The Local Strong and Weak Solutions for a Generalized Pseudoparabolic Equation

Abstract

The Cauchy problem for a nonlinear generalized pseudoparabolic equation is investigated. The well-posedness of local strong solutions for the problem is established in the Sobolev space 𝐶([0,𝑇);𝐻^𝑠(𝑅))⋂𝐶¹([0,𝑇);𝐻^𝑠−1(𝑅)) with 𝑠>3/2, while the existence of local weak solutions is proved in the space 𝐻^𝑠(𝑅) with 1≤𝑠≤3/2. Further, under certain assumptions of the nonlinear terms in the equation, it is shown that there exists a unique global strong solution to the problem in the space 𝐶([0,∞);𝐻^𝑠(𝑅))⋂𝐶¹([0,∞);𝐻^𝑠−1(𝑅)) with 𝑠≥2.

1. Introduction

Davis [1] investigated the pseudoparabolic equation𝑢_𝑡()=𝜕𝑡,𝑥𝜑𝑢𝜕𝑥^𝑥+𝛼𝑢_𝑡𝑥𝑥, (1.1) where the constant 𝛼≥0, the function 𝜑∈𝐶²(−∞,∞), 𝜑(0)=0 and 𝜑′(𝜉)>0, and the subscripts 𝑥 and 𝑡 indicate partial derivatives. Equation (1.1) arises from the study of shearing flows of incompressible simple fluids. The quantity 𝜑(𝑢_𝑥)+𝛼𝑢_𝑡𝑥 is viewed as an approximation to the stress functional during such a flow. Much attention has been given to this approximation when the function 𝜑 is linear (see [2, 3]). The existence and uniqueness of the global weak solution of the initial value problem for (1.1) were established in [1].

Recently, Chen and Xue [4] investigated the Cauchy problem for the nonlinear generalized pseudoparabolic equation𝑢_𝑡−𝛼𝑢_𝑡𝑥𝑥−𝜆𝑢_𝑥𝑥+𝛾𝑢_𝑥+𝑓(𝑢)_𝑥=𝜕𝜑𝑢𝜕𝑥^𝑥+𝑔(𝑢)−𝛼𝑔(𝑢)_𝑥𝑥,𝑥∈𝑅,𝑡>0, (1.2) where 𝑢(𝑡,𝑥) is an unknown function, 𝛼>0, 𝜆≥0, 𝛾 is a real number, 𝑓(𝑠), 𝜑(𝑠), and 𝑔(𝑠) denote given nonlinear functions. The well-posedness of global strong solution in a Sobolev space, the global classical solution and its asymptotic behavior are studied in [4] in which several key assumptions are imposed on the functions 𝜑(𝑠) and 𝑔(𝑠). In fact, various dynamic properties for many special cases of (1.2) have been established in [5–7]. For example, when 𝜑(𝑠)=𝑔(𝑠)=0, (1.2) becomes the generalized regularized long wave Burger equation.

Motivated by the works in [1, 4], we study the problem𝑢_𝑡−𝛼𝑢_𝑡𝑥𝑥=𝜕𝜑𝑢𝜕𝑥^𝑥+𝛽𝑢^2𝑚𝑢_𝑥𝑥𝑢(),𝑥∈𝑅,𝑡>0,0,𝑥=𝑢₀(𝑥),𝑥∈𝑅, (1.3) where 𝛼>0 and 𝛽≥0, 𝑚 is a nature number, 𝜑(𝑠) is a given function, and 𝑢₀(𝑥) is a given initial value function. Here we should address that (1.2) does not include the first equation of problem (1.3) due to the term 𝛽𝑢^2𝑚𝑢_𝑥𝑥. Letting 𝛽=0, the first equation of problem (1.3) reduces to (1.1).

The objectives of this work are threefold. The first objective is to establish the local well-posedness of system (1.3) in the space 𝐶([0,𝑇);𝐻^𝑠(𝑅))⋂𝐶¹([0,𝑇);𝐻^𝑠−1(𝑅)) with 𝑠>3/2. We should address that the Sobolev index 𝑠≥2 is required to guarantee the local well-posedness of (1.1) and (1.2) in the works of Davis [1] and Chen and Xue [4]. The second aim is to study the existence of local weak solutions for system (1.3). The third aim is to discuss the well-posedness of the global strong solution for problem (1.3). Under the assumptions of the function 𝜑(𝑠) and the initial value 𝑢₀(𝑥) similar to those presented in [1, 4], problem (1.3) is shown to have a unique global solution in the space 𝐶([0,∞);𝐻^𝑠(𝑅))⋂𝐶¹([0,∞);𝐻^𝑠−1(𝑅)).

The organization of this paper is as follows. The well-posedness of local strong solutions for problem (1.3) is investigated in Section 2, and the existence of local weak solutions is established in Section 3. Section 4 deals with the well-posedness of the global strong solution.

