# Blow-Up for a Stochastic Viscoelastic Lamé Equation with Logarithmic Nonlinearity

Authors: Amina Benramdane; Nadia Mezouar; Mohammed Sulaiman Alqawba; Salah Mahmoud Boulaaras; Bahri Belkacem Cherif
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we consider an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and logarithmic nonlinear source terms. We proved a blow-up result for the solution with decreasing kernel.


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## Body

## 1. Introduction

In recent years, stochastic partial differential equations in a separable Hilbert space have been studied by many authors, and various results on the existence, uniqueness, stability, blow-up, and other quantitative and qualitative properties of solutions have been established.In this work, we consider the following problem of stochastic wave equation:(1)utt−μΔu−λ+μ∇divu+∫0tht−sΔusds+utq−2ut=uup−2lnuk+εσx,tWtx,tinD×0,+∞,ux,t=0on∂D×0,+∞,ux,0=u0x,utx,0=u1xinD¯,where D is a bounded domain in IRn, n∈IN∗, with a smooth boundary ∂D; μ,λ are the Lamé constants which satisfy μ>0, λ+μ≥0; h is a positive function, p>q≥2; the constant k is a small nonnegative real number; and L2D is the set of square integrable function on D equipped with the inner product .,. and its norm .2.Wx,t is an infinite dimensional Wiener process, σx,t is L2D valued progressively measurable, and ε is a positive constant which measures the strength of noise.It is common to observe a wave motion as a physical phenomenon which is mathematically modeled by a partial differential equation of hyperbolic type. Much has been written about such equations regarding their widespread applications to engineering and sciences. However, for more realistic models, the random fluctuation had been taken into consideration which led to introduced stochastic wave equation in 1960’s. Several examples of linear stochastic wave propagation and applications can be found in [1]. Mueller [2] was the first who investigate the existence of explosive solutions for some stochastic wave equation. Motivated by Mueller [2], Chow [3] was interested by knowing how does a random perturbation affect the solution behavior for a wave equation with a polynomial nonlinearity. He was concerned with the existence of local and global solutions of the stochastic equation:
(2)utt=Δu+fu+σuWtx,tinx∈ℝd,t>0,ux,0=gx,utx,0=hx,where the initial data g and h are given functions and the nonlinear terms fu and σu are assumed to be polynomials in u. Four years later, he [4] established an energy inequality and the exponential bound for a linear stochastic equation and gave the existence theorem for a unique global solution for the randomly perturbed wave equation:
(3)utt+2αut−Ax,∂xux,t=fx,t+σx,tWtx,tinx∈ℝd,t>0,ux,0=u0x,utx,0=v0x.In 2009, Chow [5] studied the problem of explosive solutions for a class of nonlinear stochastic wave equation in a domain D⊂ℝd for d≥3,
(4)utt=c2Δ−αu+fu+σu,x,tWtx,tinx∈D,t>0,ux,0=gx,utx,0=hx.We can mention some other works such as Cheng et al. [6] who studied the existence of a global solution and blow-up solutions for the nonlinear stochastic viscoelastic wave equation with nonlinear damping and source terms:
(5)utt−Δu+∫0tht−sΔusds+ut∣q−2ut=uup−2+εσx,tWtx,tinD×0,+∞,ux,t=0on∂D×0,+∞,ux,0=u0x,utx,0=u1xinD¯.The authors proved that finite time blow-up with nonnegative probability is explosive or it is explosive in energy sense forp>q.Moreover, Kim et al. [7] considered the stochastic quasilinear viscoelastic wave equation with nonlinear damping and source terms:
(6)∣utρutt−Δu−Δutt+∫0tht−sΔusds+ut∣q−2ut=uup−2+εσx,tWtx,tinD×0,+∞,ux,t=0on∂D×0,+∞,ux,0=u0x,utx,0=u1xinD¯.They showed the existence of a global solution and blow-up in finite time.Recently, Yang et al. [8] treated the following stochastic nonlinear viscoelastic wave equation:
