1.Introduction MengWu TheLocalStrongSolutionsandGlobalWeakSolutionsforaNonlinearEquation ResearchArticle

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Volume 2013, Article ID 619068,5pages http://dx.doi.org/10.1155/2013/619068

Research Article

The Local Strong Solutions and Global Weak Solutions for a Nonlinear Equation

Meng Wu

Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Correspondence should be addressed to Meng Wu; [email protected] Received 31 March 2013; Accepted 26 April 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Meng Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The existence and uniqueness of local strong solutions for a nonlinear equation are investigated in the Sobolev space𝐶([0, 𝑇);

𝐻^𝑠(𝑅)) ∩ 𝐶¹([0, 𝑇); 𝐻^𝑠−1(𝑅))provided that the initial value lies in𝐻^𝑠(𝑅)with𝑠 > 3/2. Meanwhile, we prove the existence of global weak solutions in𝐿^∞([0, ∞); 𝐿²(𝑅))for the equation.

1. Introduction

Coclite and Karlsen [1] investigated the well posedness in classes of discontinuous functions for the generalized Degasperis-Procesi equation:

𝑢_𝑡− 𝑢_𝑡𝑥𝑥+ 4ℎ^󸀠(𝑢) 𝑢_𝑥

= ℎ^󸀠󸀠󸀠(𝑢) 𝑢³_𝑥+ 3ℎ^󸀠󸀠(𝑢) 𝑢_𝑥𝑢_𝑥𝑥+ ℎ^󸀠(𝑢) 𝑢_𝑥𝑥𝑥, (1) which is subject to the condition

󵄨󵄨󵄨󵄨󵄨ℎ^󸀠(𝑢)󵄨󵄨󵄨󵄨󵄨 ≤ 𝑐 |𝑢| , |ℎ (𝑢)| ≤ 𝑐|𝑢|², (2) or

󵄨󵄨󵄨󵄨󵄨ℎ^󸀠(𝑢)󵄨󵄨󵄨󵄨󵄨 ≤ 𝑐, |ℎ (𝑢)| ≤ 𝑐 |𝑢| , (3) where𝑐is a positive constant. The existence and𝐿¹stability of entropy weak solutions belonging to the class𝐿¹(𝑅) ∩ 𝐵𝑉(𝑅) are established for (1) in paper [1].

In this work, we study the following model:

𝑢_𝑡− 𝑢_𝑡𝑥𝑥+ 𝑚ℎ^󸀠(𝑢) 𝑢_𝑥

= ℎ^󸀠󸀠󸀠(𝑢) 𝑢³_𝑥+ 3ℎ^󸀠󸀠(𝑢) 𝑢_𝑥𝑢_𝑥𝑥+ ℎ^󸀠(𝑢) 𝑢_𝑥𝑥𝑥, (4) where 𝑚is a positive constant andℎ(𝑢) ∈ 𝐶³. If 𝑚 = 4 and ℎ(𝑢) = 𝑢²/2, (4) reduces to the classical Degasperis- Procesi model (see [2–13]). Here, we notice that assumptions

(2) and (3) do not include the caseℎ(𝑢) = 𝑢³. In this paper, we will study the caseℎ(𝑢) = 𝑢³, and𝑚is an arbitrary positive constant.

In fact, the Cauchy problem of (4) in the caseℎ(𝑢) = 𝑢³ is equivalent to the following system:

𝑢_𝑡− 𝑢_𝑡𝑥𝑥+ 3𝑚𝑢²𝑢_𝑥= 6𝑢³_𝑥+ 18𝑢𝑢_𝑥𝑢_𝑥𝑥+ 3𝑢²𝑢_𝑥𝑥𝑥,

𝑢 (0, 𝑥) = 𝑢₀(𝑥) . (5)

Using the operator(1 − 𝜕²_𝑥)⁻¹to multiply the first equation of the problem (5), we obtain