2. Local Well-Posedness

Let 𝐿^𝑝=𝐿^𝑝(𝑅)  (1≤𝑝<+∞) be the space of all measurable functions ℎ such that ‖ℎ‖^𝑝_𝐿𝑝=∫_𝑅|ℎ(𝑡,𝑥)|^𝑝𝑑𝑥<∞. We define 𝐿^∞=𝐿^∞(𝑅) with the standard norm ‖ℎ‖_𝐿∞=inf_𝑚(𝑒)=0sup_{𝑥∈𝑅⧵𝑒}|ℎ(𝑡,𝑥)|. For any real number 𝑠, 𝐻^𝑠=𝐻^𝑠(𝑅) denotes the Sobolev space with the norm defined by‖ℎ‖_𝐻𝑠=_𝑅(1+|𝜉|²)^𝑠|ℎ(𝑡,𝜉)|²𝑑𝜉^1/2<∞, (2.1) where ℎ(𝑡,𝜉)=∫_𝑅𝑒^{−𝑖𝑥𝜉}ℎ(𝑡,𝑥)𝑑𝑥.

For 𝑇>0 and nonnegative number 𝑠, 𝐶([0,𝑇);𝐻^𝑠(𝑅)) denotes the Frechet space of all continuous 𝐻^𝑠-valued functions on [0,𝑇). We set Λ=(1−𝜕²_𝑥)^1/2. For simplicity, throughout this paper, we let 𝑐 denote any positive constants.

The local well-posedness theorem is stated as follows.

Theorem 2.1. Provided that 𝑠≥3/2, 𝑢₀∈𝐻^𝑠(𝑅), 𝜑 is a polynomial of order 𝑁 with 𝜑(0)=0. Then problem (1.3) admits a unique local solution: 𝑢(𝑡,𝑥)∈𝐶[0,𝑇);𝐻^𝑠(𝑅)𝐶¹[0,𝑇);𝐻^𝑠−1(𝑅). (2.2)

Proof. In fact, the first equation of problem (1.3) is equivalent to the equation 𝑢_𝑡=Λ⁻²𝜕𝜑𝑢𝜕𝑥^𝑥+𝛽𝑢^2𝑚𝑢_𝑥𝑥, (2.3) which leads to 𝑢=𝑢₀+^𝑡₀Λ⁻²𝜕𝜑𝑢𝜕𝑥^𝑥+𝛽𝑢^2𝑚𝑢_𝑥𝑥𝑑𝜏. (2.4) Suppose that both 𝑢 and 𝑣 are in the closed ball 𝐵_𝑀0(0) of radius 𝑀₀>1 about the zero function in 𝐶([0,𝑇];𝐻^𝑠(𝑅)) and 𝐴 is the operator in the right-hand side of (2.4), for fixed 𝑡∈[0,𝑇], we get ^𝑡₀Λ⁻²𝜑𝑢_𝑥𝑥+𝛽𝑢^2𝑚𝑢_𝑥𝑥𝑑𝑡−^𝑡₀Λ⁻²𝜑𝑣_𝑥𝑥+𝛽𝑣^2𝑚𝑣_𝑥𝑥𝑑𝑡_𝐻𝑠≤𝑇sup_{0≤𝑡≤𝑇}𝜑𝑢_𝑥−𝜑𝑣_{𝑥𝐻𝑠−1}+sup_{0≤𝑡≤𝑇}𝑢^2𝑚𝑢_𝑥𝑥−𝑣^2𝑚𝑣_{𝑥𝑥𝐻𝑠−2}. (2.5) The algebraic property of 𝐻^𝑠0(𝑅) with 𝑠₀>1/2 (see [8–10]) and 𝑠>3/2 derives that 𝑢^𝑗_𝑥−𝑣^𝑗_{𝑥𝐻𝑠−1}=𝑢_𝑥−𝑣_𝑥𝑢_𝑥^𝑗−1+𝑢_𝑥^𝑗−2𝑣_𝑥+⋯+𝑢_𝑥𝑣_𝑥^𝑗−2+𝑣_𝑥^𝑗−1_𝐻𝑠−1≤𝑢_𝑥−𝑣_𝑥‖_𝐻𝑠−1‖𝑢_𝑥^𝑗−1+𝑢_𝑥^𝑗−2𝑣_𝑥+⋯+𝑢_𝑥𝑣_𝑥^𝑗−2+𝑣_𝑥^𝑗−1_𝐻𝑠−1≤𝑐𝑢_𝑥−𝑣_{𝑥𝐻𝑠−1}^𝑗−1_𝑖=0𝑢_𝑥𝐻^{𝑗−1−𝑖}_𝑠−1𝑣_𝑥^𝑖_𝐻𝑠−1≤𝑐𝑀₀^𝑗−1‖𝑢−𝑣‖_𝐻𝑠,‖𝜑𝑢_𝑥−𝜑𝑣_𝑥‖_𝐻𝑠−1≤𝑐^𝑁_𝑗=0‖𝑢_𝑥^𝑗−𝑣_𝑥^𝑗‖_𝐻𝑠−1≤𝑐𝑀₀^𝑁−1‖𝑢−𝑣‖_𝐻𝑠. (2.6) Using 𝑢^2𝑚𝑢_𝑥𝑥=𝜕_𝑥[𝑢^2𝑚𝑢_𝑥]−2𝑚𝑢^2𝑚−1(𝑢_𝑥)² and 𝑣^2𝑚𝑣_𝑥𝑥=𝜕_𝑥[𝑣^2𝑚𝑣_𝑥]−2𝑚𝑣^2𝑚−1(𝑣_𝑥)², we get 𝑢^2𝑚𝑢_𝑥𝑥−𝑣^2𝑚𝑣_{𝑥𝑥𝐻𝑠−2}≤‖𝜕_𝑥𝑢^2𝑚𝑢_𝑥−𝑣^2𝑚𝑣_𝑥‖_𝐻𝑠−2+𝑐‖𝑢^2𝑚−1𝑢²_𝑥−𝑣^2𝑚−1𝑣²_𝑥‖_𝐻𝑠−2≤𝑐𝑢^2𝑚−𝑣^2𝑚𝑣_𝑥+𝑢^2𝑚𝑢_𝑥−𝑣_{𝑥𝐻𝑠−1}+𝑐𝑢^2𝑚−1𝑢²_𝑥−𝑣²_𝑥+𝑢^2𝑚−1−𝑣^2𝑚−1𝑣²_{𝑥𝐻𝑠−1}≤𝑐‖𝑢^2𝑚−𝑣^2𝑚𝑣_𝑥‖_𝐻𝑠−1+𝑢^2𝑚𝑢_𝑥−𝑣_{𝑥𝐻𝑠−1}+𝑢^2𝑚−1𝑢²_𝑥−𝑣²_{𝑥𝐻𝑠−1}+𝑢^2𝑚−1−𝑣^2𝑚−1𝑣²_{𝑥𝐻𝑠−1}≤𝑐𝑀₀^2𝑚‖𝑢−𝑣‖_𝐻𝑠, (2.7) in which 𝑠>3/2 is used.
From (2.5)–(2.7), we obtain ‖𝐴𝑢−𝐴𝑣‖_𝐻𝑠≤‖𝑢−𝑣‖_𝐻𝑠, (2.8) where 𝜃=max(𝑐𝑇𝑀₀^𝑁−1,𝑐𝑇𝑀₀^2𝑚) and 𝑐 is independent of 𝑇. Choosing 𝑇 sufficiently small such that 𝜃<1, we know that 𝐴 is a contractive mapping. Applying the above inequality and (2.4) yields ‖𝐴𝑢‖_𝐻𝑠≤𝑢_0𝐻𝑠+𝜃‖𝑢‖_𝐻𝑠. (2.9) Choosing 𝑇 sufficiently small such that 𝜃𝑀₀+‖𝑢₀‖_𝐻𝑠<𝑀₀, we know that 𝐴 maps 𝐵_𝑀0(0) to itself. It follows from the contractive mapping principle that the mapping 𝐴 has a unique fixed point 𝑢 in 𝐵_𝑀0(0). This completes the proof of Theorem 2.1.

