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

Authors: Huafei Di; Zefang Song
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

This paper is devoted to the initial and boundary value problems for a class of nonlinear metaparabolic equationsut−βuxx−kuxxt+γuxxxx=fuxx. At low initial energy level (Ju0<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.


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## Body

## 1. Introduction

In this paper, we study the initial and boundary value problems for the following nonlinear metaparabolic equations(1)ut−βuxx−kuxxt+γuxxxx=fuxx,x∈Ω,t>0,(2)ux,0=u0x,x∈Ω,(3)u0,t=u1,t=0,uxx0,t=uxx1,t=0,t≥0,in a bounded domain Ω=0,1, where u0x 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 fs satisfies the following assumptions:
(4)ifs≤αsq,α>0,1<q<+∞,∀s∈R,iip+1Fs≥sfsforsomep>1,∀s∈R,Fs=∫0sfτdτ.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 uxxxx. The Sobolev-Galpern equation appear in the study of various problems of fluid mechanics, solid mechanics, and heat conduction theory [3–5]. 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 γuxxxx denotes the capillarity-driven surface diffusion, and the nonlinear term fuxx is an intrinsic chemical potential. For example, differentiating (1) with respect to x and taking v=ux, β=0, then Equation (1) reduces to the well-known viscous Cahn-Hilliard equation
(5)vt−vxxt+vxxxx=φvxx,x∈Ω,t>0.Equation (5) appears in the dynamics of viscous first-order phase transitions in cooling binary solutions such as glasses, alloys, and polymer mixtures [8–10]. On the other hand, Equation (5) appears in the study of the regularization of nonclassical diffusion equations by adding a fourth-order term vxxxx. There have been many outstanding results about the qualitative theory for this type of equations [11–15]. For example, Liu and Yin [13] studied Equation (5) for φv=−v+γ1v2+γ2v3 in R3; they proved the existence and nonexistence of global classical solutions and pointed out that the sign of γ2 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 [18–25]. Bakiyevich and Shadrin [21] studied the Cauchy problem of the metaparabolic equation
(6)ut−αuxx−γuxxt+βuxxxx=ft,x,x∈ℝ,t>0,ux,0=φx,x∈ℝ,where α>0, β≥0, and γ>0 are constants. They proved that the solutions are expressed through the sum of convolutions of functions φx and ft,x with corresponding fundamental solutions.In [22], Liu considered the metaparabolic equation
(7)ut−kuxxt+Auxxxx=fx,t,0≤x≤1,0≤t≤T<+∞,where Au=∫0uasds, a0+a1sb≤as, and a″s≤a2sb(a0,a1,a2, 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
(8)ut−αuxxt+uxxxx=ft,x,u,ux,uxx,uxxx,0≤x≤1,0≤t≤T<+∞,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 fixed-point principle.In [24], Zhao and Xuan studied the generalized BBM-Burgers equation
(9)ut−αuxx−γuxxt+βuxxxx+fux=0,x∈ℝ,t>0.They obtained the existence and convergence behavior of the global smooth solutions for Equation (9).Philippin [25] studied the following fourth-order parabolic equation
(10)ut−k1tΔu+k2tΔ2u=k3tuup−1,x∈Ω,t>0,where ki, 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 [26–29]. For example, Yang [26] considered the initial and boundary value problems of the following equation
(11)utt+λut+αuxxxx=fuxx,x∈0,1,t>0.He studied the asymptotic property of the solution and gave some sufficient conditions of the blow-up. When the weak damping termut of Equation (11) is replaced by the strong damping term −uxxt, we have the following fourth-order wave equation
(12)utt−2buxxt+αuxxxx=fuxx,x∈0,1,t>0.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 uxxt and nonlinearity fuxx 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 Section2, 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 (Ju0<d). Especially, the threshold result between global existence and nonexistence 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.


