# Blow-Up of Certain Solutions to Nonlinear Wave Equations in the Kirchhoff-Type Equation with Variable Exponents and Positive Initial Energy

Authors: Loay Alkhalifa; Hanni Dridi; Khaled Zennir
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper is concerned with the blow-up of certain solutions with positive initial energy to the following quasilinear wave equation:utt−MNutΔp·u+gut=fu. This work generalizes the blow-up result of solutions with negative initial energy.


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## Body

## 1. Introduction

LetΩ be an open bounded Lipschitz domain in ℝnn≥1,T>0,QT=Ω×0,T. We consider the following nonlinear hyperbolic equation:
(1)utt−MNutΔpxu+gut=fu,x,t∈QT,ux,t=0,x∈∂Ω,t∈0,T,ux,0=u0x,utx,0=u1x,x∈Ω.Here,∂Ω is a Lipschitz continuous boundary. The initial conditions meet the following:
(2)u0∈W01,p·Ω,u1∈L2Ω.The Kirchhoff functionM:ℝ+⟶ℝ+ is continuous and has the standard form:
(3)Mτ=a+bγτγ−1,a,b≥0,γ≥1,a+b>0,γ>1ifb>0.The elliptic nonhomogeneouspx-Laplacian operator is defined by
(4)Δpxu=∇·∇upx−2∇u,where ∇· is the vectorial divergence and ∇ is the gradient of u. The functional
(5)Nut=∫Ω∇upxpxdx,is the naturally associated px-Dirichlet energy integral. The term with a variable exponent
(6)fu=cx,tuqx−2u,plays the role of a source, and the dissipative term with a variable exponent
(7)gut=dx,tutrx−2ut,is a strong damping term.The coefficientsc and d are continuous in QT and satisfy
(8)0<c−=infx,t∈QTcx,t≤cx,t≤c+=supx,t∈QTcx,t<σ<∞,(9)0<d−=infx,t∈QTdx,t≤dx,t≤d+=supx,t∈QTdx,t<∞,where σ is a constant defined in (38). We assume that the Kirchhoff function M, defined by (3), satisfies the following hypotheses:
(i)
For1<α≤β<minn/p+,np−/p+n−p−, there exist m2≥m1>0 such that(10)m1τα−1≤Mτ≤m2τβ−1,τ∈ℝ+.(ii)
For allτ∈ℝ+, it holds that(11)∫0τMsds=M^τ≥Mττ.The exponentsp·, q·, and r· are continuous and satisfy
(12)2≤minp−,r−≤px,rx≤maxp+,r+<q−≤qx≤q+<p−α≤p−β≤p+α≤p+β≤p∗x,where the constants α and β are given in (10) and
(13)p−=essinfx∈Ωpx,p+=esssupx∈Ωpx,q−=essinfx∈Ωqx,q+=esssupx∈Ωqx,r−=essinfx∈Ωrx,r+=esssupx∈Ωrx.Also, we can definep∗x by
(14)p∗x=npxesssupx∈Ωn−rx,ifp+<n,+∞,ifp+≥n.We also assume thatp·, q·, and r· satisfy the log-Hölder continuity condition
(15)ξx−ξy≤−Llogx−y,fora.e.,x,y∈Ω,x−y<δ,for L>0,0<δ<1.Problem (1) models several physical and biological systems such as viscoelastic fluids, filtration processes through a porous medium, and fluids with viscosity dependent on temperature. In the intention of problem (1), we can see that it is linked to the following equation presented by Kirchhoff and Hensel [1] in 1883:
(16)ρ∂2u∂t2−P0h+E2L∫0L∂u∂xdx∂2u∂x2+gut=fu.The parametersL, h, E, ρ, and P0 represent, respectively, the length of the string, the area of the cross-section, Young’s modulus of the material, the mass density, and the initial tension. This equation is an extension of the classic d’Alembert’s wave equation by looking at the effects of changes in the length of the string during the vibrations. As for this problem, it has been studied. More precisely, for gut=ut, the global existence and nonexistence results can be found in [2, 3], and for gut=utput,p>0, the main results of existence and nonexistence are in the paper [4]. In recent years, hyperbolic problems with a constant exponent have been studied by many authors; we refer to interesting works [5–7]. However, only a little research has been done regarding hyperbolic problems with nonlinearities of the variable exponent type; some interesting works can be found in [8–13].Recently, in [14], Piskin studied the following wave equation with variable exponent nonlinearities:
(17)utt−M∥∇u∥2Δu+utpx−2ut=uqx−2u.The author proved, by using the modified energy functional method, the existence of solutions. We have also looked at the asymptotic behavior of the Kirchhoff wave equation problems. We can say that the investigation into the determination of the type, as well as the rate of decay, was the focus of attention of many researchers whose work was represented in [15, 16]. Motivated by previous studies, in this work, we consider problem (1), which is more interesting and applicable in the real approach of sciences, so a finite-time blow-up for certain solutions with positive and also negative initial energy has been proved. More precisely, our aim here is to find sufficient conditions on the variable exponents p·, q·, and r· and the initial data for which the blow-up occurs. This paper is organized as follows. After the introduction in the first section, we will give some preliminaries in Section 2. Then, in Section 3, we state the main results which will be proved in Sections 4 and 5.