3. Existence of Local Weak Solutions

In this section, we assume that 𝜑(𝜂)=𝜂^2𝑁+1 where 𝑁 is a nature number. In order to establish the existence of local weak solution, we need the following lemmas.

Lemma 3.1 (see Kato and Ponce [8]). If 𝑟≥0, then 𝐻^𝑟⋂𝐿^∞ is an algebra. Moreover, ‖𝑢𝑣‖_𝑟≤𝑐‖𝑢‖_𝐿∞‖𝑣‖_𝑟+‖𝑢‖_𝑟‖𝑣‖_𝐿∞, (3.1) where 𝑐 is a constant depending only on 𝑟.

Lemma 3.2 (see Kato and Ponce [8]). Let 𝑟>0. If 𝑢∈𝐻^𝑟⋂𝑊^1,∞ and 𝑣∈𝐻^𝑟−1⋂𝐿^∞, then ‖Λ^𝑟,𝑢𝑣‖_𝐿2≤𝑐‖𝜕_𝑥𝑢‖_𝐿∞‖Λ^𝑟−1𝑣‖_𝐿2+‖Λ^𝑟𝑢‖_𝐿2‖𝑣‖_𝐿∞. (3.2)

Lemma 3.3. Let 𝑠≥3/2, 𝜑(𝑢_𝑥)=𝑢_𝑥^2𝑁+1, and the function 𝑢(𝑡,𝑥) is a solution of problem (1.3) and the initial data 𝑢₀(𝑥)∈𝐻^𝑠. Then the following results hold.
For 𝑞∈(0,𝑠−1], there is a constant 𝑐 such that _𝑅Λ^𝑞+1𝑢²𝑑𝑥≤_𝑅Λ^𝑞+1𝑢₀²𝑑𝑥+𝑐^𝑡₀‖𝑢‖²_𝐻𝑞+1𝑢_𝑥𝐿^2𝑁_∞+𝑢_𝑥²_𝐿∞‖𝑢‖_𝐿^2𝑚−2_∞+𝑢_𝑥𝐿∞‖𝑢‖_𝐿^2𝑚−1_∞+‖𝑢‖_𝐿^2𝑚_∞𝑑𝜏. (3.3)
For 𝑞∈[0,𝑠−1], there is a constant 𝑐 such that 𝑢_𝑡𝐻𝑞≤𝑐‖𝑢‖_𝐻𝑞+1‖𝑢‖_𝐿^2𝑚−1_∞+𝑢_𝑥𝐿^2𝑁−1_∞. (3.4)

