Persistence Property and Asymptotic Description for DGH Equation with Strong Dissipation

Abstract

The present work is mainly concerned with the Dullin-Gottwald-Holm (DGH) equation with strong dissipation. We establish a sufficient condition to guarantee global-in-time solutions, then present persistence property for the Cauchy problem, and describe the asymptotic behavior of solutions for compactly supported initial data.

1. Introduction

Dullin et al. [1] derived a new equation describing the unidirectional propagation of surface waves in a shallow water regime:𝑢_𝑡−𝛼²𝑢_𝑥𝑥𝑡+𝑐₀𝑢_𝑥+3𝑢𝑢_𝑥+𝛾𝑢_𝑥𝑥𝑥=𝛼²2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥,𝑥∈ℝ,𝑡>0, (1) where the constants 𝛼² and 𝛾/𝑐₀ are squares of length scales and the constant 𝑐₀>0 is the critical shallow water wave speed for undisturbed water at rest at spatial infinity. Since this equation is derived by Dullin, Gottwald, and Holm, in what follows, we call this new integrable shallow water equation (1) DGH equation.

If 𝛼=0, (1) becomes the well-known KdV equation, whose solutions are global as long as the initial data is square integrable. This is proved by Bourgain [2]. If 𝛾=0 and 𝛼=1, (1) reduces to the Camassa-Holm equation which was derived physically by Camassa and Holm in [3] by approximating directly the Hamiltonian for Euler's equations in the shallow water regime, where 𝑢(𝑥,𝑡) represents the free surface above a flat bottom. The properties about the well-posedness, blow-up, global existence, and propagation speed have already been studied in recent works [4–13], and the generalized version of a family of dispersive equations related to Camassa-Holm equation was discussed in [14].

It is very interesting that (1) preserves the bi-Hamiltonian structure and has the following two conserved quantities: 𝐸(𝑢)=12_ℝ𝑢²+𝛼²𝑢²_𝑥𝐹(𝑢)=1𝑑𝑥,2_ℝ𝑢³+𝛼³𝑢𝑢²_𝑥+𝑐₀𝑢²−𝛾𝑢²_𝑥𝑑𝑥. (2)

Recently, in [15], local well-posedness of strong solutions to (1) was established by applying Kato's theory [16], and some sufficient conditions were found to guarantee finite time blow-up phenomenon. Moreover, Zhou [17] found the best constants for two convolution problems on the unit circle via variational method and applied the best constants on (1) to give some blow-up criteria. Later, Zhou and Guo improved the results and got some new criteria for wave breaking [18].

In general, it is quite difficult to avoid energy dissipation mechanism in the real world. Ghidaglia [19] studied the long time behavior of solutions to the weakly dissipative KdV equation as a finite dimensional dynamic system. Moreover, some results on blow-up criteria and the global existence condition for the weakly dissipative Camassa-Holm equation are presented in [20], and very related work can be found in [21, 22]. In this work, we are interested in the following model, which can be viewed as the DGH equation with dissipation 𝑢_𝑡−𝛼²𝑢_𝑥𝑥𝑡+𝑐₀𝑢_𝑥+3𝑢𝑢_𝑥+𝛾𝑢_𝑥𝑥𝑥+𝜆1−𝛼²𝜕²_𝑥𝑢=𝛼²2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥, (3) where 𝑥∈ℝ, 𝑡>0,  𝜆(1−𝛼²𝜕²_𝑥)𝑢 is the weakly dissipative term and 𝜆 is a positive dissipation parameter. Set 𝑄=(1−𝛼²𝜕²_𝑥)^1/2, then the operator 𝑄⁻² can be expressed by 𝑄⁻²𝑓=𝐺∗𝑓=_ℝ𝐺𝑥−𝑦𝑓𝑦𝑑𝑦, (4) for all 𝑓∈𝐿²(ℝ) with 𝐺(𝑥)=(1/2𝛼)𝑒^{−|𝑥|/𝛼}. With this in hand, we can rewrite (3) as a quasilinear equation of hyperbolic type 𝑢_𝑡+𝛾𝑢−𝛼²𝑢_𝑥+𝜕_𝑥𝐺∗𝑢²+𝛼²2𝑢²_𝑥+𝑐₀+𝛾𝛼²𝑢+𝜆𝑢=0. (5) It is the dissipative term that causes the previous conserved quantities 𝐸(𝑢) and 𝐹(𝑢) to be no longer conserved for (3), and this model could also be regarded as a model of a type of a certain rate-dependent continuum material called a compressible second grade fluid [23]. Our consideration is based on this fact. Furthermore, we will show how the dissipation term affects the behavior of solutions in our forthcoming paper. As a whole, the current dissipation model is of great importance mathematically and physically, and it is worthy of being considered. In what follows, we assume that 𝑐₀+𝛾/𝛼²=0 and 𝛼>0 just for simplicity. Since 𝑢(𝑥,𝑡) is bounded by its 𝐻¹-norm, a general case with 𝑐₀+𝛾/𝛼²≠0 does not change our results essentially, but it would lead to unnecessary technical complications. So the above equation is reduced to a simpler form as follows: 𝑢_𝑡+𝑢+𝑐₀𝑢_𝑥+𝜕_𝑥𝐺∗𝐹(𝑢)+𝜆𝑢=0,𝑥∈ℝ,𝑡>0, (6) where 𝐹(𝑢)=𝑢²+𝛼²2𝑢²_𝑥. (7)

The rest of this paper is organized as follows. In Section 2, we list the local well-posedness theorem for (6) with initial datum 𝑢₀∈𝐻^𝑠,  𝑠>3/2 and collect some auxiliary results. In Section 3, we establish the condition for global existence in view of the initial potential. Persistence properties of the strong solutions are explored in Section 4. Finally, in Section 5, we give a detailed description of the corresponding solution with compactly supported initial data.