𝑢_𝑡+ 3𝑢²𝑢_𝑥+ (𝑚 − 1) (1 − 𝜕_𝑥²)⁻¹𝜕_𝑥(𝑢³) = 0,

𝑢 (0, 𝑥) = 𝑢₀(𝑥) . (6)

It is shown in this work that there exists a unique local strong solution in the Sobolev space𝐶([0, 𝑇); 𝐻^𝑠(𝑅)) ∩ 𝐶¹([0, 𝑇); 𝐻^𝑠−1(𝑅))by assuming that the initial value 𝑢₀(𝑥) belongs to𝐻^𝑠(𝑅)with𝑠 > 3/2. In addition, we prove the existence of global weak solutions in𝐿^∞([0, ∞); 𝐿²(𝑅))for the system (6).

This paper is organized as follows.Section 2investigates the existence and uniqueness of local strong solutions. The result about global weak solution is given inSection 3.

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2. Local Existence

In this section, we will use the Kato theorem in [14] for abstract differential equation to establish the existence of local strong solution for the problem (6). Let us consider the following problem:

𝑑V

𝑑𝑡 + 𝐻 (V)V= 𝑔 (V) , 𝑡 ≥ 0, V(0) =V₀. (7) Let𝑋and𝑌be Hilbert spaces such that𝑌is continuously and densely embedded in𝑋, and let𝑄 : 𝑌 → 𝑋be a topological isomorphism. Let𝐿(𝑌, 𝑋)be the space of all bounded linear operators from𝑌to 𝑋. In the case of𝑋 = 𝑌, we denote this space by𝐿(𝑋). We illustrate the following conditions in which𝜎₁, 𝜎₂, 𝜎₃, and𝜎₄ are constants depending only on max{‖𝑦‖_𝑌, ‖ 𝑧‖_𝑌}.

(i)𝐻(𝑦) ∈ 𝐿(𝑌, 𝑋)for𝑦 ∈ 𝑋with

󵄩󵄩󵄩󵄩(𝐻(𝑦) − 𝐻(𝑧))𝑤󵄩󵄩󵄩󵄩^𝑋≤ 𝜎₁󵄩󵄩󵄩󵄩𝑦 − 𝑧󵄩󵄩󵄩󵄩^𝑋‖𝑤‖_𝑌, 𝑦, 𝑧, 𝑤 ∈ 𝑌, (8) and 𝐻(𝑦) ∈ 𝐺(𝑋, 1, 𝛽) (i.e., 𝐻(𝑦) is quasi-m- accretive), uniformly on bounded sets in𝑌.

(ii)𝑄𝐻(𝑦)𝑄⁻¹ = 𝐻(𝑦) + 𝐴(𝑦), where𝐴(𝑦) ∈ 𝐿(𝑋)is bounded, uniformly on bounded sets in𝑌. Moreover,

󵄩󵄩󵄩󵄩(𝐴(𝑦) − 𝐴(𝑧))𝑤󵄩󵄩󵄩󵄩𝑋≤ 𝜎₂󵄩󵄩󵄩󵄩𝑦 − 𝑧󵄩󵄩󵄩󵄩𝑌‖𝑤‖_𝑋, 𝑦, 𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋.

(9) (iii)𝑔 : 𝑌 → 𝑌extends to a map from𝑋 into𝑋, is

bounded on bounded sets in𝑌, and satisfies

󵄩󵄩󵄩󵄩𝑔(𝑦) − 𝑔(𝑧)󵄩󵄩󵄩󵄩^𝑌≤ 𝜎₃󵄩󵄩󵄩󵄩𝑦 − 𝑧󵄩󵄩󵄩󵄩^𝑌, 𝑦, 𝑧 ∈ 𝑌, (10)

󵄩󵄩󵄩󵄩𝑔(𝑦) − 𝑔(𝑧)󵄩󵄩󵄩󵄩^𝑋≤ 𝜎₄󵄩󵄩󵄩󵄩𝑦 − 𝑧󵄩󵄩󵄩󵄩^𝑋, 𝑦, 𝑧 ∈ 𝑋. (11) Kato Theorem (see [14]). Assume that conditions (i), (ii), and (iii) hold. IfV₀ ∈ 𝑌, there is a maximal𝑇 > 0depending only on‖V₀‖_𝑌and a unique solutionVto the problem(7)such that