(7)utρutt−Δu−Δutt+∫0tht−sΔusds=σx,tWtx,tinD×0,+∞,ux,t=0on∂D×0,+∞,ux,0=u0x,utx,0=u1xinD¯.They established the existence of global solution and asymptotic stability of the solution by using some properties of the convex function.However, it was noticed that the logarithmic nonlinearity appears naturally in many branches of physics such as nuclear physics, optics, and geophysics (see [9, 10]). These specific applications in physics and other fields attract a lot of mathematical scientists to work with such problems. In the deterministic case, Al-Gharabli [11] investigated the stability of the solution of a viscoelastic plate equation with a logarithmic nonlinearity source term for the following problem:
(8)utt+Δ2u+u+∫0tht−sΔu2sds=ulnukinD×0,+∞,u=∂u∂ν=0in∂D×0,+∞,ux,0=u0x,utx,0=u1xinD¯,where D⊆ℝ2 is a bounded domain with a smooth boundary ∂D. The vector ν is the unit outer normal to ∂D, and h is the nondecreasing nonnegative function.Mezouar et al. [12] treated a more general problem where they considered the following nonlinear viscoelastic Kirchhoff equation with a time-varying delay term:
(9)utlutt−M∥∇u∥2Δu−Δutt+∫0tht−sΔusds+μ1g1utx,t+μ2g2utx,t−τt=kulnuinD×0,+∞,ux,t=0on∂D×0,+∞,ux,0=u0x,utx,0=u1xinD,utx,t−τ0=f0x,t−τ0inD×0,τ0.The paper is organized as follows: in Section2, we introduce some basic definitions, necessary assumptions, and lemmas that are helpful in proving our main result. Section 3 is devoted to show the blow-up of the solution of our problem.


## 2. Preliminaries

LetΩ,F,P be a complete probability space for which a filtration Ft,t≥0 of increasing sub σ−fields Ft is given and Wx,t be a continuous Wiener random field in this space with a mean zero and the covariance operator Q satisfying
(10)TrQ=∑i≥1λi<∞.Wx,t is defined by
(11)Wx,t=∑j=1∞λjβjtejt,j∈IN∗,t≥0,where βjt is a sequence of real-valued standard Brownian motions mutually independent on the probability space Ω,F,P, λj are the eigenvalues of Q, and ej are the corresponding eigenvectors. That is,
(12)Qej=λjej.NoteE. stands for expectation with respect to probability measure P. Let H be the set of L20=L2Q1/2V,V-valued processes with the norm
(13)∥ϕt∥H2=E∫0t∥ϕs∥L202ds=E∫0tTrϕsQϕ∗sds<∞,where ϕ∗s denotes the adjoint operator of ϕs and V=H01D which is equivalent to H1D. For any process ϕs∈H, we can define the stochastic integral with respect to the Q-Wiener process as ∫0tϕsdWs which is a martingale. For more details about the infinite dimension Wiener process and stochastic integral, we refer to Da Prato and Zabczyk (pp. 90-96, [13]).To state and prove our result, we need some assumptions.A1. Assume that h:IR+⟶IR+ is a C1 nonincreasing function satisfying
(14)h0>0,μ−∫0∞hsds=l>0,and there exist tow nonnegative constants ς1 and ς2 such that
(15)−ς1ht≤h′t≤−ς2ht,t≥0.A2.
(16)∫0∞hsds<μp−2pp−12.A3. p>q≥2 and
(17)2<p≤2n−1n−2,ifn≥3,2<k≤+∞,ifn=1,2.The following theorem states the existence and uniqueness of a local solution of our problem; the proof can be established by combining the proof given in [6, 12].Theorem 1.
Assume that (A1) and (A3) hold. Ifu0,u1∈H01D×L2DandE∫0t∥σt∥22dt<∞, then there exists a solution in whichuholds (1) on the interval 0,T in the sense of distributions over 0,T×D for almost all w a test function such that
(18)u,ut∈L2Ω;L∞0,T;H2D∩H01D×H01D∩L2Ω;C0,T;H01D×L2D.We define the energy associated to the solution of system (1) by
(19)et=12∥ut∥22+12μ−∫0thsds∥∇u∥22+λ+μ2∥divu∥22+12ho∇ut+kp2∥u∥pp−1p∫Du∣plnukdx,where
(20)hovt=∫0tht−s∥v.,t−v.,s∥2ds.We rewrite (1) as an equivalent Itô’s system
(21)du=vdt,dv=μΔu+λ+μ∇divu−∫0tht−sΔusds−vq−2v+uup−2lnukdt+εσx,tdWtx,tinD×0,+∞,ux,t=0on∂D×0,+∞,ux,0=u0x,vx,0=u1xinD¯,which can be written as the integral equation
(22)ut=u0+∫0tvsds,vt=v0+∫0tμΔu+λ+μ∇divu−∫0ths−rΔurdr−v∣q−2v+uup−2lnukds+∫0tεσx,sdWsx,tinD×0,+∞,ux,t=0on∂D×0,+∞,ux,0=u0x,vx,0=u1xinD¯.Lemma 2 [14] (Sobolev-Poincaré’s inequality).