## 2. 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:(13)LqΩ=Lq,Wm,qΩ=Wm,q,HmΩ=Wm,2Ω=Hm,H0mΩ=H0m,D=u∈H2Ωu0,t=u1,t=0,uxx0,t=uxx1,t=0,uLqΩ=uq,uWm,qΩ=uWm,q,uH0mΩ=uH0m.And the notation·,· for the L2-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)–(3) are defined by
(14)Et≔Eut=∫0tutτH12dτ+12uxH12+∫ΩFuxdx,Ju=12uxH12+∫ΩFuxdx,Iu=uxH12+∫Ωfuxuxdx.Then, by simple calculation, it follows that(15)J′u=ddtJu=−uttH12≤0,(16)Et=Ju+∫0tutτH12dτ=Ju0=E0.The corresponding “Nehari manifold” and “potential well depth” are given by(17)N=u∈D∩W1,q+1ΩIu=0,u≠0,d=infu∈NJu.In addition, we define(18)N+=u∈D∩W1,q+1ΩIu>0∪0,N−=u∈D∩W1,q+1ΩIu<0.To obtain the results of this paper, we also introduce so called stable and unstable sets:(19)W=u∈D∩W1,q+1ΩIu>0,Ju<d∪0,V=u∈D∩W1,q+1ΩIu<0,Ju<d.Next, we shall give the following some essential lemmas which are important to obtain the main results of this paper.Lemma 1.
Letfs satisfy (4), u∈D∩W1,q+1Ω, then the following hold:
(1)
If0<uxH1<γ0, then u∈N+(u≠0);(2)
Ifu∈N−, then uxH1>γ0(3)
Ifu∈N, then uxH1≥γ0, where
(20)γ0=1αC∗q+11/q−1,C∗=supuxq+1uxH1,u∈D∩W1,q+1Ω,u≠0.Proof.
(1)
If0<uxH1<γ0, then
(21)−∫Ωfuxuxdx≤∫Ωfuxuxdx≤α∫Ωuxq+1dx≤αC∗q+1uxH1q+1=αC∗q+1uxH1q−1uxH12<uxH12,which gives Iu>0 or u∈N+(u≠0).(2)
Ifu∈N−, then u≠0 and
(22)uxH12<−∫Ωfuxuxdx≤αC∗q+1uxH1q−1uxH12,which gives ∥ux∥H1>γ0.(3)
Ifu∈N, then from
(23)∥ux∥H12=−∫Ωfuxuxdx≤αC∗q+1∥ux∥H1q−1∥ux∥H12,we have ∥ux∥H1≥γ0.Lemma 2.
Letfs satisfy (4) and u∈N, then
(24)d≥d0=p−12p+1γ02=p−12p+11αC∗q+12/q−1.Proof.
For anyu∈N, we have by Lemma 1 (3) that ∥ux∥H1≥γ0. Hence, from
(25)Ju=12uxH12+∫ΩFuxdx≥12uxH12+1p+1∫Ωfuxuxdx=p−12p+1uxH12+1p+1Iu=p−12p+1uxH12≥p−12p+1γ02,and the definition of potential depth d, we get d≥d0.For simplicity, we define the weak solution of (1)–(3) over the interval Ω×0,T, but it is to be understood that T is either infinity or the limit of the existence interval.Definition 3.
We say thatux,t is called a weak solution of the problems (1)–(3) on the interval Ω×0,T. If u∈L∞0,T;D∩W1,q+1Ω, with ut∈L20,T;H01Ω satisfy the following conditions
(i)
For anyv∈D∩W1,q+1Ω, such that
(26)ut,v+uxt,vx+ux,vx+uxx,vxx=−fux,vx.(ii)