## 2. Preliminaries

Regarding some definitions and basic properties of the generalized Lebesgue-Sobolev spacesLpxΩ and W1,pxΩ, where Ω is an open subset of ℝn, we refer to the book of Musielak [17] and the papers [18, 19]. Let
(18)C+Ω¯≔h:h∈CΩ¯,hx>1,forallx∈Ω¯.For anyh∈CΩ¯, we write
(19)h+=supx∈Ωhx,h−=infx∈Ωhx.Then, for anypx∈C+Ω¯, we define the variable exponent Lebesgue space as follows:
(20)Lp·Ω=u:Ω⟶ℝmeasurable:ϱp·μu<+∞,forsomeμ>0,where ϱp· is the p· modular of u, and it is defined by
(21)ϱp·u=∫Ωupxdx.It is equipped with the following so-called Luxemburg norm on this space defined by the formula(22)up·=infμ>0:∫Ωuxμpxdx≤1.Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many aspects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if1<p−≤p+<∞, and their continuous functions are dense if p+<∞.Lemma 1.
Suppose thatp· satisfies (15); then,
(23)up·≤C∇up·,∀u∈W01,p·Ω,where C>0 is a constant that depends only on p−, p+, and Ω.Lemma 2.
Ifp·∈CΩ¯ and q:Ω⟶1,∞ are measurable functions such that
(24)essinfx∈Ωp∗x−qx>0,withp∗x=npxesssupx∈Ωn−px,ifq−<n,∞,ifq−≥n,then the embedding W01,p·Ω°Lq·Ω is continuous and compact.Lemma 3.
Let1<p−≤p+<+∞. The spaces Lp·Ω and W1,p·Ω are separable, uniformly convex, and reflexive Banach spaces. The conjugate space of Lp·Ω is Lp′·Ω, where
(25)1px+1p′x=1,∀x∈Ω.
Foru∈Lp·Ω and v∈Lp′·Ω, we have
(26)∫Ωuxvxdx≤1p−+1p′−∥u∥p·∥v∥p′·.Lemma 4.
Ifp≥1 is a measurable function on Ω and u∈Lp·Ω, then up·≤1 and ρp·u≤1 are equivalent. For u∈Lp·Ω, we have
(27)up·≤1impliesρp·u≤up·,up·>1impliesρp·u≥up·.Lemma 5.
Ifpx∈1,∞ is a measurable function on Ω, then
(28)min∥u∥p·p−,∥u∥p·p+≤ρp·u≤max∥u∥p·p−,∥u∥p·p+,for all u∈Lp·Ω.