Letm be a number with
(23)2≤m≤+∞n=1,2or
(24)2≤m≤2nn−2n≥3.
Then there exists a constantCs=CsD,m such that
(25)∥u∥m≤Cs∥∇u∥2,foru∈H01D.Lemma 3 [15].
Forh,φ∈C10,+∞,IR, we have
(26)∫Dh∗φφtdx=−12ht∥φt∥22+12h′oφt−12ddthoφt−∫0thsds∥φ∥2.Lemma 4.
Letu,v be a solution of the problem (21) with the initial data u0,v0∈H01D×L2D, E∫0t∥σs∥22ds<∞. Then, the energy functional defined by (19) satisfies
(27)et=e0−∫0t∥v∥qqds−12∫0ths∥∇us∥22ds+12∫0th′o∇usds+∫0tvs,εσx,sdWs+ε22∑j=1∞∫0t∫Dλjej2xσ2x,sdxds.Proof.
We can apply the Itô’s formula to (21) for each x∈D after integrating the above equation over D to get
(28)∥vt∥22=∥v0∥22+2∫D∫0tvsμΔu+λ+μ∇divu−∫0shs−τΔuτdτ−vq−2v+uup−2lnukdsdx+2∫0tvs,εσx,sdWs+ε2∑j=1∞∫0t∫Dλjej2xσ2x,sdxds.
By using integration by parts, we get(29)μ∫D∫0tΔuvsdxds=−μ∫D∫0t∇u∇vdsdx=−μ2∥∇ut∥22−∥∇u0∥22,(30)∫D∫0tλ+μ∇divusvsdsdx=−λ+μ∫D∫0tdivusdivvsdsdx=−λ+μ2∥divut∥22−∥divu0∥22.
By applying Lemma3, we have
(31)∫0t∫D∫0shs−τΔuτvsdτdxds=−∫0t∫D∫0shs−τ∇uτ∇vsdτdxds=∫0t12hs∥∇us∥22−12h′o∇us+12ddsho∇us−∫0shτdτ∥∇us∥22ds.
We have(32)∫D∫0tuu∣p−2lnukusdsdx=∫0t∫D1pddsusplnukdxds=∫0t∫D1pddsusplnuk−1puspddslnukdxds=∫D1puplnukdx−kp2∥u∥pp.
By replacing (29)–(32) in (28) and multiplying equation (28) by 1/2, we arrive at (27).


## 3. Blow-Up

We prove our main result forp>q; we purpose
(33)E∫0∞∫Dσ2x,tdxdt<∞,(34)Gt=ε22∑j=1∞E∫0t∫Dλjej2xσ2x,sdxds,(35)G∞=ε22∑j=1∞E∫0∞∫Dλjej2xσ2x,sdxds≤ε22TrQc02E∫0∞∫Dσ2x,sdxds≔E1<∞,where
(36)TrQ=∑j=1∞λj<∞andc0=supj≥1∥ej∥∞<∞.Lemma 5.
Letu,v be a solution of system (21) with initial data u0,v0∈H01D×L2D. Then, we have
(37)ddtEet=−E∥vt∥qq−12htE∥∇ut∥22+12Eh′o∇ut+ε22∑j=1∞E∫Dλjej2xσ2x,tdx,(38)Eut,vt=Eu0,u1−μ∫0tE∥∇us∥22ds−λ+μ∫0tE∥divus∥22ds+E∫0t∫0shs−r∇ur,∇usdrds−E∫0tus,vsq−2vsds+E∫0tus,ususp−2lnuskds+E∫0t∥vs∥22ds.Proof.
Using the Itô’s formula and by following the same way as our discussions in Lemma4 with taking the expectations, we obtain (37).
We multiply the second equation in (22) by u and integrate the result over D, and we take expectation; we obtain (38).