ux,0=u0 in D∩W1,q+1Ω.(iii)
The following energy inequality holds(27)Ju≤Ju0,for any 0≤t<T.Lemma 4.
Letfs satisfy (4) and ux,t be a solution of (1)–(3) over the interval 0,T. If there exists a time t0∈0,T such that ut0∈W, then ut∈W for any t∈t0,T, where T is either infinity or the limit of the existence interval.Proof.
Arguing by contradiction and considering the time continuity ofIu and Ju, we suppose that there exists a time t1∈t0,T such that ux,t∈W for any t∈t0,t1, but ut1∈∂W, which means that (1) Jut1=d or (2) Iut1=0, ut1H2≠0. By (15) and ut0∈W, we have Jut1≤Jut0<d. It follows that case (1) is impossible. If Iut1=0, ut1H2≠0, then by the definition of d, we have Jut1>d which contradicts Jut1≤Jut0<d. The case (2) is also impossible.Lemma 5.
Letfs satisfy (4) and ux,t be a solution of (1)–(3) over the interval 0,T. If there exists a time t0∈0,T such that ut0∈V, then ut∈V for any t∈t0,T, where T is either infinity or the limit of the existence interval.Proof.
The proof of Lemma5 is similar to Lemma 4.Lemma 6 (see [29, 30]).
Assume that the functionϕt∈C2, ϕt≥0 satisfies
(28)ϕtϕ″t−1+δϕ′2t≥0,for certain real number δ>0, ϕ0>0, and ϕ′0>0. Then, there exists a real number T~ with 0<T~≤ϕ0/αϕ′0 such that
(29)ϕt⟶∞,ast⟶T~−.We construct an approximate weak solution of the problems (1)-(3) by the Galerkin¡¯s method. Let wjj=1∞ be the eigenfunction system of problem
(30)−wjxx=λwj,inΩ,wj0=wj1=0,j=1,2,⋯.Obviously, there exist some basis such thatwjj=1∞⊆D∩W1,q+1Ω, and it is dense in D∩W1,q+1Ω. Now, suppose that the approximate weak solution of the problems (1)–(3) can be written
(31)umx,t=∑j=1mdmjtwjx.According to Galerkin’s method, these coefficientsdmjt need to satisfy the following initial value problem of the nonlinear differential equations
(32)umt,wj+umxt,wjx+umx,wjx+umxx,wjxx=−fumx,wjx,umt=0=u0mx,where u0mx=∑j=1mdmj0wjx, u0mx⟶u0x, in D∩W1,q+1Ω.The initial value problem (32) possesses a local solution in 0,tm, 0<tm<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 0,T. Furthermore, we show that the solutions of the problems (1)–(3) can be approximated by the functions umx,t.