2. Preliminaries

In this section, we make some preparations for our consideration. Firstly, the local well-posedness of the Cauchy problem of (6) with initial data 𝑢₀∈𝐻^𝑠 with 𝑠>3/2 can be obtained by applying Kato's theorem [16]. More precisely, we have the following local well-posedness result.

Theorem 1. Given that 𝑢₀(𝑥)∈𝐻^𝑠, 𝑠>3/2, there exist 𝑇=𝑇(𝜆,‖𝑢₀‖_𝐻𝑠)>0 and a unique solution 𝑢 to (6), such that 𝑢=𝑢⋅,𝑢₀∈𝐶[0,𝑇);𝐻^𝑠∩𝐶¹[0,𝑇);𝐻^𝑠−1. (8) Moreover, the solution depends continuously on the initial data; that is, the mapping 𝑢₀→𝑢(⋅,𝑢₀)∶𝐻^𝑠→𝐶([0,𝑇);𝐻^𝑠)∩𝐶¹([0,𝑇);𝐻^𝑠−1) is continuous, and the maximal time of existence 𝑇>0 is independent of 𝑠.

Proof. Set 𝐴(𝑢)=(𝑢+𝑐₀)𝜕_𝑥,  𝑓(𝑢)=−𝜕_𝑥(1−𝛼²𝜕²_𝑥)⁻¹𝐹(𝑢)−𝜆𝑢,  𝑌=𝐻^𝑠,𝑋=𝐻^𝑠−1,   𝑠>3/2, and 𝑄=(1−𝛼²𝜕²_𝑥)^1/2. Applying Kato's theory for abstract quasilinear evolution equation of hyperbolic type, we can obtain the local well-posedness of (6) in 𝐻^𝑠, 𝑠>3/2 and 𝑢∈𝐶([0,𝑇);𝐻^𝑠)∩𝐶¹([0,𝑇);𝐻^𝑠−1).

The maximal value of 𝑇 in Theorem 1 is called the lifespan of the solution in general. If 𝑇<∞, that is limsup_𝑡→𝑇‖𝑢(⋅,𝑡)‖_𝐻^𝑠=∞, we say that the solution blows up in finite time, otherwise, the solution exists globally in time. Next, we show that the solution blows up if and only if its first-order derivative blows up.

Lemma 2. Given that 𝑢₀∈𝐻^𝑠,   𝑠>3/2, the solution 𝑢=𝑢(⋅,𝑢₀) of (3) blows up in finite time 𝑇<+∞ if and only if liminf_𝑡→𝑇inf_𝑥∈ℝ𝑢_𝑥(𝑥,𝑡)=−∞. (9)

Proof. We first assume that 𝑢₀∈𝐻^𝑠 for some 𝑠∈ℕ,   𝑠≥4. Equation (6) can be written into the following form in terms of 𝑦=(1−𝛼²𝜕²_𝑥)𝑢𝑦_𝑡+𝑦𝑢_𝑥+12𝑢²−𝛼²𝑢²_𝑥𝑥+𝑐₀𝑦_𝑥+𝜆𝑦=0,𝑥∈ℝ,𝑡>0. (10) Multiplying (10) by 𝑦=(1−𝛼²𝜕²_𝑥)𝑢, and integrating by parts, we have 𝑑𝑑𝑡_ℝ𝑦²𝑑𝑥=2_ℝ𝑦𝑦_𝑡𝑑𝑥=−3_ℝ𝑢_𝑥𝑦²𝑑𝑥−2𝜆_ℝ𝑦²𝑑𝑥. (11) Differentiating (10) with respect to the spatial variable 𝑥, then multiplying by 𝑦_𝑥=(1−𝛼²𝜕²_𝑥)𝑢_𝑥, and integrating by parts again, we obtain 𝑑𝑑𝑡_ℝ𝑦²_𝑥𝑑𝑥=2_ℝ𝑦_𝑥𝑦_𝑥𝑡𝑑𝑥=−5_ℝ𝑢_𝑥𝑦²_𝑥+2𝑑𝑥𝛼²_ℝ𝑢_𝑥𝑦²𝑑𝑥−2𝜆_ℝ𝑦²_𝑥𝑑𝑥. (12) Summarizing (11) and (12), we obtain 𝑑𝑑𝑡_ℝ𝑦²+𝑦²_𝑥2𝑑𝑥=−3−𝛼²_ℝ𝑢_𝑥𝑦²𝑑𝑥−5_ℝ𝑢_𝑥𝑦²_𝑥𝑑𝑥−2𝜆_ℝ𝑦²+𝑦²_𝑥𝑑𝑥. (13) If 𝑢_𝑥 is bounded from below on [0,𝑇), for example, 𝑢_𝑥≥−𝐶, 𝐶 is a positive constant, then we get by (13) and Gronwall's inequality the following: 𝑦²_𝐻1≤exp{(𝐾𝐶−2𝜆)𝑡}𝑦₀²_𝐻1, (14) where 𝐾=max{5,(3−2/𝛼²)}. Therefore, the 𝐻³-norm of the solution to (10) does not blow up in finite time. Furthermore, similar argument shows that the 𝐻^𝑘-norm with 𝑘≥4 does not blow up either in finite time. Consequently, this theorem can be proved by Theorem 1 and simple density argument for all 𝑠>3/2.

