# Blow-Up of Solutions for Wave Equation Involving the Fractional Laplacian with Nonlinear Source

Authors: Y. Bidi; A. Beniani; M. Y. Alnegga; A. Moumen
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, we study the blow-up of solutions for wave equation involving the fractional Laplacian with nonlinear source.


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## Body

## 1. Introduction and Brief History of Fractional Integrodifferentiation

LetΩ⊂ℝn,n≥1 be an open domain with Lipschitz boundary ∂Ω. In this article, we consider the hyperbolic initial-boundary value problem involving the fractional Laplacian; for w=wx,t, we consider the wave equation with power nonlinearity:(1)∂t2w+−Δrw+−Δr∂tw=wwp−2,x∈Ω,t>0,w=0,x∈∂Ω,t>0,wx,0=w0x,∂twx,0=w1x,x∈Ω,where −Δr is the fractional Laplacian such that r∈0,1. The exponent p satisfies(2)2<p≤2nn−2r=2r∗,n>2r.The fractional integrodifferentiation operation is a generalization of the differentiation operations. The idea of fractional differentiation as a generalization of the concept of the derivative to the noninteger value ofa arose almost simultaneously with the very concept of differentiation. The first mention of this idea occurs in the correspondence of G. W. Leibniz and Marquis de l’Hospital in 1695 (see [1]). The idea of fractional integrodifferentiation was further developed in the works of L. Euler, who in 1738 noticed that an expression can be given meaning even for noninteger values (see [2]). An explicit calculation formula was given in the treatise by S. Lacroix in 1820 (see [3]). Also in 1812, P.S. Laplace put forward the idea of the possibility of differentiating noninteger order for some functions. The first definition of the derivative of noninteger order was given by J. Fourier in 1822. In its modern form, fractional integrodifferentiation was formed in the works of N.H. Abel and J. Liouville. In 1823, in connection with the problem of tautochrone—a curve, when sliding along which, under the influence of gravitational forces, a body reaches its lowest point in the same time, regardless of its initial position. The idea of considering fractional differentiation as an operation inverse to fractional integration was first proposed by Holmgren in 1865 (see [4–6]). A year later, Grunwald, who was not familiar with Holmgren’s work, came to the same idea of Letnikov in 1868 (see [7–13]).In [14], an efficient novel technique, namely, the q-homotopy analysis transform method (q-HATM), is applied to find the solution for the time-fractional Kaup–Kupershmidt (KK) equation and the study of fractional Emden–Fowler (FEF) equations by utilizing a new adequate procedure; specifically, the q-homotopy analysis transform method (q-HATM) is considered in [15].Fractional wave systems with continuous nonlinearities are possessed by a large number of researchers. In [16], the authors considered initial-boundary value problem of degenerate Kirchhoff-type(3)∂t2w+wr2θ−1−Δrw=wp−1w,inΩ×ℝ+,where Ω⊂ℝn,n≥1 is a bounded domain with Lipschitz boundary, θ∈1,2∗, and wr is the Gagliardo seminorm of w defined by(4)wr=∫Ω∫Ωwx−wz2x−zn+2rdxdz1/2.The authors obtained, under appropriate conditions, the global existence in time and finite blow-up of solutions for (3) owing to the Galerkin method combined with the potential wells. They also showed the global existence of solutions under critical initial conditions. In [17], the authors studied the following damped degenerate Kirchhoff equation:(5)∂t2w+wr2θ−1−Δrw+∂twα−1∂tw+w=wp−2w,inΩ×ℝ+,where 2<α<2θ<p<2∗<r. The global existence, behavior of solutions, and blow-up in time for (4) are obtained, under appropriate assumptions. In [18], the IBVP of Kirchhoff wave equation is considered. Under some sufficient conditions, the blow-up in finite time is shown by using a modified concavity method; for more details, see [19–27].(6)∂t2w+wr2θ−1−Δrw=wp−2w,inΩ×ℝ+.We highlight here the novelty of the problem:(1)
It is interesting to note that simultaneously with the theoretical developments of classical nonlinear wave operation, practical applications of fractional integrodifferentiation operation can also be found(2)
It is shown that when the nonlinear source dominates the fractional Laplacian in (2), this ensures the global nonexistence in time (blow-up) of solutions(3)
Our results extend many recent results in the literature


