# Nonexistence of Global Solutions for Coupled System of Pseudoparabolic Equations with Variable Exponents and Weak Memories

Authors: Kh. Zennir; H. Dridi; S. Alodhaibi; S. Alkhalaf
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

The most important behavior for evolution system is the blow-up phenomena because of its wide applications in modern science. The article discusses the finite time blowup that arise under an appropriate conditions. The nonsolvability of boundary value problem for damped pseudoparabolic differential equations with variable exponents is investigated. Such problem has been previously studied in the case ifp and q are constants. New here is the case of variables of nonlinearity p and q which make the problem has a scientific interest.


---

## Body

## 1. Introduction and Overview

Boundary value problems for evolutionary equations of parabolic in degenerate sense are well studied (see, for example, [1–4]). In this article, we study boundary value problems for coupled system of pseudoparabolic equations with px-Laplacian in the presence of weak viscoelasticities. Such problems have not been studied in depth. To begin with, let Ω is an open bounded domain in ℝn for n≥2 with smooth boundary ∂Ω; we then consider in Ω×0,T(1)∂tw−Δqxw−Δx∂tw−σt∫0tϖ1t−swds=f1w,z,∂tz−Δqxz−Δx∂tz−σt∫0tϖ2t−szds=f2w,z,for 0<T<∞ with initial condition
(2)wx,0=w0x∈W01,q·Ω,(3)zx,0=z0x∈W01,q·Ω,and boundary condition
(4)w=z=0on∂Ω×0,T.The lack of stability of solutions of partial differential equations is a huge restriction for qualitative studies. The terms responsible for the blow-up phenomenon in our system (1) is that of more complicated nonlinearities when they dominate the damped terms, especially when it comes with the existence of a large class of Laplacian operator
(5)Δqxw=div∇xwqx−2∇xw.The functionsfj:ℝ2⟶ℝ,j=1,2 are given by the nonlinearities
(6)f1ξ1,ξ2=ξ1+ξ2px−2ξ1+ξ2+ξ1px−4/2ξ1ξ2px/2,f2ξ1,ξ2=ξ1+ξ2px−2ξ1+ξ2+ξ2px−4/2ξ2ξ1px/2,respectively. The weak-viscoelastic term is σt∫0tϖjt−susds.There exists a functionF∈C1ℝ2,ℝ such that
(7)ξ1f1ξ1,ξ2+ξ2f2ξ1,ξ2=pxFξ1,ξ2,∀ξ1,ξ2∈ℝ2,pxFξ1,ξ2=ξ1+ξ2px+2ξ1ξ2px/2,where f1=∂F/∂w and f2=∂F/∂z. There exist two positive constants c0 and c1 such that
(8)c0ξ1px+ξ2px≤pxFξ1,ξ2≤c1ξ1px+ξ2px.For more details, see [5–8] and references therein.Withpx-Laplacian, which is nonlinear differential operator, in [9], a problem of elliptic equation is considered as
(9)−Δpxu−upx−2u=fx,u,x∈ℝn.The variable exponentsq· and p· are two continuous functions on Ω¯ such that
(10)2<q−≤qx≤q+<p−≤px≤p+<q∗x,with
(11)q∗x=nqxn−qxifn>qx,+∞ifn≤qx,p−=essinfx∈Ωpx,p+=esssupx∈Ωpx.We assume thatqx satisfies the Zhikov-Fan condition, i.e., for all x,y∈Ω,
(12)qx−qy≤K−logx−y,withx−y<κ,with K>0,0<κ<1 and
(13)essinfx∈Ωq∗x−px>0.We state assumptions onϖj and σ as follows:σ,ϖj∈C1ℝ+,ℝ+ satisfies
(14)0<σ′t<σt,ϖj′t<−ϖjt<0,∫0+∞ϖjsds<1q+−1p−q−σ∞−1.Define positive constantsα0,α1,E1, and E2 by
(15)α1=q−C1p+q−/p+−q−,E1=p+−q−p+α1−ρ1,α2=q−C2p+q−/p+−q−,E2=p+−q−p+α2−ρ2,for some constants C1,C2,ρ1,ρ2>0 which will be specified later.Fan et al. discussed the existence and multiplicity of solutions of (9) for u∈W1,pxℝn, where n≥2,px is a function defined on ℝn.Regarding nonlinear parabolic equation, we mention the work by [1]. The author proposed the problem.
(16)∂tu−Δpu−uq−2u=fx,t,x∈Ω,0<t≤T,u=0,x∈∂Ω,0<t≤T,ux,0=u0,x∈Ω,where Ω is a bounded domain of ℝn with smooth boundary ∂Ω and p,q≥2. Time existence of solutions of system (16) was proved.Whereas, in [10], nonlinear pseudoparabolic was considered
(17)∂tw−Δqxw−Δx∂tw−σt∫0tϖ1t−swds=wpx−2w.Global in time nonexistence of (17) was shown under an appropriate conditions on ϖ,px, and qx.System (1), where the exponents qx=q and px=p, the existence/nonexistence results have been extensively studied (please see [11–15]).The paper is organized as follows. In Section2, we state the properties of the px-growth conditions and present assumptions of the kernel functions. In Section 3, we state our main results and prove some auxiliary lemmas. In Section 4, we prove the global nonexistence of solutions given in Theorem 9. The paper is concluded by explanatory commentaries.


