Initial and Boundary Value Problems for a Class of Nonlinear Metaparabolic Equations

This paper is devoted to the initial and boundary value problems for a class of nonlinear metaparabolic equations ut − βuxx − k uxxt + γuxxxx = f ðuxÞx . At low initial energy level (Jðu0Þ < d), we not only prove the existence of global weak solutions for these problems by the combination of the Galerkin approximation and potential well methods but also obtain the finite time blow-up result by adopting the potential well and improved concavity skills. Finally, we also discussed the finite time blow-up phenomenon for certain solutions of these problems with high initial energy.


Introduction
In this paper, we study the initial and boundary value problems for the following nonlinear metaparabolic equations in a bounded domain Ω = ð0, 1Þ, where u 0 ðxÞ is the initial value function defined on Ω, k > 0 is the viscosity coefficient, γ > 0 is the interfacial energy parameter, and the nonlinear smooth function f ðsÞ satisfies the following assumptions: i ð Þ f s ð Þ j j≤ α s j j q , α > 0, 1 < q < +∞, ∀s ∈ R, Equation (1) is a typical higher-order metaparabolic equation [1,2], which has extensive physical background and rich theoretical connotation. This type of equation can be regarded as the regularization of Sobolev-Galpern equation by adding a fourth-order term u xxxx . The Sobolev-Galpern equation appear in the study of various problems of fluid mechanics, solid mechanics, and heat conduction theory [3][4][5]. There have been many outstanding results about the qualitative theory for Sobolev-Galpern which include the existence, nonexistence, asymptotic behavior, regularities, and other some special properties of solutions. We also refer the reader to see [6,7] and the papers cited therein. In (1), u is the concentration of one of the two phases, the fourth-order term γu xxxx denotes the capillaritydriven surface diffusion, and the nonlinear term f ðu x Þ x is an intrinsic chemical potential. For example, differentiating (1) with respect to x and taking v = u x , β = 0, then Equation (1) reduces to the well-known viscous Cahn-Hilliard equation Equation (5) appears in the dynamics of viscous firstorder phase transitions in cooling binary solutions such as glasses, alloys, and polymer mixtures [8][9][10]. On the other hand, Equation (5) appears in the study of the regularization of nonclassical diffusion equations by adding a fourth-order term v xxxx . There have been many outstanding results about the qualitative theory for this type of equations [11][12][13][14][15]. For example, Liu and Yin [13] studied Equation (5) for φðvÞ = −v + γ 1 v 2 + γ 2 v 3 in R 3 ; they proved the existence and nonexistence of global classical solutions and pointed out that the sign of γ 2 is crucial to the global existence of solutions. In [14], Grinfeld and Novick-Cohen studied a Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn-Hilliard equation by explicit energy calculations. They also proved a partial picture of the variation in the structure of the attractor (n = 1 ) for the viscous Cahn-Hilliard equation as the mass constraint and homotopy parameter are varied. Zhao and Liu [15] considered the initial boundary problem for the viscous Cahn-Hilliard Equation (5). In their paper, the optimal control under boundary condition was given, and the existence of optimal solution was proved.