## 3. Main Results

Now, we state without proof the following existence result.Proposition 6.
Assume that (2) holds and the coefficients a, b, c, and d satisfy (3) and (9) and the exponents p, q, and r satisfy (12). Then, problem (1) has a unique weak solution such that
(29)u∈L∞0,T,W01,p·Ω,ut∈L∞0,T,L2Ω,utt∈L∞0,T,W0−1,p′·Ω,where p′· is the conjugate exponent of p·.Remark 7.
The proof can be established by employing the Galerkin method as in the work of Antontsev [8].
We first define the energy function. Let(30)Et≔12utL2Ω2+M^Nut−Ψt,where
(31)M^Nut=aNut+bγNutγ,Ψt=∫Ωcx,tqxuqxdx.In order to investigate the properties ofEt, the following lemma is necessary.Lemma 8.
Suppose thatu is a solution of problem (1) that satisfies (29); then, we have
(32)Ett=−∫Ωctx,tqxuqxdx−∫Ωdx,tutrxdx.Proof.
By using the energy function (30) and problem (1), we directly deduce (32).We also introduce the following lemma.Lemma 9.
Suppose that the conditions of Lemmas1–5 hold. Then, there exists a constant C>1, which is a generic constant that depends on Ω only, such that
(33)ϱq·s/q−u≤C∇up·αp−+ϱq·u,for any u∈W01,p·Ω and αp−≤s≤αq−.Proof.
Ifϱq·u>1, then
(34)ϱq·s/q−u≤ϱq·αu≤C∇up·αp−+ϱq·u.Ifϱq·u≤1, then we deduce by Lemma 4 that uq·≤1. Then, Lemmas 2 and 5 imply
(35)ϱq·s/q−u≤ϱq·αp−/q−u≤max∥u∥q·q−,∥u∥q·q+αp−/q−=∥u∥q·αp−≤C∥∇u∥p·αp−.LetB be the best constant of the Sobolev embedding
(36)W01,p·°Lq·.We set(37)B1=max1,B,m1c+αp+α−11/αp+,(38)σ=min1,q−m1B1q−αp+αminΓ11−q−/αp+,Γ11−q+/αp+,(39)Γ1=m1p+1−αc+B1q−αp+/q−−αp+,(40)E1=m1p+−αq−αq−−p+Γ1.Now, the main results of the blow-up for certain solutions with positive/negative initial energy are given by the following theorems.Theorem 10.
Let the assumptions of Proposition6 be satisfied, and assume that
(41)c+<σ,ctx,t≥σ~≥0,∀x,t∈QT,0<E0<E1,Γ1<Γ0=∇u0p·αp+≤B1−αp+.Then, the solutions of (1) blow up in finite time:
(42)T∗≤1−λμλFλ/1−λ0,λ,μ>0.Theorem 11.
Let the assumptions of Proposition6 be satisfied, and assume that
(43)E0<0.
Then, the solution of (1) blows up in finite time (42).