Lemma 3. Let 𝑢₀∈𝐻¹_𝛼, then as long as the solution 𝑢(𝑥,𝑡) given by Theorem 1 exists, for any 𝑡∈[0,𝑇), one has ‖𝑢‖²_𝐻1𝛼=exp(−2𝜆𝑡)𝑢₀²_𝐻1𝛼, (15) where the norm is defined as ‖𝑢‖²_𝐻1𝛼=_ℝ𝑢²+𝛼²𝑢²_𝑥𝑑𝑥. (16)

Proof. Multiplying both sides of (10) by 𝑢 and integrating by parts on ℝ, we get _ℝ𝑢𝑦_𝑡𝑑𝑥+_ℝ𝑦𝑢_𝑥𝑢𝑑𝑥+_ℝ12𝑢²−𝛼²𝑢²_𝑥𝑥+𝑢𝑑𝑥_ℝ𝑐₀𝑦_𝑥𝑢𝑑𝑥+_ℝ𝜆𝑦𝑢𝑑𝑥=0. (17) Note that _ℝ𝑦𝑢_𝑥𝑢𝑑𝑥+_ℝ12𝑢²−𝛼²𝑢²_𝑥𝑥𝑢𝑑𝑥=0,_ℝ𝑐₀𝑦_𝑥𝑢𝑑𝑥=0. (18) Then, we have _ℝ𝑢𝑢_𝑡−𝛼²𝑢_𝑥𝑥𝑡𝑑𝑥+_ℝ𝜆𝑢²−𝛼²𝑢𝑢_𝑥𝑥𝑑𝑥=0. (19) Hence, _ℝ𝑢𝑢_𝑡𝑑𝑥−𝛼²_ℝ𝑢𝑢_𝑥𝑥𝑡𝑑𝑥+𝜆_ℝ𝑢²𝑑𝑥−𝜆𝛼²_ℝ𝑢𝑢_𝑥𝑥𝑑𝑥=0. (20) Thus, we easily get _ℝ𝑢𝑢_𝑡+𝛼²𝑢_𝑥𝑢_𝑥𝑡𝑑𝑥+𝜆_ℝ𝑢²+𝛼²𝑢²_𝑥𝑑𝑥=0, (21) and, therefore, 𝑑‖𝑢‖𝑑𝑡^2𝐻1𝛼+2𝜆‖𝑢‖²_𝐻1𝛼=0. (22) By integration from 0 to 𝑡, we get ‖𝑢‖²_𝐻1𝛼=exp(−2𝜆𝑡)𝑢₀²_𝐻1𝛼,forany𝑡∈[0,T). (23) Hence, the lemma is proved.

We also need to introduce the standard particle trajectory method for later use. Consider now the following initial value problem as follows: 𝑞_𝑡=𝑢𝑡,𝑞+𝑐₀[),𝑞(),𝑡∈0,𝑇0,𝑥=𝑥,𝑥∈ℝ, (24) where 𝑢∈𝐶¹([0,𝑇),𝐻^𝑠−1) is the solution to (6) with initial data 𝑢₀∈𝐻^𝑠, (𝑠>3/2) and 𝑇>0 is the maximal time of existence. By direct computation, we have 𝑞_𝑡𝑥(𝑡,𝑥)=𝑢_𝑥𝑡,𝑞(𝑡,𝑥)𝑞_𝑥(𝑡,𝑥). (25) Then, 𝑞_𝑥(𝑡,𝑥)=exp^𝑡₀𝑢_𝑥𝜏,𝑞(𝜏,𝑥)𝑑𝜏>0,𝑡>0,𝑥∈ℝ, (26) which means that 𝑞(𝑡,⋅)∶ℝ→ℝ is a diffeomorphism of the line for every 𝑡∈[0,𝑇). Consequently, the 𝐿^∞-norm of any function 𝑣(𝑡,⋅) is preserved under the family of the diffeomorphism 𝑞(𝑡,⋅), that is, ‖𝑣(𝑡,⋅)‖_𝐿^∞=𝑣𝑡,𝑞(𝑡,⋅)_𝐿∞,𝑡∈[0,𝑇). (27) Similarly, inf_𝑥∈ℝ𝑣(𝑡,𝑥)=inf_𝑥∈ℝ𝑣𝑡,𝑞(𝑡,𝑥),𝑡∈[0,𝑇),sup_𝑥∈ℝ𝑣(𝑡,𝑥)=sup_𝑥∈ℝ𝑣𝑡,𝑞(𝑡,𝑥),𝑡∈[0,𝑇). (28) Moreover, one can verify the following important identity for the strong solution in its lifespan: 𝑑𝑦𝑞()𝑞𝑑𝑡𝑥,𝑡,𝑡²_𝑥(𝑥,𝑡)=−𝜆𝑦𝑞(𝑥,𝑡),𝑡𝑞²_𝑥(𝑥,𝑡). (29) We get that 𝑦𝑞(𝑥,𝑡),𝑡𝑞²_𝑥(𝑥,𝑡)=𝑦₀(𝑥)exp(−𝜆𝑡), (30) where 𝑦(𝑥,𝑡) is defined by 𝑦(𝑥,𝑡)=(1−𝛼²𝜕²_𝑥)𝑢(𝑥,𝑡), for 𝑡≥0 in its lifespan.