Let us mention that there is an abounding literature about the initial and boundary value problems or Cauchy problem to nonlinear parabolic and hyperbolic equations. We refer the reader to the monographs [16,17] which devoted to the second-order parabolic and pseudoparabolic problems. For the fourth-order nonlinear parabolic and hyperbolic equations, there are also some results about the initial boundary value and Cauchy problems, especially on global existence/nonexistence, uniqueness/nonuniqueness, and asymptotic behavior [18][19][20][21][22][23][24][25]. Bakiyevich and Shadrin [21] studied the Cauchy problem of the metaparabolic equation where α > 0, β ≥ 0, and γ > 0 are constants. They proved that the solutions are expressed through the sum of convolutions of functions φðxÞ and f ðt, xÞ with corresponding fundamental solutions.
In [22], Liu considered the metaparabolic equation where AðuÞ = Ð u 0 aðsÞds, a 0 + a 1 jsj b ≤ aðsÞ, and ja″ðsÞj ≤ a 2 jsj b (a 0 , a 1 , a 2 , and b are positive constants). He proved the existence of weak solutions by using the method of continuity.
Khudaverdiyev and Farhadova [23] discussed the following fourth-order semilinear pseudoparabolic equation where α > 0 is a fixed number. They proved the existence in large theorem (i.e., true for sufficiently large values of T) for generalized solution by means of Schauder stronger fixedpoint principle.
In [24], Zhao and Xuan studied the generalized BBM-Burgers equation They obtained the existence and convergence behavior of the global smooth solutions for Equation (9).
Philippin [25] studied the following fourth-order parabolic equation where k i , i = 1, 2, 3 are positive constants or in general positive derivable functions of time t. Under appropriate assumptions on the data, he proved that the solutions u cannot exist for all time, and an upper bound is derived.
Equation (1) is also closely connected with many equations [26][27][28][29]. For example, Yang [26] considered the initial and boundary value problems of the following equation He studied the asymptotic property of the solution and gave some sufficient conditions of the blow-up. When the weak damping term u t of Equation (11) is replaced by the strong damping term −u xxt , we have the following fourthorder wave equation Chen and Lu [27] studied the initial and boundary value problems of Equation (12). They proved the existence and uniqueness of the global generalized solution and global classical solution by the Galerkin method. Furthermore, Xu et al. [28] considered the initial and boundary value problems and proved the global existence and nonexistence of solutions by adopting and modifying the so called concavity method under some conditions with low initial energy. Ali Khelghati and Khadijeh Baghaei [29] proved that the blow up for Equation (12) occurs in finite time for arbitrary positive initial energy.
Motivated by the above researches, in the present work, we mainly study the initial and boundary value problems (1)-(3) of metaparabolic equations. Hereafter, for simplicity, we set α = β = γ = 1. Especially, the appearance of the dispersion term u xxt and nonlinearity f ðu x Þ x for these problems cause some difficulties such that we cannot apply the normal Galerkin approximation, concavity, and potential methods directly; we have to invent some new skills and methods to overcome these difficulties.
Our paper is organized as follows. In Section 2, we introduce some functionals and potential wells and discuss the invariance of some sets which are needed for our work. In Sections 3 and 4, the existence and nonexistence of global weak solutions for problems (1)-(3) are proved by the Galerkin approximation and potential well and improved concavity methods at low initial energy (Jðu 0 Þ < d). Especially, the threshold result between global existence and nonexistence 2 Advances in Mathematical Physics is obtained under certain conditions. In the last section, we investigate the finite time blow-up for certain solutions of problems (1)-(3) with high initial energy.

