.24)
From hypothesis (H4) we have 𝑑𝑑𝑡^{𝐾(𝑡)−𝐾(𝑡)}𝑢_𝑡𝑞𝑑𝑥≤𝐶₁^{𝐾(𝑡)−𝐾(𝑡)}||𝜃_𝑥||²𝑑𝑥+𝛽32^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑥||²𝑑𝑥−𝛽4^{𝐾(𝑡)−𝐾(𝑡)}+𝐶₂||𝑢_𝑡||²𝑑𝑥||𝑢^𝑥(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||².(2.25) where 𝐶₁ and 𝐶₂ are positive constants. This concludes the demonstration of the lemma.

Now we use the above auxiliary lemmas to conclude the proof of Theorem 2.1.

Proof of Theorem 2.1. Consider the functional ℱ(𝑡)=^{𝐾(𝑡)−𝐾(𝑡)}𝛽8𝑢^𝑡𝑢+𝑢_𝑡𝑞𝑑𝑥.(2.26) From Lemmas 2.3 and 2.4 we obtain 𝑑𝑑𝑡ℱ(𝑡)≤−3𝛽2^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑥||²𝑑𝑥−𝛽8^{𝐾(𝑡)−𝐾(𝑡)}+𝐶||𝑢_𝑡||²𝑑𝑥₁+𝛼²𝛽𝐶_𝑝16^{𝐾(𝑡)−𝐾(𝑡)}||𝜃_𝑥||²𝑑𝑥+𝐶₂||𝑢^𝑥(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||².(2.27)
Finally we introduce the functional ℒ(𝑡)=ℱ(𝑡)+𝑁𝐸(𝑡),(2.28) where 𝑁∈ℕ will be chosen later.
From Lemma 2.2 and from (2.27) it follows that 𝑑𝑑𝑡ℒ(𝑡)≤−𝑁𝑘−𝛼𝛽𝐶¹−𝛼𝛽²𝐶_𝑝16𝛼𝛽^{𝐾(𝑡)−𝐾(𝑡)}−𝑁𝛿₀||𝜃_𝑥||²𝑑𝑥2−𝐶²||𝑢^𝑥−𝛽(𝐾(𝑡),𝑡)||²+||𝑢_𝑥(−𝐾(𝑡),𝑡)||²32^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑥||²𝑑𝑥−𝛽8^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥.(2.29)
Taking 𝑁 sufficiently large we find that there is a positive constant 𝐶₃ such that 𝑑𝑑𝑡ℒ(𝑡)≤−𝐶³^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑥||²𝑑𝑥+^{𝐾(𝑡)−𝐾(𝑡)}||𝑢_𝑡||²𝑑𝑥+𝛼𝛽^{𝐾(𝑡)−𝐾(𝑡)}||𝜃_𝑥||²𝑑𝑥.(2.30)
Observe that ℒ(𝑡) and 𝐸(𝑡) are equivalents, that is, there exists positive constant 𝐶₄ satisfying 𝑁2𝐸(𝑡;𝑢,𝜃)≤ℒ(𝑡)≤𝐶⁴𝐸(𝑡;𝑢,𝜃).(2.31)
Therefore, 𝑑𝑑𝑡ℒ(𝑡)≤−𝐶𝐶³₄𝐿(𝑡).(2.32)
Now, from equivalence (2.31) it follows that 𝐸(𝑡)≤𝐶𝐸(0)𝑒^−𝛾𝑡,(2.33) where 𝐶=2𝐶₄/𝑁 and 𝛾=𝐶₃/𝐶₄. The proof is now complete.