Proof. For 𝑞∈(0,𝑠−1], applying (Λ^𝑞𝑢)Λ^𝑞 to both sides of the first equation of system (1.3) and integrating with respect to 𝑥 by parts, we have the identity 12𝑑𝑑𝑡_𝑅Λ^𝑞𝑢²+𝛼Λ^𝑞𝑢_𝑥²𝑑𝑥=_𝑅Λ^𝑞𝑢Λ^𝑞𝜑𝑢_𝑥𝑥𝑑𝑥+𝛽_𝑅Λ^𝑞𝑢Λ^𝑞𝑢^2𝑚𝑢_𝑥𝑥𝑑𝑥. (3.5) We will estimate the two terms on the right-hand side of (3.5), respectively. For the first term, by using the Cauchy-Schwartz inequality and Lemmas 3.1 and 3.2, we have _𝑅Λ^𝑞𝑢Λ^𝑞𝜑𝑢_𝑥𝑥𝑑𝑥=_𝑅Λ^𝑞𝑢_𝑥Λ^𝑞𝜑𝑢_𝑥Λ𝑑𝑥≤𝑐^𝑞𝑢_𝑥𝐿2Λ^𝑞𝑢_𝑥^2𝑁+1_𝐿2≤𝑐‖𝑢‖²_𝐻𝑞+1𝑢_𝑥𝐿^2𝑁_∞. (3.6) For the second term, we have _𝑅Λ^𝑞𝑢Λ^𝑞𝑢^2𝑚𝑢_𝑥𝑥𝑑𝑥=_𝑅Λ^𝑞𝑢Λ^𝑞𝑢^2𝑚𝑢_𝑥𝑥−2𝑚𝑢^2𝑚−1𝑢²_𝑥=𝑑𝑥_𝑅Λ^𝑞𝑢_𝑥Λ^𝑞𝑢^2𝑚𝑢_𝑥𝑑𝑥−2𝑚_𝑅Λ^𝑞𝑢Λ^𝑞𝑢^2𝑚−1𝑢²_𝑥𝑑𝑥=𝐾₁+𝐾₂. (3.7) For 𝐾₁, applying Lemma 3.1 derives 𝐾₁≤𝑐‖𝑢‖²_𝐻𝑞+1‖𝑢‖_𝐿^2𝑚_∞+‖𝑢_𝑥‖_𝐿∞‖𝑢‖_𝐿^2𝑚−1_∞. (3.8) For 𝐾₂, we get 𝐾₂≤𝑐‖𝑢‖_𝐻𝑞‖𝑢^2𝑚−1𝑢²_𝑥‖_𝐻𝑞≤𝑐‖𝑢‖_𝐻𝑞‖𝑢^2𝑚−1𝑢_𝑥‖_𝐿∞‖𝑢_𝑥‖_𝐻𝑞+‖𝑢^2𝑚−1𝑢_𝑥‖_𝐻𝑞‖𝑢_𝑥‖_𝐿∞≤𝑐‖𝑢‖²_𝐻𝑞+1‖𝑢_𝑥‖_𝐿∞‖𝑢‖_𝐿^2𝑚−1_∞+‖𝑢_𝑥‖²_𝐿∞‖𝑢‖_𝐿^2𝑚−2_∞. (3.9)
It follows from (3.5)–(3.9) that there exists a constant 𝑐 such that 12𝑑𝑑𝑡_𝑅Λ^𝑞𝑢²+Λ^𝑞𝑢_𝑥²𝑑𝑥≤𝑐‖𝑢‖²_𝐻𝑞+1‖𝑢_𝑥‖_𝐿^2𝑁_∞+‖𝑢_𝑥‖²_𝐿∞‖𝑢‖_𝐿^2𝑚−2_∞+‖𝑢_𝑥‖_𝐿∞‖𝑢‖_𝐿^2𝑚−1_∞+‖𝑢‖_𝐿^2𝑚_∞. (3.10) Integrating both sides of the above inequality with respect to 𝑡 results in inequality (3.3).
To estimate the norm of 𝑢_𝑡, we apply the operator (1−𝜕²_𝑥)⁻¹ to both sides of the first equation of system (1.3) to obtain the equation 𝑢_𝑡=Λ⁻²𝜕𝜑𝑢𝜕𝑥^𝑥+𝛽𝑢^2𝑚𝑢_𝑥𝑥. (3.11) Applying (Λ^𝑞𝑢_𝑡)Λ^𝑞 to both sides of (3.11) for 𝑞∈[0,𝑠−1] gives rise to _𝑅Λ^𝑞𝑢_𝑡²𝑑𝑥=_𝑅Λ^𝑞𝑢_𝑡Λ^𝑞−2𝜕_𝑥𝜑𝑢_𝑥+𝑢^2𝑚𝑢_𝑥𝑥𝑑𝜏. (3.12) For the right-hand of (3.12), we have _𝑅Λ^𝑞𝑢_𝑡1−𝜕²_𝑥⁻¹Λ^𝑞𝜕_𝑥𝜑𝑢_𝑥𝑑𝑥≤𝑐‖𝑢_𝑡‖_𝐻𝑞𝑅1+𝜉^2𝑞−1_𝑅𝑢_𝑥^2𝑁𝜉−𝜂𝑢_𝑥𝜂𝑑𝜂^21/2≤𝑐‖𝑢_𝑡‖_𝐻𝑞‖𝑢‖_𝐻1‖𝑢‖_𝐻𝑞+1‖𝑢_𝑥‖_𝐿^2𝑁−1_∞,_𝑅Λ^𝑞𝑢_𝑡1−𝜕²_𝑥⁻¹Λ^𝑞𝜕_𝑥𝑢^2𝑚𝑢_𝑥𝑑𝑥≤𝑐‖𝑢_𝑡‖_𝐻𝑞𝑅1+𝜉^2𝑞−1_𝑅𝑢^2𝑚𝜉−𝜂𝑢_𝑥𝜂𝑑𝜂^21/2≤𝑐‖𝑢_𝑡‖_𝐻𝑞‖𝑢‖_𝐻1‖𝑢‖_𝐻𝑞+1‖𝑢‖_𝐿^2𝑚−1_∞,_𝑅Λ^𝑞𝑢_𝑡1−𝜕²_𝑥⁻¹Λ^𝑞𝑢^2𝑚−1𝑢²_𝑥𝑑𝑥≤𝑐‖𝑢_𝑡‖_𝐻𝑞𝑅1+𝜉^2𝑞−1_𝑅𝑢^2𝑚−1𝑢_𝑥𝜉−𝜂𝑢_𝑥𝜂𝑑𝜂^21/2≤𝑐‖𝑢_𝑡‖_𝐻𝑞‖𝑢‖_𝐻1‖𝑢‖_𝐻𝑞+1‖𝑢‖_𝐿^2𝑚−1_∞. (3.13)
Applying (3.13) into (3.12) yields the inequality ‖𝑢_𝑡‖_𝐻𝑞≤𝑐‖𝑢‖_𝐻1‖𝑢‖_𝐻𝑞+1‖𝑢‖_𝐿^2𝑚−1_∞+‖𝑢_𝑥‖_𝐿^2𝑁−1_∞ (3.14) for a constant 𝑐>0. This completes the proof of Lemma 3.3.