We setHt=Gt−Eet. As h is a positive decreasing function so
(39)H′t=G′t−ddtEet=E∥v∥qq+12htE∥∇ut∥22−12Eh′o∇ut≥E∥v∥qq.
Consequently,(40)H′t≥0.Lemma 6.
Letu,v be a solution of system (21). Assume that (A1) holds. Then, there exists a positive constant C such that
(41)E∥ut∥p+1s≤CGt−Ht−12E∥v∥22+1pE∫Du∣plnukdx−12Eho∇ut−λ+μ2E∥divu∥22+E∥u∥p+1p+1,where 2≤s≤p+1.Proof.
(42)Gt−Ht−12E∥v∥22+1pE∫Duplnukdx−12Eho∇ut−λ+μ2E∥divu∥22+E∥u∥p+1p+1=Eet−12E∥v∥22+1pE∫Duplnukdx−12Eho∇ut+E∥u∥p+1p+1−λ+μ2∥divu∥22=12E∥ut∥22+12Eμ−∫0thsds∥∇u∥22+λ+μ2E∥divu∥22+12Eho∇ut+kp2E∥u∥pp−1pE∫Duplnukdx−12E∥v∥22+1pE∫Duplnukdx−12Eho∇ut−λ+μ2E∥divu∥22+E∥u∥p+1p+1=12μ−∫0thsdsE∥∇u∥22+kp2E∥u∥pp+E∥u∥p+1p+1≥12lE∥∇u∥22+E∥u∥p+1p+1.The last inequality is getting from (A1).Case 7.
If∥u∥p+1≤1, then ∥u∥p+1s≤∥u∥p+12.
By applying Lemma2, we obtain ∥u∥p+1s≤c∥∇u∥22, then
(43)12lE∥∇u∥22+kp2E∥u∥p+1p+1≥12lE∥u∥p+1s+kp2E∥u∥p+1p+1≥E∥u∥p+1s.Case 8.
If∥u∥p+1≥1, then ∥u∥p+1p+1≥∥u∥p+1s.
Hence,(44)12lE∥∇u∥22+E∥u∥p+1p+1≥12lE∥∇u∥22+E∥u∥p+1s≥E∥u∥p+1s.
Consequently, we obtain (41).
We are ready to state and prove our main result forp>q. For this purpose, we define
(45)Lt≔H1−αt+δEu,v,where
(46)0<α<minp−12p+1,p+1−qp+1qand δ is a very small constant determined later.Theorem 9.
Assume (A1) and (A2) hold. Letu,v be a solution of system (21) with initial data u0,v0∈H01D×L2D satisfying
(47)Ee0≤−1+βE1,where β is a nonnegative constant and E1 is given in (35). If p>q, then there exists a positive time T0∈0,T such that
(48)limt⟶T0−Eet=+∞,where
(49)T0=1−ααKLα/1−α0,L0=H1−α0+δEu0,u1>0,and K is given later.Proof.
Let(50)Lt=H1−αt+δEu,v.
A direct differentiation ofLt gives
(51)L′t=1−αH−αtH′t+δ−μE∥∇ut∥22−λ+μE∥divu∥22+E∫0tht−r∇ur,∇utdr−Eut,vtq−2vt+Eut,ututp−2lnutk+E∥vt∥22=1−αH−αtH′t+δ−μE∥∇ut∥22−λ+μE∥divu∥22+E∫0tht−r∇ur,∇utdr−Eut,vtq−2vt+Eut,ututp−2lnutk+E∥vt∥22+δpHt−Gt+Eet.
Recalling (39) and (19), (51) leads to
(52)L′t≥1−αH−αtE∥v∥qq+δpHt−Gt+δμp2−μE∥∇ut∥22+δp2+1E∥v∥22−δEut,vtq−2vt−δp2E∫0thsds∥∇u∥22+δE∫0tht−r∇ur,∇utdr+λ+μδp2−1E∥divu∥22+δkpE∥u∥pp+δp2Eho∇ut.
By using Young’s and Hölder’s inequalities, we get(53)E∫0tht−r∇ur,∇utdr=E∫0tht−r∇ur−∇ut,∇utdr+E∫0thsds∥∇ut∥22≥−p2Eho∇ut−12pE∫0thsds∥∇ut∥22+E∫0thsds∥∇ut∥22.