V=V(⋅,V₀) ∈ 𝐶 ([0, 𝑇) ; 𝑌) ∩ 𝐶¹([0, 𝑇) ; 𝑋) . (12) Moreover, the mapV₀ → V(⋅,V₀)is a continuous map from𝑌 to the following space:

𝐶 ([0, 𝑇) ; 𝑌) ∩ 𝐶¹([0, 𝑇) ; 𝑋) . (13) In order to apply the Kato theorem to establish the local well posedness for the problem (6), we let𝐻(𝑢) = 3𝑢²𝜕_𝑥,𝑌 = 𝐻^𝑠(𝑅),𝑋 = 𝐻^𝑠−1(𝑅),Λ = (1−𝜕_𝑥²)^1/2,𝑔(𝑢) = (𝑚−1)Λ⁻²𝜕_𝑥(𝑢³), and𝑄 = Λ^𝑠. We know that𝑄is an isomorphism of𝐻^𝑠onto 𝐻^𝑠−1. Now, we cite the following Lemmas.

Lemma 1. The operator𝐴(𝑢) = 𝑢²𝜕_𝑥with𝑢 ∈ 𝐻^𝑠(𝑅),𝑠 > 3/2 belongs to𝐺(𝐻^𝑠−1(𝑅), 1, 𝛽).

Lemma 2. Assume that𝐻(𝑢) = 3𝑢²𝜕_𝑥with𝑢 ∈ 𝐻^𝑠(𝑅)and 𝑠 > 3/2. Then,𝐻(𝑢) ∈ 𝐿(𝐻^𝑠(𝑅), 𝐻^𝑠−1(𝑅))for all𝑢 ∈ 𝐻^𝑠(𝑅).

Moreover,

‖(𝐻 (𝑢) − 𝐻 (𝑧)) 𝑤‖_𝐻^𝑠−1 ≤ 𝜎₁‖𝑢 − 𝑧‖_𝐻^𝑠−1‖𝑤‖_𝐻^𝑠,

𝑢, 𝑧, 𝑤 ∈ 𝐻^𝑠(𝑅) . (14)

Lemma 3. For𝑠 > 3/2,𝑢, 𝑧 ∈ 𝐻^𝑠(𝑅)and𝑤 ∈ 𝐻^𝑠−1, it holds that𝐴(𝑢) = [Λ^𝑠, 3𝑢²𝜕_𝑥]Λ^−𝑠∈ 𝐿(𝐻^𝑠−1)for𝑢 ∈ 𝐻^𝑠and

‖(𝐴(𝑢) − 𝐴(𝑧))𝑤‖_𝐻^𝑠−1 ≤ 𝜎₂‖𝑢 − 𝑧‖_𝐻^𝑠‖𝑤‖_𝐻^𝑠−1. (15) The above three Lemmas can be found in Ni and Zhou [15].

Lemma 4. Let 𝑢, 𝑧 ∈ 𝐻^𝑠 with𝑠 > 3/2 and𝑔(𝑢) = (𝑚 − 1)Λ⁻²𝜕_𝑥(𝑢³). Then,𝑔is bounded on bounded sets in𝐻^𝑠 and satisfies

󵄩󵄩󵄩󵄩𝑔(𝑢) − 𝑔(𝑧)󵄩󵄩󵄩󵄩𝐻^𝑠≤ 𝜎₃‖𝑢 − 𝑧‖_𝐻^𝑠,

󵄩󵄩󵄩󵄩𝑔(𝑢) − 𝑔(𝑧)󵄩󵄩󵄩󵄩^𝐻^𝑠−1 ≤ 𝜎₄‖𝑢 − 𝑧‖_𝐻^𝑠−1. (16) Proof. For 𝑠₀ > 1/2, we know that ‖𝑢V‖_𝐻𝑠0(𝑅) ≤ 𝑐‖𝑢‖_𝐻𝑠0(𝑅)‖V‖_𝐻𝑠0(𝑅). Consequently, we have