From the expression of 𝑢(𝑥,𝑡) in terms of 𝑦(𝑥,𝑡), for all 𝑡∈[0,𝑇), 𝑥∈ℝ, we can rewrite 𝑢(𝑥,𝑡) and 𝑢_𝑥(𝑥,𝑡) as follows: 𝑢()=1𝑥,𝑡𝑒2𝛼^{−𝑥/𝛼𝑥}_−∞𝑒^𝜉/𝛼𝑦()+1𝜉,𝑡𝑑𝜉𝑒2𝛼^𝑥/𝛼∞_𝑥𝑒^−𝜉/𝛼𝑦(𝜉,𝑡)𝑑𝜉, (31) from which we get that 𝑢_𝑥()1𝑥,𝑡=−2𝛼²𝑒^{−𝑥/𝛼𝑥}_−∞𝑒^𝜉/𝛼𝑦()+1𝜉,𝑡𝑑𝜉2𝛼²𝑒^𝑥/𝛼∞_𝑥𝑒^−𝜉/𝛼𝑦(𝜉,𝑡)𝑑𝜉. (32)

3. Global Existence

It is shown that it is the sign of initial potential not the size of it that can guarantee the global existence of strong solutions.

Theorem 4. Assume that 𝑢₀∈𝐻^𝑠,  𝑠>3/2, and  𝑦₀=𝑢₀−𝛼²𝑢_0𝑥𝑥 satisfies 𝑦₀(𝑥)≤0,𝑥∈−∞,𝑥₀,𝑦₀(𝑥)≥0,𝑥∈𝑥₀,∞, (33) for some point 𝑥₀∈ℝ. Then, the solution 𝑢(𝑥,𝑡) to (6) exists globally in time.

Proof. From the hypothesis and (30), we obtain that 𝑦(𝑥,𝑡)≥0, 𝑞(𝑥₀,𝑡)≤𝑥<∞; 𝑦(𝑥,𝑡)≤0, −∞<𝑥≤𝑞(𝑥₀,𝑡). According to (31) and (32), one can get that when 𝑥>𝑥₀, 𝑢𝑞(𝑥,𝑡),𝑡+𝛼𝑢_𝑥𝑞()=1𝑥,𝑡,𝑡𝛼𝑒^{𝑞(𝑥,𝑡)/𝛼∞}_{𝑞()𝑥,𝑡}𝑒^−𝜉/𝛼𝑦(𝜉,𝑡)𝑑𝜉≥0, (34) it follows that −𝛼𝑢_𝑥𝑞(𝑥,𝑡),𝑡≤𝑢𝑞(𝑥,𝑡),𝑡≤‖𝑢‖_𝐿^∞≤exp(−𝜆𝑡)√𝑢2𝛼^0𝐻1𝛼≤1√𝑢2𝛼^0𝐻1𝛼, (35) that is, 𝑢_𝑥(𝑥,𝑡) is bounded below. Similarly, when 𝑥<𝑥₀, 𝑢𝑞(𝑥,𝑡),𝑡−𝛼𝑢_𝑥𝑞()=1𝑥,𝑡,𝑡𝛼𝑒^{−𝑞(𝑥,𝑡)/𝛼𝑞(𝑥,𝑡)}_−∞𝑒^𝜉/𝛼𝑦(𝜉,𝑡)𝑑𝜉≤0, (36) so −𝛼𝑢_𝑥(𝑞(𝑥,𝑡),𝑡)≤−𝑢(𝑞(𝑥,𝑡),𝑡). We also get the bounded below result as above. Therefore, the theorem is proved by Lemma 2.

Corollary 5. Assume that 𝑢₀∈𝐻^𝑠,  𝑠>3/2, and  𝑦₀=𝑢₀−𝛼²𝑢_0𝑥𝑥 is of one sign, then the corresponding solution 𝑢(𝑥,𝑡) to (6) exists globally.

In fact, if 𝑥₀ is regarded as ±∞, we prove this corollary immediately from Theorem 4.

4. Persistence Properties

In this section, we will investigate the following property for the strong solutions to (6) in 𝐿^∞-space which behave algebraically at infinity as their initial profiles do. The main idea comes from the recent work of Himonas and his collaborators [7].

Theorem 6. Assume that for some 𝑇>0 and 𝑠>3/2,   𝑢∈𝐶([0,𝑇];𝐻^𝑠) is a strong solution of the initial value problem associated to (6), and that 𝑢₀(𝑥)=𝑢(𝑥,0) satisfies 𝑢₀(𝑥),𝑢_0𝑥(𝑥)∼𝑂𝑥^−𝜃/𝛼𝑥↑∞, (37) for some 𝜃∈(0,1) and 𝛼≥1. Then, |𝑢(𝑥,𝑡)|,𝑢_𝑥(𝑥,𝑡)∼𝑂𝑥^−𝜃/𝛼𝑥↑∞, (38) uniformly in the time interval [0,𝑇].