Hence,(54)L′t≥1−αH−αtE∥v∥qq+δpHt−Gt+δμp2−μE∥∇ut∥22+δp2+1E∥v∥22−δEut,vtq−2vt+δ1−12p−p2E∫0thsds∥∇ut∥22−δp2Eho∇ut+δp2Eho∇ut+λ+μδp2−1E∥divu∥22+δkpE∥u∥pp≥1−αH−αtE∥v∥qq+δpHt−Gt+δμp2−μE∥∇ut∥22+δp2+1E∥v∥22−δEut,vtq−2vt+δ1−12p−p2E∫0thsds∥∇ut∥22+λ+μδp2−1E∥divu∥22+δkpE∥u∥pp.
Asq<p+1, then E∥ut∥qq≤cE∥ut∥p+1p+1 so by using Young’s and Hölder’s inequality; we obtain
(55)Eut,vtq−2vt≤E∥vt∥qqq−1/qE∥ut∥qq1/q≤cE∥vt∥qqq−1/qE∥ut∥p+1p+11/p+1≤cE∥vt∥qqq−1/qE∥ut∥p+1p+11/p+1−1/qE∥ut∥p+1p+11/q≤cq−1qξE∥vt∥qq+ξ1−qqE∥ut∥p+1p+1E∥ut∥p+1p+11/p+1−1/q,where ξ and c are constants.
We consider the following partition ofD:
(56)D1=x∈D:u>1,D2=x∈D:∣u∣≤1.
We have(57)E∫Duplnukdx=E∫D1uplnukdx+E∫D2uplnukdx≤E∫D1uplnukdx≤E∫D1kup+1dx≤kE∥u∥p+1p+1.
By (40), (47), and −Ee0=H0, we have
(58)1+βGt<1+βE1≤H0≤Ht≤Gt+1pE∫Duplnukdx.
Therefore,(59)Gt≤11+βHt.
From (57), (58), and (59), we get
(60)kE∥ut∥p+1p+1≥E∫D1kup+1dx≥pHt−Gt≥pβ1+βHt.
AsH is increasing positive nonnegative function and by recalling (46), we get
(61)E∥ut∥p+1p+11/p+1−1/q≤pβk1+β1/p+1−1/qH1/p+1−1/qt≤pβk1+β1/p+1−1/qH−αt≤pβk1+β1/p+1−1/qH−α0.
Taking into account (61) in (55), we find
(62)Eut,vtq−2vt≤cpβk1+β1/p+1−1/qq−1qξE∥vt∥qqH−αt+cpβk1+β1/p+1−1/qξ1−qqE∥ut∥p+1p+1H−α0.
Substituting (62) into (54), we get
(63)L′t≥1−αH−αtE∥v∥qq+δpHt−Gt+δμp2−1E∥∇ut∥22+δp2+1E∥v∥22+δ1−12p−p2E∫0thsds∥∇ut∥22+λ+μδp2−1E∥divu∥22−δa1q−1qξE∥v∥qqH−αt−δa1qξ1−qE∥u∥p+1p+1H−α0+δkpE∥u∥pp,where a1=cpβ/k1+β1/p+1−1/q.
Using Lemma6, we arrive at
(64)L′t≥1−α−δa1q−1qξH−αtE∥v∥qq+δpHt−Gt+δμp2−1E∥∇ut∥22+δp2+1E∥v∥22+δ1−12p−p2E∫0thsds∥∇ut∥22+λ+μδp2−1∥divu∥22−δa1qξ1−qH−α0CGt−Ht−12E∥v∥22+1pE∫Duplnukdx+E∥u∥p+1p+1−12Eho∇ut−λ+μ2E∥divu∥22+δkpE∥u∥pp.
Onceξ is fixed, we pick δ small enough so that
(65)1−α−δa1q−1qξ≥0,
It implies that(66)L′t≥δp+a2ξ1−qHt−Gt+δp2+1+a212ξ1−qE∥v∥22−δa2ξ1−q1pE∫Duplnukdx+δλ+μξ1−qa22+p2−1E∥divu∥22+δa2ξ1−q12Eho∇ut+δa3E∥∇ut∥22+δkpE∥u∥pp−δa2ξ1−qE∥u∥p+1p+1.where a2=Ca1/qH−α0 and a3=μp/2−1+1−1/2p−p/2∫0∞hsds which is positive from (A2).