󵄩󵄩󵄩󵄩𝑔(𝑢) − 𝑔(𝑧)󵄩󵄩󵄩󵄩𝐻^𝑠≤ 𝑐󵄩󵄩󵄩󵄩󵄩𝑢³− 𝑧³󵄩󵄩󵄩󵄩󵄩𝐻^𝑠−1

≤ 𝑐‖𝑢 − 𝑧‖_𝐻^𝑠−1(‖𝑢‖²_𝐻^𝑠−1+ ‖V‖²_𝐻^𝑠−1)

≤ 𝜎₃‖𝑢 − 𝑧‖_𝐻^𝑠,

󵄩󵄩󵄩󵄩𝑔(𝑢) − 𝑔(𝑧)󵄩󵄩󵄩󵄩𝐻^𝑠−1 ≤ 𝑐󵄩󵄩󵄩󵄩󵄩𝑢³− 𝑧³󵄩󵄩󵄩󵄩󵄩𝐻^𝑠−2

≤ 𝑐󵄩󵄩󵄩󵄩󵄩𝑢³− 𝑧³󵄩󵄩󵄩󵄩󵄩𝐻^𝑠−1

≤ 𝑐‖𝑢 − 𝑧‖_𝐻^𝑠−1(‖𝑢‖²_𝐻^𝑠−1+ ‖V‖²_𝐻^𝑠−1)

≤ 𝜎₄‖𝑢 − 𝑧‖_𝐻^𝑠−1.

(17) Using the Kato Theorem, Lemmas1–4, we immediately obtain the local well-posedness theorem.

Theorem 5. Assume that𝑢₀ ∈ 𝐻^𝑠(𝑅)with𝑠 > 3/2. Then, there exists a𝑇 > 0such that the system(5)or the problem(6) has a unique solution𝑢(𝑡, 𝑥)satisfying

𝑢 (𝑡, 𝑥) ∈ 𝐶 ([0, 𝑇) ; 𝐻^𝑠(𝑅)) ∩ 𝐶¹([0, 𝑇) ; 𝐻^𝑠−1(𝑅)) . (18)

3. Weak Solutions

In this section, our aim is to establish the existence of global weak solutions for the system (6). Firstly, we prove that the solution of the problem (5) is bounded in the space𝐿²(𝑅)and 𝐿^∞(𝑅).

Lemma 6. The solution of the problem(5)with𝑚 > 0satisfies

∫𝑅𝐾₁𝐾 𝑑𝑥 = ∫

𝑅

1 + 𝜉²

𝑚 + 𝜉²󵄨󵄨󵄨󵄨̂𝑢(𝜉)󵄨󵄨󵄨󵄨²𝑑𝜉 = ∫

𝑅

1 + 𝜉²

𝑚 + 𝜉²󵄨󵄨󵄨󵄨̂𝑢⁰(𝜉)󵄨󵄨󵄨󵄨²𝑑𝜉, (19) where𝐾₁= 𝑢 − 𝜕_𝑥𝑥² 𝑢, and𝐾 = (𝑚 − 𝜕_𝑥𝑥² )⁻¹𝑢. Moreover, there exist two constants𝑐₁> 0and𝑐₂> 0depending only on𝑚such that

𝑐₁󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐿²(𝑅) ≤ 𝑐₁‖𝑢‖_𝐿²_(𝑅)≤ 𝑐₂󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐿²(𝑅). (20)

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Proof. Setting𝐾₁= 𝑢 − 𝜕²_𝑥𝑥𝑢and𝐾 = (𝑚 − 𝜕_𝑥𝑥² )⁻¹𝑢and using the first equation of the problem (5), we obtain𝑢 = 𝑚𝑦 − 𝑦_𝑥𝑥 and