Lemma 3.4. If 𝑢(𝑡,𝑥) is a solution of problem (1.3), 𝛼>0, 𝜑(𝜂)=𝜂^2𝑁+1, then ‖𝑢‖_𝐿∞≤𝑐‖𝑢‖_𝐻1(𝑅)≤𝑐‖𝑢₀‖_𝐻1(𝑅), (3.15) where 𝑐 is a constant.

Proof. Multiplying both sides of the first equation of (1.3) by 𝑢(𝑡,𝑥) and integrating with respect to 𝑥 over 𝑅, we have 12𝑑𝑑𝑡_𝑅𝑢(𝑡,𝑥)²+𝛼𝑢_𝑥(𝑡,𝑥)²𝑑𝑥=_𝑅𝜑𝑢_𝑥𝑥𝑢(𝑡,𝑥)𝑑𝑥+𝛽_𝑅𝑢^2𝑚+1𝑢_𝑥𝑥𝑑𝑥. (3.16)
Since _𝑅𝜑𝑢_𝑥𝑥𝑢(𝑡,𝑥)𝑑𝑥+𝛽_𝑅𝑢^2𝑚+1𝑢_𝑥𝑥𝑑𝑥=−_𝑅𝑢_𝑥^2𝑁+2𝑑𝑥−𝛽(2𝑚+1)_𝑅𝑢^2𝑚𝑢²_𝑥𝑑𝑥<0, (3.17) we derive that 12𝑑𝑑𝑡_𝑅𝑢(𝑡,𝑥)²+𝛼𝑢_𝑥(𝑡,𝑥)²𝑑𝑥<0, (3.18) which results in _𝑅𝑢(𝑡,𝑥)²+𝛼𝑢_𝑥(𝑡,𝑥)²<_𝑅𝑢(0,𝑥)²+𝛼𝑢_𝑥(0,𝑥)²≤𝑐‖𝑢₀‖²_𝐻1. (3.19)
From (3.19), we know that (3.15) holds. This completes the proof.

Defining𝜙(𝑥)=𝑒^{1/(𝑥2−1)},|𝑥||𝑥|<1,0,≥1 (3.20) and setting 𝜙_𝜀(𝑥)=𝜀^−1/4𝜙(𝜀^−1/4𝑥) with 0<𝜀<1/4 and 𝑢_𝜀0=𝜙_𝜀⋆𝑢₀, we know that 𝑢_𝜀0∈𝐶^∞ for any 𝑢₀∈𝐻^𝑠(𝑅) and 𝑠>0.