## 3. 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.Theorem 7.
Assume thatf satisfy (4), and u0∈W, then the problems (1)–(3) admits a global weak solution u∈L∞0,∞;D∩W1,q+1Ω, with ut∈L20,∞;H01Ω and u∈W for all 0≤t<∞.Proof.
Multiplying (32) by dmjt′ and summing for j=1,⋯,m, then we have
(33)umt,umt+umx,umxt+umxt,umxt+umxx,umxxt=−fumx,umxt.By a direct calculation, it follows that(34)∫0tumτH12dτ+Jum=Ju0m,where
(35)Jum=12umxH12+∫ΩFumxdx.Utilizing the strong convergence ofu0m in D∩W1,q+1Ω, we note that Ju0m⟶Ju0<d. Hence, we get Ju0m<d for sufficiently large m. On the other hand, from u0∈W and u0mx⟶u0x in D∩W1,q+1Ω, it follows that u0m∈W for sufficiently large m. Similar to the proof of Lemma 4, we have that the solution um constructed by (31) remains in W for 0≤t<∞ and sufficiently large m.Thus, from (4) and
(36)d>Jum=12umxH12+∫ΩFumxdx≥12umxH12+1p+1∫Ωfumxumxdx≥p−12p+1umxH12+1p+1Ium≥0,we obtain
(37)∫0tumτH12dτ<d,0≤t<∞,umxH12<2p+1p−1d,0≤t<∞,umxq+12≤C∗2umxH12<C∗22p+1p−1d,0≤t<∞,fumxrr≤αrumxr+1r+1<αrC∗r+12p+1p−1dr+1/2,0≤t<∞,where r=q+1/q, 1/q+1+1/r=1. Therefore, there exist a subsequence of um which from now on will be also denoted by um such that as m⟶∞(38)um⟶uinL∞0,∞;D∩W1,q+1Ωweaklystar,(39)um⟶ua.e.Q=Ω×0,∞,(40)umt⟶utinL20,∞;H01Ωweakly,(41)umx⟶uxinL∞0,∞;Lq+1Ωweaklystar,(42)fumx⟶χ=fuxinL∞0,∞;LrΩweaklystar.Convergences (38)–(42) permit us to pass to the limit in (32). Taking m⟶∞, we obtain
(43)ut,wj+uxt,wjx+ux,wjx+uxx,wjxx=−fux,wjx,forj=1,2⋯. Considering that the basis wjj=1∞ are dense in D∩W1,q+1Ω, we choose a function v∈L∞0,∞;D∩W1,q+1Ω having the form vt=∑j=1∞djtwj, where djtj=1∞ are given functions. Multiplying (43) by djt and summing j=1,2,⋯, then we have
(44)ut,v+uxt,vx+ux,vx+uxx,vxx=−fux,vx.Moreover, (32) gives ux,0=u0x in D∩W1,q+1Ω. Next, we will prove that u satisfies (27). Taking into account the nonlinear term of the functional Ju, we deduce
(45)∫ΩFumxdx−∫ΩFuxdx=∫Ωfux+θmumxumx−uxdx≤∥fux+θmumx∥q+1/q∥umx−u∥q+1≤C∥umx−ux∥q+1⟶0,asm⟶∞, where 0<θm<1. Hence, we have
(46)limm⟶∞∫ΩFumxdx=∫ΩFuxdx.Then, making use of Fatou’s Lemma and (34), (46), we deduce
(47)12∥ux∥H12≤liminfm⟶∞12∥umx∥H12=liminfm⟶∞Jum−∫ΩFumxdx≤limm⟶∞Ju0m−∫ΩFumxdx=Ju0−∫ΩFuxdx,which yields (27). Thus, we obtain that u is a global weak solution of problems (1)–(3). Finally, making use of Lemma 4 again, we get ut∈W for 0≤t<∞.