Proof. The proof is organized as follows. Firstly, we will estimate ‖𝑢(𝑥,𝑡)‖_𝐿^∞ and ‖𝑢_𝑥(𝑥,𝑡)‖_𝐿∞. Then, we apply the weight function to obtain the desired result. In the following proof, we denote some constants by 𝑐; they may be different from instance to instance, changing even within the same line.
Multiplying (6) by 𝑢^2𝑛−1 with 𝑛∈ℤ⁺, then integrating both sides with respect to 𝑥 variable, we can get _ℝ𝑢^2𝑛−1𝑢_𝑡𝑑𝑥+_ℝ𝑢^2𝑛−1𝑢+𝑐₀𝑢_𝑥+𝑑𝑥_ℝ𝑢^2𝑛−1𝜕_𝑥𝐺∗𝐹(𝑢)𝑑𝑥=−𝜆_ℝ𝑢^2𝑛𝑑𝑥. (39) The first term of the above identity is _ℝ𝑢^2𝑛−1𝑢_𝑡1𝑑𝑥=𝑑2𝑛‖𝑢(𝑡𝑑𝑡)‖_𝐿^2𝑛_2𝑛=‖𝑢(𝑡)‖_𝐿^2𝑛−1_2𝑛𝑑‖𝑢(𝑡𝑑𝑡)‖_𝐿^2𝑛, (40) and the estimates of the second term is _ℝ𝑢^2𝑛−1𝑢𝑢_𝑥𝑑𝑥≤𝑢_𝑥(𝑡)_𝐿∞‖𝑢(𝑡)‖_𝐿^2𝑛_2𝑛,𝑐_0ℝ𝑢^2𝑛−1𝑢_𝑥𝑑𝑥=𝑐_0ℝ𝑢^2𝑛2𝑛_𝑥𝑑𝑥=0. (41) In view of Hölder's inequality, we can obtain the following estimate for the third term in (39) _ℝ𝑢^2𝑛−1𝜕_𝑥𝐺∗𝐹(𝑢)𝑑𝑥≤‖𝑢(𝑡)‖_𝐿^2𝑛−1_2𝑛𝜕_𝑥𝐺∗𝐹(𝑢)_𝐿2𝑛. (42) For the last term _ℝ𝑢^2𝑛−1𝜆𝑢𝑑𝑥≤𝜆‖𝑢(𝑡)‖_𝐿^2𝑛_2𝑛, (43) putting all the inequalities above into (39) yields 𝑑‖𝑢(𝑡𝑑𝑡)‖_𝐿^2𝑛≤𝑢_𝑥(𝑡)_𝐿2𝑛+𝜆‖𝑢(𝑡)‖_𝐿^2𝑛+𝜕_𝑥𝐺∗𝐹(𝑢)_𝐿2𝑛. (44) Using the Sobolev embedding theorem, there exists a constant 𝑀=sup_{𝑡∈[0,𝑇]}‖𝑢(𝑥,𝑡)‖_𝐻^𝑠, (45) such that we have by applying Gronwall's inequality ‖𝑢(𝑡)‖_𝐿^2𝑛≤𝑐𝑒^𝑀𝑡‖𝑢(0)‖_𝐿^2𝑛+^𝑡₀𝜕_𝑥𝐺∗𝐹(𝑢)_𝐿2𝑛𝑑𝜏. (46) For any 𝑓∈𝐿¹(ℝ)∩𝐿^∞(ℝ), we know that lim_𝑞↑∞𝑓_𝐿𝑞=𝑓_𝐿∞. (47) Taking the limits in (46) (notice that 𝐺∈𝐿¹ and 𝐹(𝑢)∈𝐿¹∩𝐿^∞) from (47), we get ‖𝑢(𝑡)‖_𝐿^∞≤𝑐𝑒^𝑀𝑡‖𝑢(0)‖_𝐿^∞+^𝑡₀𝜕_𝑥𝐺∗𝐹(𝑢)_𝐿∞𝑑𝜏. (48) Then, differentiating (6) with respect to variable 𝑥 produces the following equation: 𝑢_𝑥𝑡+𝑢𝑢_𝑥𝑥+𝑐₀𝑢_𝑥𝑥+𝑢²_𝑥+𝜕²_𝑥𝐺∗𝐹(𝑢)+𝜆𝑢_𝑥=0. (49) Again, multiplying (49) by 𝑢_𝑥^2𝑛−1 with 𝑛∈ℤ⁺, integrating the result in 𝑥 variable, and considering the second term and the third term in the above identity with integration by parts, one gets _ℝ𝑢𝑢_𝑥𝑥𝑢_𝑥^2𝑛−1𝑑𝑥=_ℝ𝑢𝑢_𝑥^2𝑛−12𝑛_𝑥1𝑑𝑥=−2𝑛_ℝ𝑢_𝑥𝑢_𝑥^2𝑛𝑐𝑑𝑥,_0ℝ𝑢_𝑥𝑥𝑢_𝑥^2𝑛−1𝑑𝑥=𝑐_0ℝ𝑢_𝑥^2𝑛−12𝑛_𝑥𝑑𝑥=0, (50) so, we have _ℝ𝑢_𝑥𝑡𝑢_𝑥^2𝑛−11𝑑𝑥−2𝑛_ℝ𝑢_𝑥𝑢_𝑥^2𝑛𝑑𝑥+_ℝ𝑢_𝑥^2𝑛+1𝑑𝑥=−_ℝ𝑢_𝑥^2𝑛−1𝜕²_𝑥𝐺∗𝐹(𝑢)𝑑𝑥−𝜆_ℝ𝑢_𝑥^2𝑛−1𝑢_𝑥𝑑𝑥. (51) Similarly, the following inequality holds 𝑑𝑢𝑑𝑡^𝑥(𝑡)_𝐿2𝑛≤2𝑢_𝑥(𝑡)_𝐿∞+𝜆𝑢_𝑥(𝑡)_𝐿2𝑛+𝜕²_𝑥𝐺∗𝐹(𝑢)(𝑡)_𝐿2𝑛, (52) and therefore as before, we obtain 𝑢_𝑥(𝑡)_𝐿2𝑛≤𝑐𝑒^2𝑀𝑡𝑢_𝑥(0)_𝐿2𝑛+^𝑡₀𝜕²_𝑥𝐺∗𝐹(𝑢)_𝐿2𝑛𝑑𝜏. (53) Taking the limits in(53), we obtain𝑢_𝑥(𝑡)_𝐿∞≤𝑐𝑒^2𝑀𝑡𝑢_𝑥(0)_𝐿∞+^𝑡₀𝜕²_𝑥𝐺∗𝐹(𝑢)_𝐿∞𝑑𝜏. (54)