It follows from Theorem 2.1 that for each 𝜀 the Cauchy problem𝑢_𝑡−𝑢_𝑡𝑥𝑥=𝜕𝜑𝑢𝜕𝑥^𝑥+𝛽𝑢^2𝑚𝑢_𝑥𝑥,𝑢(0,𝑥)=𝑢_𝜀0(𝑥),𝑥∈𝑅 (3.21) has a unique solution 𝑢_𝜀(𝑡,𝑥)∈𝐶^∞([0,𝑇);𝐻^∞).

Lemma 3.5. Under the assumptions of problem (3.21), the following estimates hold for any 𝜀 with 0<𝜀<1/4, 𝑢₀∈𝐻^𝑠(𝑅) and 𝑠>0∶‖𝑢_𝜀0𝑥‖_𝐿∞≤𝑐₁‖𝑢_0𝑥‖_𝐿∞,‖𝑢_𝜀0‖_𝐻𝑞≤𝑐₁,𝑖𝑓𝑞≤𝑠, (3.22) where 𝑐₁ is a constant independent of 𝜀.

Proof. Using the definition of 𝑢_𝜀0 and 𝑢_𝜀0𝑥 results in the conclusion of the lemma.

Lemma 3.6. Suppose that 𝑢₀(𝑥)∈𝐻^𝑠(𝑅) with 𝑠∈[1,3/2] such that ‖𝑢_0𝑥‖_𝐿∞<∞. Let 𝑢_𝜀0 be defined as in system (3.21) and let 𝜑(𝜂)=𝜂^2𝑁+1. Then there exist two positive constants 𝑇 and 𝑐, independent of 𝜀, such that the solution 𝑢_𝜀 of problem (3.21) satisfies ‖𝑢_𝜀𝑥‖_𝐿∞≤𝑐 for any 𝑡∈[0,𝑇).

Proof. Using notation 𝑢=𝑢_𝜀 and differentiating both sides of the first equation of problem (3.11) with respect to 𝑥 give rise to 𝑢_𝑡𝑥=−𝜑𝑢_𝑥−𝛽𝑢^2𝑚𝑢_𝑥+Λ⁻²𝜑𝑢_𝑥+𝛽𝑢^2𝑚𝑢_𝑥−2𝑚𝛽𝑢^2𝑚−1𝑢_𝑥𝑥. (3.23) Letting 𝑝>0 be an integer and multiplying the above equation by (𝑢_𝑥)^2𝑝+1 and then integrating the resulting equation with respect to 𝑥 yield the equality 1𝑑2𝑝+2𝑑𝑡_𝑅𝑢_𝑥^2𝑝+2𝑑𝑥=−_𝑅𝜑𝑢_𝑥𝑢_𝑥^2𝑝+1𝑑𝑥−𝛽_𝑅𝑢^2𝑚𝑢_𝑥^2𝑝+2𝑑𝑥+_𝑅𝐽𝑢_𝑥^2𝑝+1𝑑𝑥, (3.24) where 𝐽=Λ⁻²𝜑𝑢_𝑥+𝛽𝑢^2𝑚𝑢_𝑥−2𝑚𝛽𝑢^2𝑚−1𝑢_𝑥𝑥. (3.25)
Applying the Hölder’s inequality to (3.24) and noting Lemmas 3.4 and 3.5, we obtain 1𝑑2𝑝+2𝑑𝑡_𝑅𝑢_𝑥^2𝑝+2𝑑𝑥≤𝑐𝑢_𝑥𝐿^2𝑁_∞𝑅𝑢_𝑥^2𝑝+2𝑑𝑥+𝑐_𝑅𝑢_𝑥^2𝑝+2+𝑑𝑥_𝑅|𝐽|^2𝑝+2𝑑𝑥^1/(2𝑝+2)𝑢_𝑥^2𝑝+2𝑑𝑥^{2(𝑝+1)/2(𝑝+2)} (3.26) or 𝑑𝑑𝑡_𝑅𝑢_𝑥^2(𝑝+2)𝑑𝑥^1/(2𝑝+2)≤𝑐‖𝑢_𝑥‖_𝐿^2𝑁_∞𝑅𝑢_𝑥^2𝑝+2𝑑𝑥^1/(2𝑝+2)+𝑐_𝑅𝑢_𝑥^2𝑝+2𝑑𝑥^1/(2𝑝+2)+_𝑅|𝐽|^2𝑝+2𝑑𝑥^1/(2𝑝+2). (3.27)
Since ‖𝑓‖_𝐿𝑝→‖𝑓‖_𝐿∞ as 𝑝→∞ for any 𝑓∈𝐿^∞⋂𝐿², integrating both sides of the inequality (3.27) with respect to 𝑡 and taking the limit as 𝑝→∞ result in the estimate ‖𝑢_𝑥‖_𝐿∞≤‖𝑢_0𝑥‖_𝐿∞+^𝑡₀𝑐‖𝑢_𝑥‖_𝐿∞+‖𝑢_𝑥‖_𝐿^2𝑁+1_∞+‖𝐽‖_𝐿∞𝑑𝜏. (3.28) Using the algebra property of 𝐻^𝑠0(𝑅) with 𝑠₀>1/2 yields (‖𝑢_𝜀‖_𝐻1/2+ means that there exists a sufficiently small 𝛿>0 such that ‖𝑢_𝜀‖_1/2+=‖𝑢_𝜀‖_𝐻1/2+𝛿) ‖𝐽‖_𝐿∞≤𝑐‖𝐽‖_𝐻1/2+≤𝑐‖Λ⁻²𝜑𝑢_𝑥+𝛽𝑢^2𝑚𝑢_𝑥−2𝑚𝛽𝑢^2𝑚−1𝑢_𝑥𝑥‖_𝐻1/2+≤𝑐‖𝜑𝑢_𝑥‖_𝐻0+‖𝑢‖_𝐻1+‖𝑢^2𝑚−1𝑢_𝑥‖_𝐻0≤𝑐‖𝑢_𝑥‖^2𝑁‖𝑢‖_𝐻1+‖𝑢‖_𝐻1+‖𝑢‖_𝐿^2𝑚−1_∞‖𝑢‖_𝐻1≤𝑐‖𝑢_𝑥‖^2𝑁+1, (3.29) in which Lemmas 3.4 and 3.5 are used. From (3.28) and (3.29), one has ‖𝑢_𝑥‖_𝐿∞≤‖𝑢_0𝑥‖_𝐿∞+𝑐^𝑡₀1+‖𝑢_𝑥‖_𝐿∞+‖𝑢_𝑥‖_𝐿^2𝑁_∞+‖𝑢_𝑥‖_𝐿^2𝑁+1_∞𝑑𝜏. (3.30)
From Lemma 3.5, it follows from the contraction mapping principle that there is a 𝑇>0 such that the equation ‖𝑊‖_𝐿∞=‖𝑢_0𝑥‖_𝐿∞+𝑐^𝑡₀1+‖𝑊‖_𝐿∞+‖𝑊‖_𝐿^2𝑁_∞+‖𝑊‖_𝐿^2𝑁+1_∞𝑑𝜏 (3.31) has a unique solution 𝑊∈𝐶[0,𝑇]. Using the result presented on page  51 in [11] yields that there are constants 𝑇>0 and 𝑐>0 independent of 𝜀 such that ‖𝑢_𝑥‖_𝐿∞≤‖𝑊(𝑡)‖_𝐿∞≤𝑐 for arbitrary 𝑡∈[0,𝑇], which leads to the conclusion of Lemma 3.6.