## 4. Finite Time Blow-up of Solutions withJu0<d0

In this section, we consider the finite time blow up of solutions withE0=Ju0<d0 for the problems (1)–(3).Theorem 8.
Letf satisfy (4), and u0∈D∩W1,q+1Ω. Assume that Iu0<0 and E0=Ju0<d0, where d0 is defined in Lemma 2, then the weak solution ut of problems (1)–(3) blow-up in finite time.Proof.
Letut be any weak solution of the problems (1)–(3) with Iu0<0 and E0=Ju0<d0, T~ be the maximal existence time of ut. Next, we will prove T~<+∞. Arguing by contradiction, we suppose T~=+∞. We define the function Ψ:0,T1⟶R+ by
(48)Ψt=∫0t∥uτ∥H12dτ+T1−t∥u0∥H12+bt+T02,
whereb,T0, and T1 are positive constants to be chosen later. By simple calculation, we have
(49)Ψ′t=∥ut∥H12−∥u0∥H12+2bt+T0=2∫0t∫Ωuτutτdxdτ+2∫0t∫Ωuxτuxtτdxdτ+2bt+T0,Ψ″t=2∫Ωututtdx+2∫Ωuxtuxttdx+2b.By (1), we obtain
(50)Ψ″t=2∫Ωuuxx−uxxxx+fuxxdx+2b=−2∥ux∥H12−2∫Ωuxfuxdx+2b.Therefore, we can get(51)ΨtΨ″t−p+34Ψ′t2=Ψt−2∥ux∥H12−2∫Ωuxfuxdx+2b−p+3∫0t∫Ωuτutτdxdτ+∫0t∫Ωuxτuxtτdxdτ+bt+T02=Ψt−2∥ux∥H12−2∫Ωuxfuxdx+2b+p+3ηt−Ψt−T1−t∥u0∥H12∫0t∥utτ∥H12dτ+b,where
(52)ηt=∫0t∥uτ∥H12dτ+bt+T02∫0t∥utτ∥H12dτ+b−∫0t∫Ωuτutτdxdτ+∫0t∫Ωuxτuxtτdxdτ+bt+T02.Using the Schwarz and Young inequalities, we have(53)∫0tuτ,utτdτ2≤∫0t∥uτ∥22dτ∫0t∥utτ∥22dτ,(54)∫0tuxτ,uxtτdτ2≤∫0t∥uxτ∥22dτ∫0t∥uxtτ∥22dτ,(55)∫0tuτ,utτdτ∫0tuxτ,uxtτdτ≤12∫0tuτ22dτ∫0tuxtτ22dτ+12∫0t∥ut∥22dτ∫0t∥∇u∥22dτ.Inserting (53)–(55) into (52), we have
(56)ηt≥0,t∈0,T1.Thus,(57)Ψ″tΨt−p+34Ψ′t2≥Ψtξt,where
(58)ξt=−2∥ux∥H12−2∫Ωuxfuxdx+2b−p+3∫0t∥utτ∥H12dτ+b≥−p+3∫0t∥utτ∥H12dτ−p+1b−2p+1∫ΩFuxdx−2∥ux∥H12=−2p+1Et−12∥ux∥H12−∫0t∥utτ∥H1dτ−p+3∫0t∥utτ∥H12dτ−p+1b−2∥ux∥H12≥−2p+1E0+p−1∥ux∥H12+p−1∫0t∥utτ∥H12dτ−p+1b.FromIu0<0, E0=Ju0<d0 and Lemma 5, we have Iu<0 for all t∈0,∞. Hence, by Lemma 1, it follows that ∥ux∥H12>γ02=2p+1/p−1d0=1/αC∗p+12/p−1. Thus, we have from (58) that
(59)ξt>2p+1d0−Ju0−p+1b.We chooseb small enough such that b≤2d0−Ju0, then we have
(60)Ψ″tΨt−p+34Ψ′t2≥Ψtξt>0,for allt∈0,T1. From Iu<0 for all t∈0,T1 and (50), we get Ψ″t>0. Hence, we have Ψ′t>Ψ′0=2bT0>0 for all t∈0,T1.From what has been discussed above, using Lemma6 and letting δ=p−1/4, we can obtain that there exists a finite time T~>0 such that
(61)limt⟶T~−Ψt=+∞,or
(62)limt⟶T~−∫0t∥uτ∥H12dτ+T1−t∥u0∥H12+bt+T02=+∞,where
(63)T~≤Ψ0γΨ′0=2T1∥u0∥H12+bT02p−1bT0.which contradicts T~=+∞. Hence, the desired assertion immediately follows.From the discussed above in Sections3 and 4, a threshold result of global existence and nonexistence of solutions for problems (1)–(3) has been obtained as follows.Corollary 9.
Letf satisfy (4), and u0∈D∩W1,q+1Ω. Assume that E0=Ju0<d0. Then, problems (1)–(3) admits a global weak solution provided Iu0≥0 (includes u0=0); Problems (1)–(3) dose not admit any global solution provided Iu0<0.