𝑑 𝑑𝑡∫

𝑅𝐾₁𝐾 𝑑𝑥

= ∫𝑅

𝜕𝐾₁

𝜕𝑡 𝐾 𝑑𝑥 + ∫

𝑅𝐾₁𝜕𝐾

𝜕𝑡𝑑𝑥 = 2 ∫

𝑅

𝜕𝐾₁

𝜕𝑡 𝐾 𝑑𝑥

= 2 ∫

𝑅[−3𝑚𝑢²𝑢_𝑥+ 6𝑢³_𝑥+ 18𝑢_𝑥𝑢_𝑥𝑥+ 3𝑢²𝑢_𝑥𝑥𝑥] 𝐾 𝑑𝑥

= 2 ∫

𝑅[−𝑚𝜕_𝑥(𝑢³) + (𝑢³)_𝑥𝑥𝑥] 𝐾 𝑑𝑥

= ∫𝑅(𝑚𝑢³) 𝐾_𝑥− 𝑢³𝐾_𝑥𝑥𝑥𝑑𝑥

= ∫𝑅[𝑚𝑢³] 𝐾_𝑥− 𝑢³(𝑚𝐾_𝑥− 𝑢_𝑥) 𝑑𝑥

= ∫𝑅𝑢³𝑢_𝑥𝑑𝑥,

= 0.

(21) Using the Parseval identity and (21), we obtain (19) and (20).

FromTheorem 5, we know that for any𝑢₀ ∈ 𝐻^𝑠(𝑅)with 𝑠 > 3/2, there exists a maximal𝑇 = 𝑇(𝑢₀) > 0and a unique strong solution𝑢to the problem (6) such that

𝑢 ∈ 𝐶 ([0, 𝑇) ; 𝐻^𝑠(𝑅)) ∩ 𝐶¹([0, 𝑇) ; 𝐻^𝑠−1(𝑅)) . (22) Firstly, we study the following differential equation:

𝑝_𝑡= 3𝑢²(𝑡, 𝑝) , 𝑡 ∈ [0, 𝑇) ,

𝑝 (0, 𝑥) = 𝑥. (23)

Lemma 7. Let𝑢₀ ∈ 𝐻^𝑠,𝑠 > 3, and let𝑇 > 0be the maximal existence time of the solution to the problem(6). Then, the problem(23) has a unique solution𝑝 ∈ 𝐶¹([0, 𝑇) × 𝑅, 𝑅).

Moreover, the map𝑝(𝑡, ⋅)is an increasing diffeomorphism of 𝑅with𝑝_𝑥(𝑡, 𝑥) > 0for(𝑡, 𝑥) ∈ [0, 𝑇) × 𝑅.

Proof. UsingTheorem 5, we obtain𝑢 ∈ 𝐶¹([0, 𝑇); 𝐻^𝑠−1(𝑅)) and𝐻^𝑠−1 ∈ 𝐶¹(𝑅). Therefore, we know that functions𝑢(𝑡, 𝑥) and𝑢_𝑥(𝑡, 𝑥)are bounded, Lipschitz in space, and𝐶¹in time.

Using the existence and uniqueness theorem for ordinary differential equations derives that the problem (23) has a unique solution𝑝 ∈ 𝐶¹([0, 𝑇) × 𝑅, 𝑅).

Differentiating (23) with respect to 𝑥 gives rise to the following:

𝑑

𝑑𝑡𝑝_𝑥= 6𝑢𝑢_𝑥(𝑡, 𝑝) 𝑝_𝑥, 𝑡 ∈ [0, 𝑇) , 𝑝_𝑥(0, 𝑥) = 1,

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from which we obtain 𝑝_𝑥(𝑡, 𝑥) =exp(∫^𝑡

06𝑢𝑢_𝑥(𝜏, 𝑝 (𝜏, 𝑥)) 𝑑𝜏) . (25) For every𝑇^󸀠 < 𝑇, using the Sobolev imbedding theorem yields that

sup

(𝜏,𝑥)∈[^0,𝑇^󸀠)^×𝑅󵄨󵄨󵄨󵄨𝑢^𝑥(𝜏, 𝑥)󵄨󵄨󵄨󵄨 < ∞. (26) It is inferred that there exists a constant𝐾₀> 0such that 𝑝_𝑥(𝑡, 𝑥) ≥ 𝑒^−𝐾⁰^𝑡for(𝑡, 𝑥) ∈ [0, 𝑇) × 𝑅. It completes the proof.