Next, we will introduce the weight function to get our desired result. This function 𝜑_𝑁(𝑥) with 𝑁∈ℤ⁺ is independent of 𝑡 as the following: 𝜑_𝑁(𝑥)=𝑥1,𝑥≤1,_𝜃/𝛼(),𝑁,𝑥∈1,𝑁_𝜃/𝛼,𝑥≥𝑁. (55) From (6) and (49), we get the following two equations: 𝜑_𝑁𝑢_𝑡+𝜑_𝑁𝑢𝑢_𝑥+𝜑_𝑁𝑐₀𝑢_𝑥+𝜑_𝑁𝜕_𝑥𝐺∗𝐹(𝑢)+𝜆𝜑_𝑁𝜑𝑢=0,_𝑁𝑢_𝑥𝑡+𝜑_𝑁𝑢𝑢_𝑥𝑥+𝜑_𝑁𝑐₀𝑢_𝑥𝑥+𝜑_𝑁𝑢²_𝑥+𝜑_𝑁𝜕²_𝑥𝐺∗𝐹(𝑢)+𝜆𝜑_𝑁𝑢_𝑥=0. (56) We need some tricks to deal with the following term as in [18]: _ℝ𝜑_𝑁^2𝑛−1𝑢^2𝑛−1𝜑_𝑁𝑢_𝑥=𝑑𝑥_ℝ𝜑_𝑁𝑢^2𝑛−1𝑢𝜑_𝑁𝑥−𝑢𝜑_𝑁𝑥=𝑑𝑥_ℝ𝜑_𝑁𝑢^2𝑛−1𝑑𝜑_𝑁𝑢−_ℝ𝜑_𝑁𝑢^2𝑛−1𝑢𝜑_𝑁𝑥≤𝑑𝑥_ℝ𝜑_𝑁𝑢^2𝑛𝑑𝑥, (57) where we have used the fact 0≤𝜑^′_𝑁(𝑥)≤𝜑_𝑁(𝑥), a.e. 𝑥∈ℝ. Similar technique is used for the term ∫_ℝ(𝜑_𝑁)^2𝑛−1𝑢_𝑥^2𝑛−1𝜑_𝑁𝑢_𝑥𝑥𝑑𝑥. Hence, as in the weightless case, we get the following inequality in view of (48) and (54) as follows: 𝑢(𝑡)𝜑_𝑁𝐿∞+𝑢_𝑥(𝑡)𝜑_𝑁𝐿∞≤𝑐𝑒^2𝑀𝑡𝑢(0)𝜑_𝑁𝐿∞+𝑢_𝑥(0)𝜑_𝑁𝐿∞+𝑐𝑒^2𝑀𝑡×^𝑡₀𝜑_𝑁𝜕_𝑥𝐺∗𝐹(𝑢)_𝐿∞+𝜑_𝑁𝜕²_𝑥𝐺∗𝐹(𝑢)_𝐿∞𝑑𝜏. (58) On the other hand, a simple calculation shows that there exists 𝐶>0, depending only on 𝛼 and 𝜃 such that for any 𝑁∈ℤ⁺, 𝜑_𝑁(𝑥)_ℝ𝑒^{−|𝑥−𝑦|/𝛼}1𝜑_𝑁𝑦𝑑𝑦≤𝐶. (59) Therefore, for any appropriate function 𝑔, one obtains that 𝜑_𝑁𝜕_𝑥𝐺∗𝑔²(𝑥=1)𝜑2𝛼^𝑁(𝑥)_ℝ𝑒^{−|𝑥−𝑦|/𝛼}𝑔²𝑦≤1𝑑𝑦𝜑2𝛼^𝑁(𝑥)_ℝ𝑒^{−|𝑥−𝑦|/𝛼}1𝜑_𝑁𝑦𝜑_𝑁𝑦𝑔𝑦𝑔𝑦≤1𝑑𝑦𝜑2𝛼^𝑁(𝑥)_ℝ𝑒^{−|𝑥−𝑦|/𝛼}1𝜑_𝑁𝑦𝑑𝑦𝑔𝜑_𝑁𝐿∞𝑔_𝐿∞≤𝐶𝛼𝑔𝜑_𝑁𝐿∞𝑔_𝐿∞, (60) and similarly, |𝜑_𝑁𝜕²_𝑥𝐺∗𝑔²(𝑥)|≤(𝐶/𝛼)‖𝑔𝜑_𝑁‖_𝐿∞‖𝑔‖_𝐿∞. Using the same method, we can estimate the following two terms 𝜑_𝑁(𝑥≤𝐶𝐺∗𝑔)𝛼𝑔𝜑_𝑁𝐿∞,𝜑_𝑁𝜕_𝑥(𝑥≤𝐶𝐺∗𝑔)𝛼𝑔𝜑_𝑁𝐿∞. (61) Therefore, it follows that there exists a constant 𝐶₁(𝑀,𝑇,𝛼,𝜆)>0 such that 𝑢(𝑡)𝜑_𝑁𝐿∞+𝑢_𝑥(𝑡)𝜑_𝑁𝐿∞≤𝐶₁𝑢(0)𝜑_𝑁𝐿∞+𝑢_𝑥(0)𝜑_𝑁𝐿∞+𝐶₁^𝑡₀‖𝑢(𝜏)‖_𝐿^∞+𝑢_𝑥(𝜏)_𝐿∞⋅𝜑_𝑁𝑢(𝜏)_𝐿∞+𝜑_𝑁𝑢_𝑥(𝜏)_𝐿∞𝑑𝜏≤𝐶₁𝑢(0)𝜑_𝑁𝐿∞+𝑢_𝑥(0)𝜑_𝑁𝐿∞+^𝑡₀𝜑_𝑁𝑢(𝜏)_𝐿∞+𝜑_𝑁𝑢_𝑥(𝜏)_𝐿∞𝑑𝜏. (62) Hence, the following inequality is obtained for any 𝑁∈ℤ⁺ and any 𝑡∈[0,𝑇]: 𝑢(𝑡)𝜑_𝑁𝐿∞+𝑢_𝑥(𝑡)𝜑_𝑁𝐿∞≤𝐶₁𝑢(0)𝜑_𝑁𝐿∞+𝑢_𝑥(0)𝜑_𝑁𝐿∞≤𝐶₁𝑢(0)max1,𝑥^𝜃/𝛼_𝐿∞+𝑢_𝑥(0)max1,𝑥^𝜃/𝛼_𝐿∞. (63) Finally, taking the limit as 𝑁 goes to infinity in the above inequality, we can find that for any 𝑡∈[0,𝑇], 𝑢(𝑥,𝑡)𝑥^𝜃/𝛼+𝑢_𝑥(𝑥,𝑡)𝑥^𝜃/𝛼≤𝐶₁𝑢(0)max1,𝑥^𝜃/𝛼_𝐿∞+𝑢_𝑥(0)max1,𝑥^𝜃/𝛼_𝐿∞, (64) which completes the proof of Theorem 6.