Preliminaries
In this section, we introduce some functionals, potential wells, and important lemmas that will be needed in this paper. Throughout this paper, the following abbreviations are used for precise statement: And the notation ð·, · Þ for the L 2 -inner product will also be used for the notation of duality paring between dual spaces.
First of all, let us consider the functionals as follows. The "total energy" and "potential energy" associated with the problems (1) Then, by simple calculation, it follows that The corresponding "Nehari manifold" and "potential well depth" are given by In addition, we define To obtain the results of this paper, we also introduce so called stable and unstable sets: Next, we shall give the following some essential lemmas which are important to obtain the main results of this paper.

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Proof. For any u ∈ N, we have by Lemma 1 (3) that ∥u x ∥ H 1 ≥ γ 0 . Hence, from and the definition of potential depth d, we get d ≥ d 0 .
For simplicity, we define the weak solution of (1)-(3) over the interval Ω × ½0, TÞ, but it is to be understood that T is either infinity or the limit of the existence interval.
(iii) The following energy inequality holds for any 0 ≤ t < T.
Proof. The proof of Lemma 5 is similar to Lemma 4.
Lemma 6 (see [29,30]). Assume that the function ϕðtÞ ∈ C 2 , ϕðtÞ ≥ 0 satisfies for certain real number δ > 0, ϕð0Þ > 0, and ϕ ′ ð0Þ > 0. Then, there exists a real numberT with 0 <T ≤ ðϕð0ÞÞ/ðαϕ ′ ð0ÞÞ such that We construct an approximate weak solution of the problems (1)-(3) by the Galerkin¡¯s method. Let fw j g ∞ j=1 be the eigenfunction system of problem Obviously, there exist some basis such that fw j g ∞ j=1 ⊆ D ∩ W 1,q+1 ðΩÞ, and it is dense in D ∩ W 1,q+1 ðΩÞ. Now, suppose that the approximate weak solution of the problems (1)-(3) can be written According to Galerkin's method, these coefficients d j m ðtÞ need to satisfy the following initial value problem of the nonlinear differential equations where u 0m ðxÞ = ∑ m j=1 d j m ð0Þw j ðxÞ, u 0m ðxÞ ⟶ u 0 ðxÞ, in D ∩ W 1,q+1 ðΩÞ.
The initial value problem (32) possesses a local solution in ½0, t m Þ, 0 < t m < T for an arbitrary T > 0. Under some appropriate assumptions on the nonlinear terms and the initial data, we prove that the system (32) has global weak solutions in the interval ½0, T. Furthermore, we show that the solutions of the problems (1)-(3) can be approximated by the functions u m ðx, tÞ.

Existence of Global Weak Solutions
In this section, we shall prove the existence of global weak solution by the combination of the Galerkin approximation and potential well methods.

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Proof. Multiplying (32) by d j m ðtÞ ′ and summing for j = 1, ⋯, m, then we have By a direct calculation, it follows that where Utilizing the strong convergence of u 0m in D ∩ W 1,q+1 ðΩÞ, we note that Jðu 0m Þ ⟶ Jðu 0 Þ < d. Hence, we get Jðu 0m Þ < d for sufficiently large m. On the other hand, from u 0 ∈ W and u 0m ðxÞ ⟶ u 0 ðxÞ in D ∩ W 1,q+1 ðΩÞ, it follows that u 0m ∈ W for sufficiently large m. Similar to the proof of Lemma 4, we have that the solution u m constructed by (31) remains in W for 0 ≤ t < ∞ and sufficiently large m.

Conclusion and Future Work
In our work, we mainly study the qualitative properties of the solutions for the initial and boundary value problems (1)-(3). It is well known that Equation (1) is a typical higher-order metaparabolic equation, which has extensive practical background and rich theoretical connotation. For example, the solutions u of (1) can be used to denote the concentration of one of the two phases, the fourth-order term γu xxxx presents the capillarity-driven surface diffusion, and the nonlinear term f ðu x Þ x is an intrinsic chemical potential. Especially, the interaction between the dispersion term u xxt and nonlinearity f ðu x Þ x of these problems cause some difficulties such that we cannot apply the normal Galerkin approximation, concavity, and potential methods directly. Considering the above situation, at low initial energy level, we first prove the existence of global weak solutions for these problems by the Galerkin approximation and potential well methods and obtain the finite time blow-up result by the potential well and improved concavity skills. In addition, we establish the 8 Advances in Mathematical Physics finite time blow-up result for certain solutions with high initial energy. However, as far as we know, there is little information on the long-time behavior of global solutions for above problems. Whether the global solutions will exhibit a long-time dynamic behavior at low initial energy? Do both problems (1)-(3) have the global solutions and asymptotic property at high initial energy level? These questions are all opening, and we are now working on these problems. On the other hand, we note that the fractional partial differential equations have been applied in various areas of science, and their related theoretical results and applications have been investigated by some authors (see [31][32][33] and the references therein). The study of their qualitative properties is one of the hot topics. Do the conclusions of present paper also hold for the initial and boundary value problems of the fractional nonlinear metaparabolic equations? This question is very interesting and opening.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no competing interests.