From (A1), (19), and Lemma 2, we have
(67)Ht−Gt≥−12E∥v∥22−μ2+2CsE∥∇u∥22−λ+μ2E∥divu∥22−12Eho∇ut−kp2E∥u∥pp+1pE∫Duplnukdx+E∥u∥p+1p+1.
Now we add and subtractδa4Ht−Gt in (66), and using (67), we find
(68)L′t≥δp−a4+a2ξ1−qHt−Gt+δp2+1−a42+a212ξ1−qE∥v∥22+δλ+μξ1−qa22+p2−1−a42E∥divu∥22+δa2ξ1−q12−a4Eho∇ut+δa3−a4μ+4Cs2E∥∇ut∥22+δkp1−a4pE∥u∥pp+δpa4−a2ξ1−qE∫Duplnukdx+δa4−a2ξ1−qE∥u∥p+1p+1,where a4=mina2ξ1−q,2a3/μ+4Cs>0.
Using (60), we obtain
(69)L′t≥δpβ1+βHt+δp2+1−a42+a212ξ1−qE∥v∥22+δλ+μξ1−qa22+p2−1−a42E∥divu∥22+δ2a2ξ1−q−a4Eho∇ut+δa3−a4μ+4Cs2E∥∇ut∥22+δkp1−a4pE∥u∥pp+δa4−a2ξ1−qE∥u∥p+1p+1≥γHt+E∥v∥22+E∥divu∥22+Eho∇ut+E∥∇ut∥22+E∥u∥pp+E∥u∥p+1p+1≥0,where γ>0 is the minimum of the coefficients of Ht, E∥v∥22, E∥divu∥22, Eho∇ut, E∥∇ut∥22, and E∥u∥pp in (69).
Consequently,(70)Lt≥L0>0,∀t>0.
Next, we have(71)Lt1/1−α=H1−αt+δEu,v1/1−α≤21/1−αHt+δ1/1−αE∫Duvdx1/1−α.
Therefore, by using Hölder’s and Young’s inequalities, we obtain(72)E∫Duvdx1/1−α≤cE∥u∥p+121/2E∥v∥221/21/1−α≤cE∥u∥p+121/21−αE∥v∥221/21−α≤cE∥u∥p+12η/21−αη+E∥v∥22ζ/21−αζ,with 1/η+1/ζ=1.
We choose=21−α,η=21−α/1−2α, and we use (46), so (72) becomes
(73)E∫Duvdx1/1−α≤c1−2αE∥u∥p+12/1−2α+E∥v∥22≤cE∥u∥p+12/1−2α+E∥v∥22.
By applying Lemma6 with s=2/1−2α and recalling (19), we obtain
(74)E∫Duvdx1/1−α≤cGt−Ht−12E∥v∥22+1pE∫Duplnukdx−12Eho∇ut−λ+μ2E∥divu∥22+E∥u∥p+1p+1+E∥v∥22≤c12E∥v∥22+12μ−∫0thsdsE∥∇u∥22+λ+μ2E∥divu∥22+kp2E∥u∥pp+12Eho∇ut−1pE∫Duplnukdx−12E∥v∥22+1pE∫Duplnukdx−12Eho∇ut−λ+μ2E∥divu∥22+E∥v∥22+E∥u∥p+1p+1≤cE∥v∥22+12μE∥∇u∥22+kp2E∥u∥pp+λ+μ2E∥divu∥22+12Eho∇ut+E∥u∥p+1p+1.
Hence,(75)Lt1/1−α≤21/1−αHt+δ1/1−αcE∥v∥22+12μE∥∇u∥22+λ+μ2E∥divu∥22+kp2E∥u∥pp+12Eho∇ut+E∥u∥p+1p+1≤C~Ht+E∥v∥22+E∥∇u∥22+E∥divu∥22+E∥u∥pp+Eho∇ut+E∥u∥p+1p+1,where C~=21/1−αmax1,cδ1/1−α,cδ1/1−αλ+μ/2,cδ1/1−αk/p2.
According to (69) and (75), we have
(76)Lt1/1−α≤C~γL′t≤K~L′t.
In a direct integration of (76), we get
(77)Lt1/1−α≥1L01/1−α−K~αt/1−α.
Therefore,Lt blows up in time T≤T0=1−α/αKLα/1−α0, and the proof is completed.



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*Source: 9943969-2021-04-19.xml*