Using Lemmas 3.3–3.6, notation 𝑢_𝜀=𝑢 and Gronwall’s inequality result in the inequalities ‖𝑢_𝜀‖_𝐻𝑞≤𝐶_𝑇𝑒^𝐶𝑇,‖𝑢_𝜀𝑡‖_𝐻𝑟≤𝐶_𝑇𝑒^𝐶𝑇, (3.32) where 𝑞∈(0,𝑠], 𝑟∈(0,𝑠−1](1≤𝑠≤3/2) and 𝐶_𝑇 depends on 𝑇. It follows from the Aubin’s compactness theorem that there is a subsequence of {𝑢_𝜀}, denoted by {𝑢_𝜀𝑛}, such that {𝑢_𝜀𝑛} and their temporal derivatives {𝑢_𝜀𝑛𝑡} are weakly convergent to a function 𝑢(𝑡,𝑥) and its derivative 𝑢_𝑡 in 𝐿²([0,𝑇],𝐻^𝑠) and 𝐿²([0,𝑇],𝐻^𝑠−1), respectively. Moreover, for any real number 𝑅₁>0, {𝑢_𝜀𝑛} is convergent to the function 𝑢 strongly in the space 𝐿²([0,𝑇],𝐻^𝑞(−𝑅₁,𝑅₁)) and {𝑢_𝜀𝑛𝑡} converges to 𝑢_𝑡 strongly in the space 𝐿²([0,𝑇],𝐻^𝑟(−𝑅₁,𝑅₁)) for 𝑟∈[0,𝑠−1]. Thus, we can prove the existence of a weak solution to (1.3).

Theorem 3.7. Suppose that 𝑢₀(𝑥)∈𝐻^𝑠 with 1≤𝑠≤3/2, ‖𝑢_0𝑥‖_𝐿∞<∞ and 𝜑(𝜂)=𝜂^2𝑁+1. Then there exists a 𝑇>0 such that (1.3) subject to initial value 𝑢₀(𝑥) has a weak solution 𝑢(𝑡,𝑥)∈𝐿²([0,𝑇],𝐻^𝑠) in the sense of distribution and 𝑢_𝑥∈𝐿^∞([0,𝑇]×𝑅).