## 5. Finite Time Blow-up of Solutions with High Initial Energy

In this section, we shall state and prove the finite time blow-up result with high initial energy for the problems (1)–(3).Theorem 10.
Letf satisfy (4), and u0∈D∩W1,q+1Ω. Assume that
(64)0≤Ju0≤p−1λ12p+11+λ1∥u0∥H12,where λ1 is the optimal constant satisfying the Poincáre inequality ∥ux∥22≥λ1∥u∥22, then the weak solution ut of problems (1)–(3) blow-up in finite time.Proof.
Arguing by contradiction, we suppose thatut is a global weak solution of the problems (1)–(3). Considering that
(65)∫0tutτdτ=ut−u0,∀t∈0,∞,so we have
(66)∫0t∥utτ∥H1dτ≥∥∫0tutτdτ∥H1=∥ut−u0∥H1≥∥ut∥H1−∥u0∥H1.From (16), (66) and Hölder’s inequality, we obtain
(67)∥ut∥H1≤∥u0∥H1+∫0t∥utτ∥H1dτ≤∥u0∥H1+t1/2∫0t∥utτ∥H12dτ1/2≤∥u0∥H1+t1/2Ju0−Jut1/2.Since assume thatut is a global weak solution of the problems (1)–(3), we get Jut≥0 for all t∈0,∞. Otherwise, there exists a time t0∈0,∞ such that Jut0<0. Hence, from
(68)0>Jut0=12∥uxt0∥H12+∫ΩFuxt0dx≥12∥uxt0∥H12+1p+1∫Ωfuxt0uxt0dx≥p−12p+1∥uxt0∥H12+1p+1Iut0,we have Iut0<0 and Jut0<0 which implies that ut0∈V. Therefore, by the results of Theorem 8, we obtain that ux,t;ut0=ux,t−t0;u0 blows up in finite time, which is a contradiction. Thus, we have
(69)Ju0≥Jut≥0,forallt∈0,∞.Next, combining (67) and (69), we get
(70)∥ut∥H1≤∥u0∥H1+t1/2Ju0−Jut1/2≤∥u0∥H1+t1/2Ju01/2,for allt∈0,∞.On the other hand, multiplyingu on two sides of Equation (1) and integrating by parts, we have
(71)ut,u+uxt,ux+ux,ux+uxx,uxx=−fux,ux.The Poinca’re inequality gives∥ux∥22≥λ1∥u∥22, where λ1 is the first eigenvalue of the problem
(72)wxx+λw=0,inΩ,w=0,on∂Ω.Thus, we have(73)∥u∥H12=∥u∥22+∥ux∥22≤1+λ1λ1∥ux∥22≤1+λ1λ1∥ux∥H12.By the combination of (4), (73), and Sobolev’s inequality, we can get that
(74)ddt12∥u∥H12=−∥ux∥H12−fux,ux≥−∥ux∥H12−p+1∫ΩFuxdx=p−12∥ux∥H12−p+1Jut≥p−1λ121+λ1∥u∥H12−p+1Jut.Sinced/dtJut≤0, for ∀k>0, we have
(75)ddt12∥u∥H12−kJut≥ddt12∥u∥H12≥p−1λ121+λ1∥u∥H12−p+1Jut,=p−1λ11+λ112∥u∥H12−p+11+λ1p−1λ1Jut.Takingk=p+11+λ1/p−1λ1 in (75) and Gt=1/2∥u∥H12−p+11+λ1/p−1λ1Jut, then we have
(76)ddtGt≥p−1λ11+λ1Gt.Integrating the inequality (76) from 0 to t, we see
(77)Gt≥ep−1λ1/1+λ1tG0,t∈0,∞,which means that
(78)∥u∥H12≥2p+1λ1p−11+λ1Jut+2ep−1λ1/1+λ1tG0,t∈0,∞.From the assumption condition (64), we have G0>0. Hence, we get from (69) and (78) that ∥u∥H12≥2ep−1λ1/1+λ1tG0, i.e.,
(79)∥u∥H1≥2G01/2ep−1λ1/21+λ1t,t∈0,∞.From the combination of (70) and (79), we have
(80)2G01/2ep−1λ1/21+λ1t≤∥u0∥H1+t1/2Ju01/2.Clearly, the above inequality cannot hold fort large enough, this means that the solution u of problems (1)–(3) cannot exist all time.Furthermore, by (80) and ep−1λ1/21+λ1t≥p−1λ1/21+λ1t, we can obtain the inequality
(81)2G0p−1λ121+λ1t−Ju01/2t1/2−∥u0∥H1≤0,which implies that there exists a finite time T~>0 such that
(82)limt⟶T~−∥u∥H12=+∞,and T~1/2 is the largest root of the following equation
(83)2G0p−1λ121+λ1t2−Ju01/2t−∥u0∥H1=0.This completes the proof.


## 6. 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 γuxxxx presents the capillarity-driven surface diffusion, and the nonlinear term fuxx is an intrinsic chemical potential. Especially, the interaction between the dispersion term uxxt and nonlinearity fuxx 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 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 [31–33] 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.



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