## 4. Proof of Theorem10

To prove Theorem10, we need the following lemmas.Lemma 12.
Let the assumptions of Theorem10 hold; then, there exists σ>0 such that for any c+<σ, there exist a constant Γ2>Γ1 such that
(44)∇u·,tp·αp+≥Γ2,∀t≥0.Proof.
By using the hypothesis (10) and the function (30), we obtain
(45)Et≥M^t−Ψt≥m1αp+αρp·α∇u−c+q−ρq·u≥m1αp+αmin∇up·αp−,∇up·αp+−c+q−maxuq·q−,uq·q+≥m1αp+αmin∇up·αp−,∇up·αp+−c+q−maxB1∇up·q−,B1∇up·q+=m1αp+αminΓp−/p+,Γ−c+q−maxB1αp+Γq−/αp+,B1αp+Γq+/αp+≔ψΓ,∀Γ∈ℝ+,where Γ=∇up·αp+. Let the function
(46)ϕ:0,1⟶ℝ,be defined by
(47)ϕΓ=m1αp+αΓ−c+q−B1αp+Γq−/αp+.
Notice thatϕΓ=ψΓ, for 0<Γ≤B1−αp+. It is easy to check that the function ϕ is increasing for 0<Γ<Γ1 and decreasing for Γ1<Γ≤∞. On the other hand, by (38), we deduce that, for any c+<σ, since E0<E1=ϕΓ1, there exists a positive constant Γ2∈Γ1,∞ such that ϕΓ2=E0. Then, we have ϕΓ0=ψΓ0≤E0=ϕΓ2. This implies that Γ0≥Γ2.
Now, we suppose on the contrary that∇ut0p·αp+<Γ2 for some t0>0. Then, there exists t1>0 such that Γ1<∇ut1p·αp+. Using the monotonicity of ϕΓ, we have
(48)Et1≥ϕ∇ut1p·αp+>ϕΓ2=E0,which contradicts Et<E0, for all t∈0,T.Lemma 13.
Let the assumptions of Theorem10 hold. Then, in light of Lemma 12, we have
(49)ρq·u≥κ,κ>0.Proof.
By using (30), we get
(50)c+q−ρq·u≥Ψt≥−E0+m1αp+αρp·α∇u+12utL2Ω2≥m1αp+αρp·α∇u−ψΓ2≥c+q−maxB1αp+Γ2q−/αp+,B1αp+Γ2q+/αp+≔κ.Let(51)Ht=E1−Et.Lemma 14.
Let the assumptions of Theorem10 be satisfied; then, we have
(52)0<H0≤Ht≤σq−ρq·u.Proof.
Using (30), (32), and (51), we obtain
(53)0<H0≤Ht≤E1−12utL2Ω2+M^Nut+Ψt.
Then, the use of (10) gives
(54)E1−12utL2Ω2−M^Nut≤E1−∫0NutMτdτ≤E1−m1αp+αmin∇up·αp−,∇up·αp+≤E1−m1αp+αminΓ2p−/p+,Γ2≤E1−m1αp+αminΓ1p−/p+,Γ1=E1−m1αp+αΓ1.
Now, recallingE1 in (38), we have
(55)E1−12utL2Ω2−M^Nut≤−m1p+1−αq−αΓ1<0.
On the other hand, we use (9) to get
(56)Ψt≤c+q−ρq·u≤σq−ρq·u.
Combining (55) with (56) gives (52).Corollary 15.
Under the assumptions of Lemma9, we have
(i)
uq−s≤C∇up·αp−+uq−q−(ii)
ρq·s/q−≤CHt+utL2Ω2+ρq·u(iii)
uq−s≤CHt+utL2Ω2+uq−q−for any u∈W01,p·Ω and αp−≤s≤αq−.Lemma 16.
Assume that (12) and (15) hold. Then, the solution of (1) satisfies
(57)ϱq·u≥Cuq−q−,for some C>0.Proof.
Let(58)Ω+=x∈Ω/∣u∣≥1,Ω−=x∈Ω/∣u∣<1.
We have(59)ϱq·u=∫Ω+uqxdx+∫Ω−uqxdx≥∫Ω+uqxdx+c1∫Ω−uqxdxq+/q−.
This implies(60)c2ϱq·uq−/q++ϱq·u≥uq−q−.
Now, given (52), (60) leads to
(61)ϱq·u≥1+c2q−σq−/q+−1−1uq−q−.
Thus, (57) follows.Lemma 17.
Suppose that (12) holds, and u is a solution of (1). Then,
(62)ϱr·u≤Cϱq·r−/q−u+ϱq·r+/q−u.Proof.
(63)ϱr·u≤∫Ω−ur−dx+∫Ω+ur+dx≤C∫Ω−uq−dxr−/q−+∫Ω+uq−dxr+/q−≤Cuq−r−+uq−r+≤Cϱq·r−/q−u+ϱq·r+/q−u.We set(64)Ft=H1−λt+εu,utL2Ω,for ε small, which will be specified later, and for
(65)0<λ≤minαq−−r+q−r+−1,q−−22q−.Now, we are in a position to prove Theorem10.Proof.
We differentiate (64) and use the equation in (1) to get
(66)Ftt=1−λH−λtHtt+εutL2Ω2+u,uttL2Ω≥1−λH−λtHtt+εutL2Ω2−MNutϱp·∇u+c−ϱq·u−εd+gut,uL2Ω≥1−λH−λtHtt+εutL2Ω2−m2p−β−1ϱp·β∇u+c−ϱq·u−εd+gut,uL2Ω≥1−λH−λtHtt+εutL2Ω2−m2p−β−1ϱp·α∇u+c−ϱq·u−εd+gut,uL2Ω.
Adding and subtracting the termε1−ηq−Ht, for 0<η<1, from the right-hand side of (56), by using (49) and (10), we get