Lemma 8. Assume that𝑢₀ ∈ 𝐻^𝑠(𝑅), 𝑠 > 3/2. Let𝑇be the maximal existence time of the solution𝑢to the problem(6).

Then, it has

‖𝑢 (𝑡, 𝑥)‖_𝐿^∞ ≤ 󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐿^∞𝑒^𝑐𝑡 ∀𝑡 ∈ [0, 𝑇] , (27) where𝑐 > 0is a constant independent of𝑡.

Proof. Let𝜉(𝑥) = (1/2)𝑒^−|𝑥|, we have(1 − 𝜕_𝑥²)⁻¹𝑔 = 𝜉 ⋆ 𝑓for all𝑔 ∈ 𝐿²(𝑅)and𝑢 = 𝜉 ⋆ 𝐾₁(𝑡, 𝑥). Using a simple density argument presented in [7], it suffices to consider𝑠 = 3 to prove this lemma. Let𝑇be the maximal existence time of the solution𝑢to the problem (6) with the initial value𝑢₀∈ 𝐻³(𝑅) such that𝑢 ∈ 𝐶([0, 𝑇), 𝐻³(𝑅)) ∩ 𝐶¹([0, 𝑇), 𝐻²(𝑅)). From (6), we have

𝑢_𝑡+ 3𝑢²𝑢_𝑥= − (𝑚 − 1) 𝜉 ⋆ (3𝑢²𝑢_𝑥) . (28) Since

−𝜉 ⋆ (3𝑢²𝑢_𝑥) = −1 2∫^∞

−∞𝑒^{−|𝑥−𝜂|}3𝑢²𝑢_𝜂𝑑𝜂

= −3 2∫^𝑥

−∞𝑒^−𝑥+𝜂𝑢²𝑢_𝜂𝑑𝜂 −3 2∫^+∞

𝑥 𝑒^𝑥−𝜂𝑢²𝑢_𝜂𝑑𝜂

=1 2∫^𝑥

∞𝑒^{−|𝑥−𝜂|}𝑢³𝑑𝜂 − 1 2∫^∞

𝑥 𝑒^{−|𝑥−𝜂|}𝑢³𝑑𝜂, 𝑑𝑢 (𝑡, 𝑝 (𝑡, 𝑥))

𝑑𝑡 = 𝑢_𝑡(𝑡, 𝑝 (𝑡, 𝑥)) + 𝑢_𝑥(𝑡, 𝑝 (𝑡, 𝑥))𝑑𝑝 (𝑡, 𝑥) 𝑑𝑡

= (𝑢_𝑡+ 3𝑢²𝑢_𝑥) (𝑡, 𝑝 (𝑡, 𝑥)) ,

(29) from (29), we have

𝑑𝑢 (𝑡, 𝑝 (𝑡, 𝑥))

𝑑𝑡 = 𝑚 − 1

2 ∫^{𝑝(𝑡,𝑥)}

−∞ 𝑒−|𝑝(𝑡,𝑥)−𝜂|𝑢³𝑑𝜂

−𝑚 − 1 2 ∫^∞

𝑝(𝑡,𝑥)𝑒−|𝑝(𝑡,𝑥)−𝜂|𝑢³𝑑𝜂.