## 2. Auxiliary Results and Function Spaces

The fractional Laplacian−Δrw of the function w is given by(7)−Δrwx=C∫ℝnwx−wzx−zn+2rdz,∀x∈ℝn,where(8)C−1=∫ℝn1−cosζ1ζn+2rdζ.We define the fractional-order Sobolev space by(9)Wr,2Ω=v∈L2Ω:∫Ω∫Ωvx−vz2x−zn+2rdxdz<∞,equipped with the norm(10)wWr,2Ω=∫Ωw2dx+∫Ω∫Ωvx−vz2x−zn+2rdxdz1/2.Let(11)W0r,2Ω=w∈Wr,2Ω:w=0,x∈∂Ω,be a closed linear subspace of Wr,2Ω, and its norm is given by(12)wW0r,2Ω=∫Ω∫Ωvx−vz2x−zn+2rdxdz1/2.The spaceW0r,2Ω is a Hilbert space with inner product(13)w,uW0r,2Ω=∫Ω∫Ωwx−wzux−uzx−zn+2rdxdz.


## 3. The Potential Wells

For simplicity, in this section, we consider problem (1) in stationary case. In fact, if we replace w in this section by wt for any t∈0,T, all the facts are still valid. We define(14)Jw=12wW0r,2Ω2−1pwpp.We denote(15)ℐw=wW0r,2Ω2−wpp.We introduce now the stable set as follows:(16)W=w∈W0r,2Ω:ℐw>0,Jw<d∪0,where the mountain pass level d is defined as(17)d=infw∈W0r,2Ω∖0supμ≥0Jμw.We introduce the so-calledNehari manifold:(18)N=w∈W0r,2Ω∖0:ℐw=0.Then potential depthd is characterized by(19)d=infw∈NJw,which implies that(20)dist0,N=minw∈NwW0r,2Ω.We will prove the invariance of the setW.For the reader’s convenience, we recall the main embedding results for the fractional Sobolev spaces; see [28] for details.Lemma 1.
LetΩ be bounded domain. Then(1)
The embeddingW0r,2Ω↪LpΩ is compact for any p∈1,2r∗(2)
The embeddingW0r,2Ω↪L2r∗Ω is continuousLemma 2.
(1)
For anys∈1,2r∗, there exists a positive constant C0=C0n,s,r such that for any u∈W0r,2Ω(21)wLsΩ≤C0∫Ω∫Ωux−uz2x−yn+2rdxdy.(2)
For anys∈1,2r∗ and any bounded sequence ujj in W0r,2Ω, there exists u in Lsℝn, with u=0 a.e. in ℝn−Ω, such that up to a subsequence, still denoted by ujj(22)uj⟶ustrongly inLsΩasj⟶∞.Definition 1.
A functionw=wx,t is said to be a global (weak) solution of problem (1), if(23)w∈L∞0,∞,W0r,2Ω,wt∈L∞0,∞,L2Ω,wo∈L∞0,∞,W0r,2Ω,wt∈L∞0,∞,L2Ωw1∈L∞0,∞,L2Ω,and for any ϕ∈L∞0,∞,W0r,2Ω, t∈ℝ+∗,(24)wt.,t,ϕ.,t+12∫0twt.,t,ϕ.,tW0r,2Ωdτ+∫0twt.,t,ϕ.,tW0r,2Ωdτ=w1,ϕ.,0+∫0tw.,τw.,τp−2,ϕ.,τdτ.If a (weak) global solutionw belongs to C0,∞;W0r,2Ω, we say that u is a strong global solution of problem (1).The energyE of solution at time t to (1) is given by(25)Et=12∂twt22+Jw.Lemma 3.
Letwx,t be a weak solution of problem (1). If w0∈W,w1∈L2Ω, then Et≤E0.