(67)Ftt≥1−λH−λtHtt+ε1−ηq−Ht−εd+gut,uL2Ω+εutL2Ω2−m2p−β−1ϱp·α∇u+c−ϱq·u+ε1−ηq−−E1+12utL2Ω2+M^Nut−c−q−ϱq·u≥1−λH−λtHtt+ε1−ηq−Ht−εd+gut,uL2Ω+εutL2Ω2−m2p−β−1ρp·α∇u+c−ϱq·u+ε1−ηq−−E1c+q−κ−c−q−ϱq·u+12utL2Ω2+ε1−ηq−m1αp+αϱp·α∇u.
Then, forη small enough, we have
(68)Ftt≥1−λH−λtHtt−εd+gut,uL2Ω+εδHt+utL2Ω2+ϱp·α∇u+ϱq·u,where
(69)δ=min1−ηq−,1+1−ηq−2,1−ηq−m1αp+α−m2p−β−1,ηc−−1−ηE1c+κ.
Recall Young’s inequality(70)XY≤δ1Xl1l1+δ1−l2Yl2l2,where X,Y≥0,δ1>0,l1,l2∈ℝ+, such that 1/l1+1/l2=1. Applying (70) to estimate the term gut,uL2Ω, we get
(71)∫Ωutrx−1∣u∣dx≤1r−∫Ωδ1rxurxdx+r+−1r+∫Ωδ1−rx/rx−1utrxdx,where
(72)δ1−rx/rx−1=ξH−λt,where ξ is a large constant to be specified later.
Now, by using (32) and (49), we get
(73)ϱr·ut≤Httd+−σ~q+d+ϱq·u≤Httd+−σ~κq+d+≤Httd+.
Combining (73) and (71) yields
(74)∫Ωutrx−1∣u∣dx≤1r−∫Ωξ1−rxurxHλrx−1td+r+−1ξr+d+H−λtHtt.
Substituting (74) in (68), we obtain
(75)Ftt≥1−λ−εr+−1ξr+H−λtHtt+εδHt+utL2Ω2+ϱp·α∇u+ϱq·u−εξ1−r−d+r−C1Hλr+−1tϱr·u.
To estimate the last term in (75), we use (62) and (52) to get
(76)Hλr+−1tϱr·u≤Cϱq·r−/q−+λr+−1u+ϱq·r+/q−+λr+−1u.
Then, we use (65) and Lemma 9, for
(77)s=r++λq−r+−1≤αq−,s=r−+λq−r+−1≤αq−,to deduce from (76) that
(78)Hλr+−1tϱr·u≤C∇up·αp−+ϱq·u.
By exploiting Lemmas5 and 12, we get
(79)ϱp·α∇u≥C∇up·αp−.
Combining (75), (78), and (79) leads to
(80)Ftt≥1−λ−εr+−1ξr+H−λtHtt+εδ−ξ1−r−d+r−CHt+utL2Ω2+∇up·αp−+ϱq·u.
Now, we pickξ large enough and ε so small such that
(81)δ2=δ−ξ1−r−d+r−C>0,ε≤1−λr+ξr+−1,F0=H1−λ0+εu0,u1L2Ω>0.
Then, by using (57), (80) takes the form
(82)Ftt≥δ2εHt+utL2Ω2+∇up·αp−+uq−q−.
Therefore, we get(83)Ft≥F0>0,forallt≥0.
On the other hand, the application of the Hölder inequality yields(84)u,utL2Ω1−λ≤Cuq−1/1−λutL2Ω1/1−λ,and from (70), we get
(85)u,utL2Ω1−λ≤Cuq−θ1/1−λ+utL2Ωθ2/1−λ,for 1/θ1+1/θ2=1. Setting θ1/1−λ=2/1−2λ≤q−, we get θ2=21−λ by virtue of (65). Therefore, (85) takes the form
(86)u,utL2Ω1−λ≤Cuq−s+utL2Ω2,where s=2/1−2λ. By recalling Corollary 15, we get
(87)u,utL2Ω1−λ≤CHt+utL2Ω2+uq−q−.
Now, (87) and the following Minkowski’s inequality
(88)X+Yi≤2i−1Xi+Yi,will give
(89)F1/1−λt=H1−λt+εu,utL2Ω1/1−λ≤2λ/1−λHt+ε1/1−λu,utL2Ω1−λ≤CHt+utL2Ω2+uq−q−≤CHt+utL2Ω2+∇up·αp−+uq−q−.
By combining (82) and (89), we obtain
(90)Ftt≥μFt1/1−λt,where μ is a positive constant. A simple integration of (90) over 0,t yields
(91)F1/1−λt≥1F−λ/1−λ0−μλt/1−λ,which implies that the solution blows up in finite time T∗, such that
(92)T∗≤1−λμλFλ/1−λ0.
This completes the proof of Theorem10.


## 5. Proof of Theorem11

We set(93)Ht≔−Et.To prove our main result, we first establish the following lemma.Lemma 18.
Letu be the solution of (1). Then, there exists a constant C>0 such that
(94)∇ut,xp·≥C,∀t≥0.Proof.
Suppose that, by contradiction, there exists a sequencetk such that
(95)∇ut,xp·⟶0,ask⟶∞.
Then, by using Lemmas2 and 5, we get
(96)limk→∞Etk≥0,which contradicts the fact that Et<0,∀t≥0.Using (93) and (94) and applying the same procedures used to prove Theorem 10 will give the proof of Theorem 11.



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