Proof. From Lemma 3.6, we know that {𝑢_𝜀𝑛𝑥}(𝜀_𝑛→0) is bounded in the space 𝐿^∞. Thus, the sequences {𝑢_𝜀𝑛}, {𝑢_𝜀𝑛𝑥}, {𝑢²_𝜀𝑛𝑥}, and {𝑢_𝜀^2𝑁+1_𝑛𝑥} are weakly convergent to 𝑢, 𝑢_𝑥, 𝑢²_𝑥, and 𝑢_𝑥^2𝑁+1 in 𝐿²[0,𝑇],𝐻^𝑟(−𝑅,𝑅) for any 𝑟∈[0,𝑠−1), separately. Therefore, 𝑢 satisfies the equation ^𝑇_0𝑅𝑢𝑔_𝑡−𝑔_𝑥𝑥𝑡𝑑𝑥𝑑𝑡=^𝑇_0𝑅𝑢_𝑥^2𝑁+1𝑔_𝑥+𝛽𝑢^2𝑚𝑢_𝑥𝑔_𝑥−2𝑚𝛽𝑢^2𝑚−1𝑢²_𝑥𝑔𝑑𝑥𝑑𝑡, (3.33) with 𝑢(0,𝑥)=𝑢₀(𝑥) and 𝑔∈𝐶^∞₀. Since 𝑋=𝐿¹([0,𝑇]×𝑅) is a separable Banach space and {𝑢_𝜀𝑛𝑥} is a bounded sequence in the dual space 𝑋^∗=𝐿^∞([0,𝑇]×𝑅) of 𝑋, there exists a subsequence of {𝑢_𝜀𝑛𝑥}, still denoted by {𝑢_𝜀𝑛𝑥}, weakly star convergent to a function 𝑣 in 𝐿^∞([0,𝑇]×𝑅). It derives from the weakly convergence of {𝑢_𝜀𝑛𝑥} to 𝑢_𝑥 in 𝐿²([0,𝑇]×𝑅) that 𝑢_𝑥=𝑣 almost everywhere. Thus, we obtain 𝑢_𝑥∈𝐿^∞([0,𝑇]×𝑅).

4. Well-Posedness of Global Solutions

Lemma 4.1. If 𝑢(𝑡,𝑥) is a solution of problem (1.3), 𝛼>0, 𝜑(𝜂)=𝜂^2𝑁+1, then ‖𝑢_𝑥‖_𝐿∞≤𝐴^1/2, (4.1) where 𝐴=_𝑅1+𝛼𝛼𝑢^′₀(𝑥)²+(1+𝛼)𝑢₀^′′(𝑥)²𝑑𝑥. (4.2)

Proof. Multiplying each side of the first equation of problem (1.3) by 𝑢_𝑥𝑥 and integrating over [0,𝑡]×𝑅 yields ^𝑡_0𝑅𝑢²_𝑥𝑥𝜑^′𝑢_𝑥+𝑢^2𝑚𝑢²_𝑥𝑥+𝛼2𝜕𝑢𝜕𝑡^2𝑥𝑥𝑑𝑥𝑑𝑡=^𝑡_0𝑅𝑢_𝑡𝑢_𝑥𝑥𝑑𝑥𝑑𝑡. (4.3) Integrating the right-hand side of the above identity by parts and using 𝑢_𝑥(±∞)=0, we get 2^𝑡_0𝑅𝑢_𝑡𝑢_𝑥𝑥𝑑𝑥𝑑𝑡=_𝑅𝑢^′₀(𝑥)²𝑑𝑥−_𝑅𝑢²_𝑥(𝑡,𝑥)𝑑𝑥. (4.4) From (4.3), (4.4) and the assumption of this lemma, we have 𝛼‖𝑢_𝑥𝑥‖²_𝐿2+‖𝑢_𝑥‖_𝐿2≤_𝑅𝑢^′₀(𝑥)²+𝛼𝑢₀^′′(𝑥)²𝑑𝑥, (4.5) from which we obtain (4.1).

Theorem 4.2. Suppose that 𝑠≥2, 𝑢₀∈𝐻^𝑠(𝑅), 𝜑(𝑢_𝑥)=𝑢_𝑥^2𝑁+1 with positive integer 𝑁. Then problem (1.3) has a unique global solution: 𝑢(𝑡,𝑥)∈𝐶[0,∞);𝐻^𝑠(𝑅)𝐶¹[0,∞);𝐻^𝑠−1(𝑅). (4.6)

Proof. Using the Gronwall inequality and Lemma 3.3 and choosing 𝑠=𝑞+1, we have ‖𝑢‖_𝐻𝑠≤𝑐‖𝑢₀‖_𝐻𝑠𝑒^{∫𝑡0(‖𝑢𝑥‖𝐿∞2𝑁+‖𝑢𝑥‖2𝐿∞‖𝑢‖𝐿∞2𝑚−2+‖𝑢𝑥‖𝐿∞‖𝑢‖𝐿∞2𝑚−1+‖𝑢‖𝐿∞2𝑚)𝑑𝜏}. (4.7) From Lemma 4.1, we have ‖𝑢_𝑥‖≤𝐴^1/2=_𝑅1+𝛼𝛼𝑢^′₀(𝑥)²+(1+𝛼)𝑢₀^′′(𝑥)²𝑑𝑥^1/2≤𝑐‖𝑢₀‖_𝐻2(𝑅). (4.8) Using (4.7) and (4.8) derives ‖𝑢‖_𝐻𝑠≤𝑐‖𝑢₀‖_𝐻𝑠𝑒^𝑐𝑡, (4.9) which completes the proof of Theorem 4.2.