5. Asymptotic Description

The following result is to give a detailed description on the corresponding strong solution 𝑢(𝑥,𝑡) in its lifespan with 𝑢₀(𝑥) being compactly supported.

Theorem 7. Assume that the initial datum 0≢𝑢₀(𝑥)∈𝐻^𝑠 with 𝑠>5/2 is compactly supported in [𝑎,𝑐], then the corresponding solution 𝑢(𝑥,𝑡)∈𝐶([0,𝑇);𝐻^𝑠) to (6) has the following property: for any 𝑡∈(0,𝑇), 𝑢(𝑥,𝑡)=𝐿(𝑡)𝑒^−𝑥/𝛼(),𝑢()(𝑡)𝑒as𝑥>𝑞𝑐,𝑡𝑥,𝑡=𝑙^𝑥/𝛼as𝑥<𝑞(𝑎,𝑡), (65) where 𝑞(𝑥,𝑡) is defined by (24) and 𝑇 is its lifespan. Furthermore, 𝐿(𝑡) and 𝑙(𝑡) denote continuous nonvanishing functions, with 𝐿(𝑡)>0 and 𝑙(𝑡)<0 for 𝑡∈(0,𝑇). Moreover, 𝐿(𝑡) is a strictly increasing function, while 𝑙(𝑡) is strictly decreasing.

Remark 8. This is an interesting phenomenon for our model; it implies that the strong solution does not have compact 𝑥-support for any 𝑡>0 in its lifespan anymore, although the corresponding 𝑢₀(𝑥) is compactly supported. No matter that the initial profile 𝑢₀(𝑥) is (no matter it is positive or negative), for any 𝑡>0 in its lifespan, the nontrivial solution 𝑢(𝑥;𝑡) is always positive at infinity and negative at negative infinity. Moreover, we found that the dissipative coefficient does not affect this behavior.