(30)

(4)

UsingLemma 6and (30) derives that

󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󵄨

𝑑𝑢 (𝑡, 𝑝 (𝑡, 𝑥))

𝑑𝑡 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨≤ |𝑚 − 1|

2 ∫^∞

−∞𝑒−|𝑝(𝑡,𝑥)−𝜂|𝑢²𝑑𝜂

≤ |𝑚 − 1|

2 ∫^∞

−∞𝑢³𝑑𝜂

≤ |𝑚 − 1|

2 ‖𝑢‖²_𝐿2‖𝑢‖_𝐿^∞

≤ 𝑐󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐿²(𝑅)‖𝑢‖_𝐿^∞

≤ 𝑐‖𝑢‖_𝐿^∞,

(31)

where𝑐 is a positive constant independent of𝑡. Using (31) results in the following:

−𝑐 ∫^𝑡

0‖𝑢‖_𝐿^∞_(𝑅)𝑑𝑡 + 𝑢₀≤ 𝑢 (𝑡, 𝑝 (𝑡, 𝑥)) ≤ 𝑐 ∫^𝑡

0‖𝑢‖_𝐿^∞_(𝑅)𝑑𝑡 + 𝑢₀. (32) Therefore,

󵄨󵄨󵄨󵄨𝑢(𝑡,𝑝(𝑡,𝑥))󵄨󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩󵄩𝑢(𝑡,𝑝(𝑡,𝑥))󵄩󵄩󵄩󵄩𝐿^∞

≤ 󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐿^∞+ 𝑐 ∫^𝑡

0‖𝑢‖_𝐿^∞_(𝑅)𝑑𝑡. (33) Using the Sobolev embedding theorem to ensure the uniform boundedness of𝑢_𝑥(𝑠, 𝜂)for(𝑠, 𝜂) ∈ [0, 𝑡] × 𝑅with𝑡 ∈ [0, 𝑇^󸀠), fromLemma 7, for every𝑡 ∈ [0, 𝑇^󸀠), we get a constant𝐶(𝑡) such that

𝑒^{−𝐶(𝑡)}≤ 𝑝_𝑥(𝑡, 𝑥) ≤ 𝑒^𝐶(𝑡), 𝑥 ∈ 𝑅. (34) We deduce from (34) that the function 𝑝(𝑡, ⋅) is strictly increasing on𝑅with lim_{𝑥 → ±∞}𝑝(𝑡, 𝑥) = ±∞as long as𝑡 ∈ [0, 𝑇^󸀠). It follows from (33) that

‖𝑢 (𝑡, 𝑥)‖_𝐿^∞ = 󵄩󵄩󵄩󵄩𝑢 (𝑡, 𝑝 (𝑡, 𝑥))󵄩󵄩󵄩󵄩𝐿^∞ ≤ 󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩^𝐿^∞+ 𝑐 ∫^𝑡

0‖𝑢‖_𝐿^∞_(𝑅)𝑑𝑡.

(35) Using the Gronwall inequality and (35) derives that (27) holds.

For a real number𝑠with𝑠 > 0, suppose that the function 𝑢₀(𝑥)is in𝐻^𝑠(𝑅), and let𝑢_𝜀0be the convolution𝑢_𝜀0 = 𝜙_𝜀⋆ 𝑢₀ of the function𝜙_𝜀(𝑥) = 𝜀^−1/4𝜙(𝜀^−1/4𝑥)and𝑢₀such that the Fourier transform ̂𝜙of𝜙satisfies ̂𝜙 ∈ 𝐶^∞₀ ,𝜙(𝜉) ≥ 0, and̂ 𝜙(𝜉) = 1̂ for any𝜉 ∈ (−1, 1). Then, we have𝑢_𝜀0(𝑥) ∈ 𝐶^∞. It follows fromTheorem 5that for each𝜀satisfying0 < 𝜀 < 1/2, the Cauchy problem,