## 4. Blow-Up Result

In this section, we prove the blow-up result to problem (1)Lemma 4.
Letwx,t is the weak solution of problem (1). If w0∈Wandw1∈L2Ω satisfying that(26)w2≥2pp−2KE0,(27)∂tw,wW0r,2Ω<0,(28)w0∈ℳ,(29)∫Ωw0w1dx>0,then any solution of (1) belongs to ℳ.Proof.
We claim thatw∈ℳ for t∈0,T; by contradiction, we suppose that t0∈0,T is the first time such that(30)Iwt0=0,(31)ℐwt<0fort∈0,t0.
We first introduce an auxiliary function,(32)Mt=w2,and directly(33)M′t=∂tw,w+w,∂tw=2∂tw,w,(34)M″t=2∂t2w,w+2∂tw2.
Multiplying (1) 1 by w and then by integration over ℝn, we have(35)∂t2w,w+w,wWr,20Ω+∂tw,wWr,20Ω=∫Ωwp−2wdx,so that(36)∂t2w,w=−wW0r,2Ω−∂tw,wW0r,2Ω+∫Ωwwp−2wdx.
Substituting (27) into (36), we obtain(37)M″t=2∂twWr,20Ω−2wWr,20Ω−2∂tw,wWr,20Ω+2∫Ωw|wp−2wdx.
By (27), we have(38)M″t≥2∂tw2−2ℐw.
By (31), we have M″t>0 for any t∈0,t0; then, M′t is strictly increasing on 0,t0. Thus(39)M′t>M′0>0fort∈0,t0.
We haveMt is also strictly increasing on 0,t0.
We have(40)Mt>M0≥2pp−2KE0for allt∈0,t0.
From the continuity ofw at t=t0, it follows that(41)Mt0=wt02>2pp−2KE0.
On the other hand,(42)E0≥Et=12∂tw22+12wW0r,2Ω2−1pwpp=12∂tw22+12−1pwW0r,2Ω2+1pℐw.
Together with (30) and Lemma 2, we get(43)Ε0≥12∂tw.,t022+12−1pw.,t0W0r,2Ω2≥12∂tw.,t022+p−22pK−1w.,t022≥p−22pK−1w.,t022,which contradicts (41). This completes the proof.We are now ready to prove the finite time blow-up of solution to (1) when E0>0.Definition 2.
We say that the functionwx,t blows up in finite time if there exists t∗∈0,∞ such that(44)wx,tL2Ω⟶∞ast⟶t∗.Theorem 1.
Letw0∈W0r,2Ω and w1∈L2Ω. Assume that w0∈ℳ, E0>0, and ∫Ωw0w1dx>0, then any solution of (1) blows-up in finite time.Proof.
We havew∈ℳ; arguing by contradiction, we suppose that w is weak global solution , for any t∈0,∞. From (34) and Cauchy–Schwarts inequality, we get(45)M′2t=4w,∂t2≤w2∂tw2,t∈0,∞,which together with (36) implies that(46)M″tMt−1+αM′t2≥2∂tw2−2w0r,2Ω2+2∫Ωwwp−2wdxMt−41+αw,∂tw2≥2∂tw2−2w0r,2Ω2+2∫Ωwwp−2wdxMt−41+αw2∂tw2=−2w0r,2Ω2+2∫Ωwwp−2wdx−21+2α∂tw2Mt=AtMt,where α>0. We notice that(47)At≔−2wW0r,2Ω2+2wpp−21+2α∂tw2≥−21+2α∂tw2−2wW0r,2Ω2+p∂tw2+pwW0r,2Ω2−2pΕ0=−4α−p+2∂tw2+p−2wW0r,2Ω2−2pΕ0≥−4α−p+2∂tw2+p−2K−1w22−2pΕ0,for t∈t,∞. Set α=p−2/4>0, and then 4α−p+2∂tw2=0. So from (47), we get(48)At≥K−1p−2w2−2pE0≥0.
At this point, by (33)–(48), we obtain(49)M″tMt−1+αM′t2>0,t∈0,∞,where α>0. This implies that(50)M−α′=−αM−α−1M′t<0,(51)M−α″=−αM−α−2M″tMt−1+αM′t2<0,for all t∈0,∞, which means that the function M−α is concave. Obviously, M0>0; then, there must exist a T>0 such that(52)limt⟶TM−at=0,so that(53)limt⟶T−Mt=∞i.e.,limt⟶T−w2=∞.
Thus, the proof is completed.



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