Proof. First, since 𝑢₀(𝑥) has a compact support, so does 𝑦₀(𝑥)=(1−𝛼²𝜕²_𝑥)𝑢₀(𝑥). Equation (30) tells us that 𝑦=(1−𝛼²𝜕²_𝑥)𝑢(𝑥,𝑡)=((1−𝛼²𝜕²_𝑥)𝑢₀(𝑞⁻¹(𝑥,𝑡))exp(−𝜆𝑡))/(𝜕_𝑥𝑞⁻¹((𝑥,𝑡),𝑡))² is compactly supported in [𝑞(𝑎,𝑡),𝑞(𝑐,𝑡)] in its lifespan. Hence, the following functions are well defined 𝐸(𝑡)=_ℝ𝑒^𝜉/𝛼𝑦(𝜉,𝑡)𝑑𝜉,𝐹(𝑡)=_ℝ𝑒^−𝜉/𝛼𝑦(𝜉,𝑡)𝑑𝜉, (66) with 𝐸(0)=_ℝ𝑒^𝜉/𝛼𝑦₀(𝜉)=𝑑𝜉_ℝ𝑒^𝜉/𝛼𝑢₀(𝜉)𝑑𝜉−𝛼²_ℝ𝑒^𝜉/𝛼𝑢_0𝑥𝑥(𝜉)𝑑𝜉=0. (67) And 𝐹(0)=0 by integration by parts.
Then, for 𝑥>𝑞(𝑐,𝑡), we have 𝑢()=1𝑥,𝑡𝑒2𝛼^{−|𝑥|/𝛼}()=1∗𝑦𝑥,𝑡𝑒2𝛼^−𝑥/𝛼_𝑞()^{𝑞(𝑐,𝑡)}_𝑎,𝑡𝑒^𝜉/𝛼𝑦()1𝜉,𝑡𝑑𝜉=𝑒2𝛼^−𝑥/𝛼𝐸(𝑡), (68) where (66) is used.
Similarly, when 𝑥<𝑞(𝑎,𝑡), we get 𝑢()=1𝑥,𝑡𝑒2𝛼^{−|𝑥|/𝛼}()=1∗𝑦𝑥,𝑡𝑒2𝛼^𝑥/𝛼_𝑞()^{𝑞(𝑐,𝑡)}_𝑎,𝑡𝑒^−𝜉/𝛼𝑦()1𝜉,𝑡𝑑𝜉=𝑒2𝛼^𝑥/𝛼𝐹(𝑡). (69) Because 𝑦(𝑥,𝑡) has a compact support in 𝑥 in the interval [𝑞(𝑎,𝑡),𝑞(𝑐,𝑡)] for any 𝑡∈[0,𝑇], we get 𝑦(𝑥,𝑡)=𝑢(𝑥,𝑡)−𝛼²𝑢_𝑥𝑥(𝑥,𝑡)=0, for 𝑥>𝑞(𝑐,𝑡) or 𝑥<𝑞(𝑎,𝑡). Hence, as consequences of (68) and (69), we have 𝑢(𝑥,𝑡)=−𝛼𝑢_𝑥(𝑥,𝑡)=𝛼²𝑢_𝑥𝑥()=1𝑥,𝑡𝑒2𝛼^−𝑥/𝛼𝐸(𝑡)(),𝑢(),as𝑥>𝑞𝑐,𝑡𝑥,𝑡=𝛼𝑢_𝑥(𝑥,𝑡)=𝛼²𝑢_𝑥𝑥()=1𝑥,𝑡𝑒2𝛼^𝑥/𝛼𝐹(𝑡),as𝑥<𝑞(𝑎,𝑡). (70) On the other hand, 𝑑𝐸(𝑡)=𝑑𝑡_ℝ𝑒^𝜉/𝛼𝑦_𝑡(𝜉,𝑡)𝑑𝑥. (71)
Substituting the identity (10) into 𝑑𝐸(𝑡)/𝑑𝑡, we obtain 𝑑𝐸(𝑡)𝑑𝑡=−_ℝ𝑒^𝜉/𝛼𝑦𝑢_𝑥+12𝑢²−𝛼²𝑢²_𝑥𝑥+𝑐₀𝑦_𝑥=1+𝜆𝑦𝑑𝜉𝛼_ℝ𝑒^𝜉/𝛼1𝑦𝑢𝑑𝜉+2𝛼_ℝ𝑒^𝜉/𝛼𝑢²−𝛼²𝑢²_𝑥+𝑐𝑑𝜉₀𝛼_ℝ𝑒^𝜉/𝛼𝑦𝑑𝜉+𝛼²_ℝ𝑒^𝜉/𝛼𝜆𝑢_𝑥𝑥𝑑𝜉−_ℝ𝑒^𝜉/𝛼3𝜆𝑢𝑑𝜉=2𝛼_ℝ𝑒^𝜉/𝛼𝑢²𝛼𝑑𝜉+2_ℝ𝑒^𝜉/𝛼𝑢²_𝑥+𝑑𝜉_ℝ𝑒^𝜉/𝛼𝑢𝑢_𝑥𝑑𝜉=_ℝ𝑒^𝜉/𝛼1𝛼𝑢²+𝛼2𝑢²_𝑥𝑑𝜉>0, (72) where we used (70). Therefore, in the lifespan of the solution, we have that 𝐸(𝑡) is an increasing function with 𝐸(0)=0; thus, it follows that 𝐸(𝑡)>0 for 𝑡∈(0,𝑇]; that is, 𝐸(𝑡)=^𝑡_0ℝ𝑒^𝜉/𝛼1𝛼𝑢²+𝛼2𝑢²_𝑥(𝜉,𝜏)𝑑𝜉𝑑𝜏>0. (73) By similar argument, one can verify that the following identity for 𝐹(𝑡) is true: 𝐹(𝑡)=−^𝑡_0ℝ𝑒^−𝜉/𝛼1𝛼𝑢²+𝛼2𝑢²_𝑥(𝜉,𝜏)𝑑𝜉𝑑𝜏<0. (74) In order to finish the proof, it is sufficient to let 𝐿(𝑡)=(1/2𝛼)𝐸(𝑡), and to let 𝑙(𝑡)=(1/2𝛼)𝐹(𝑡), respectively.