𝑢_𝑡− 𝑢_𝑡𝑥𝑥+ 3𝑚𝑢²𝑢_𝑥= 6𝑢³_𝑥+ 18𝑢𝑢_𝑥𝑢_𝑥𝑥+ 3𝑢²𝑢_𝑥𝑥𝑥,

𝑢 (0, 𝑥) = 𝑢𝜀0(𝑥) , (36)

has a unique solution 𝑢_𝜀(𝑡, 𝑥) ∈ 𝐶^∞([0, 𝑇); 𝐻^∞). Using Lemmas6and8, for every𝑡 ∈ [0, 𝑇), we obtain

󵄩󵄩󵄩󵄩𝑢^𝜀(𝑡, 𝑥)󵄩󵄩󵄩󵄩𝐿²(𝑅)≤ 𝑐󵄩󵄩󵄩󵄩𝑢^𝜀(0, 𝑥)󵄩󵄩󵄩󵄩𝐿²(𝑅)≤ 𝑐󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐿²(𝑅),

󵄩󵄩󵄩󵄩𝑢^𝜀(𝑡, 𝑥)󵄩󵄩󵄩󵄩𝐿^∞ ≤ 󵄩󵄩󵄩󵄩𝑢^𝜀(0, 𝑥)󵄩󵄩󵄩󵄩𝐿^∞𝑒^𝑐𝑡≤ 𝑐󵄩󵄩󵄩󵄩𝑢⁰󵄩󵄩󵄩󵄩𝐿^∞𝑒^𝑐𝑡. (37)

Sending𝑡 → 𝑇, we know that inequalities (37) are still valid.

This means that for𝑡 ∈ [0, ∞), (37) hold.

Now, we state the concepts of weak solutions.

Definition 9(weak solution). We call a function𝑢 : 𝑅₊×𝑅 → 𝑅a weak solution of the Cauchy problem (5) provided that

(i)𝑢 ∈ 𝐿^∞(𝑅₊; 𝐿²(𝑅));

(ii)𝑢_𝑡− 𝑢_𝑡𝑥𝑥+ 3𝑚𝑢²𝑢_𝑥 = 6𝑢³_𝑥+ 18𝑢𝑢_𝑥𝑢_𝑥𝑥+ 3𝑢²𝑢_𝑥𝑥𝑥in 𝐷^󸀠([0, ∞) × 𝑅), that is, for all𝜑 ∈ 𝐶^∞₀ ([0, ∞) × 𝑅) there holds the following identity:

∫𝑅⁺∫

𝑅(𝑢 (𝜑_𝑡− 𝜑_𝑡𝑥𝑥) + 𝑚𝑢³𝜑_𝑥− 𝑢³𝜑_𝑥𝑥𝑥) 𝑑𝑥 𝑑𝑡 + ∫𝑅𝑢₀(𝑥) 𝜑 (0, 𝑥) 𝑑𝑥 = 0.

(38)

Theorem 10. Let𝑢₀(𝑥) ∈ 𝐿²(𝑅). Then, there exists a weak solution𝑢(𝑡, 𝑥) ∈ 𝐿^∞([0, ∞); 𝐿²(𝑅))to the problem(5).

Proof. Consider the problem (36). For an arbitrary𝑇 > 0, choosing a subsequence𝜀_𝑛 → 0, from (37), we know that 𝑢_𝜀_𝑛 is bounded in𝐿^∞ and‖𝑢_𝜀_𝑛‖_𝐿2(𝑅) is uniformly bounded in𝐿²(𝑅). Therefore, we obtain that𝑢³_𝜀_𝑛is bounded in𝐿²(𝑅).

Therefore, there exist subsequences {𝑢_𝜀_𝑛} and {𝑢³_𝜀_𝑛}, still denoted by {𝑢_𝜀_𝑛} and {𝑢³_𝜀_𝑛}, are weakly convergent toV in 𝐿²(𝑅). Noticing (38) completes the proof.

Acknowledgment

This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).

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1.Introduction MengWu TheLocalStrongSolutionsandGlobalWeakSolutionsforaNonlinearEquation ResearchArticle

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The Local Strong Solutions and Global Weak Solutions for a Nonlinear Equation

Meng Wu

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3. Weak Solutions

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