## 2. Preliminary

We try to list here some useful mathematical tools.First, letΩ be an open bounded domain in ℝn for n≥2 with smooth boundary ∂Ω and p:Ω⟶1,∞ be a measurable function. Denoting by
(18)p−=essinfx∈Ωpx,p+=esssupx∈Ωpx.We define thep· modular of a measurable function w:Ω⟶ℝ∪±∞ as
(19)Qp·w=∫Ω−Ω∞wpxdx+esssupx∈Ω∞wx,where
(20)Ω∞=x∈Ω;px=∞.The special Orlicz Musielak spaceLp·Ω is a Lebesgue space with variable-exponent, and it consists of all the measurable function w defined on Ω for which
(21)Qp·λw<∞someλ>0.Let(22)wp·=infλ>0;Qp·wλ≤1,be the Luxembourg norm on this space (see [16]).The Sobolev spaceW1,q·Ω consists of functions w∈Lq·Ω whose distributional gradient ∇xw exists and satisfies ∣∇xw∣∈Lq·Ω. This space is a Banach with respect to the norm
(23)w1,q·=wq·+∇xwq·.Lemma 1 (Corollary 8.2.5 in [17]).
(1)
If (12) holds with qx, then(24)wq·≤C∇xwq·,∀w∈W01,q·Ω,where Ω is a bounded domain and C is a positive constant. The norm of the space W01,q·Ω is given by
(25)w1,q·=∇xwq·,∀w∈W01,q·Ω.(2)
If(26)q∈CΩ¯,:Ω⟶1,∞,is a measurable function and
(27)essinfx∈Ωq∗x−px>0,with
(28)q∗=nqxn−qx+.
Then(29)W01,q·Ω°°Lp·Ω,with a continuous and compact embedding and
(30)wp·≤CS∇xwq·,where CS>0 is an embedding constant.Proposition 2 (Section 1 and Lemma 3.2.20 in [17]).
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
(31)1px+1p′x=1,∀x∈Ω.
Forw∈Lp·Ω and v∈Lp′·Ω, we have
(32)∫Ωwxvxdx≤1p−+1p′−∥w∥p·∥v∥p′·.Lemma 3 (Lemma 3.2.4 in [17]).
Ifp≥1 is a measurable function on Ω and w∈Lp·Ω, then wp·≤1 and Qp·w≤1 are equivalent. For w∈Lp·Ω, we have
(1)
wp·≤1 implies Qp·w≤wp·(2)
wp·>1 implies Qp·w≥wp·Lemma 4 (Section 2 in [18] and Lemma 3.2.5 in [17]).
Ifpx∈1,∞ is a measurable function on Ω, then
(33)min∥w∥p·p−,∥w∥p·p+≤Qp·w≤max∥w∥p·p−,∥w∥p·p+,for all w∈Lp·Ω.Lemma 5 (Lemma 3.2.20 in [17]).
Ifp1x≥p2x≥1a.e. in Ω, there is a continuous inclusion Lp1·Ω⊂Lp2·Ω and for all w∈Lp1·Ω,
(34)∥w∥p2·≤2∥1∥r·∥w∥p1·,where
(35)1rx≡1p2x−1p1x.
The following notation will be used throughout this paper(36)ϖj∘vt=∫0tϖjt−svt−vs22ds,for v∈L2Ω and t≥0. We have the following technical lemma.Lemma 6.
Letκ∈ℕ. For any Δxpv∈C10,T,H01Ω with p=0,1,⋯,κ−1, j=1,2, we have
(37)σt∫0tϖjt−sΔxκvs,∂tvtds2=−1κ+12∂tσtϖj∘∇xκvt+−1κ2∂tσt∫0tϖjsds∇xκv22+−1κ2σt∂tϖj∘∇xκvt+−1κ+12σtϖjt∇xκv22+−1κ2σ′tϖj∘∇xκvt+−1κ+12σ′t∫0tϖjsds∇xκv22.Proof.
Since(38)Δxκv,w2=−1κ∇xκv,∇xκw2,holds for any Δxpv,Δxpw∈H01Ω, we have
(39)σt∫0tϖjt−sΔxκvs,∂tvt2ds=−1κσt∫0tϖjt−s∇xκ∂tvt,∇xκvs−∇xκvt+∇xκvt2ds=−1κσt∫0tϖjt−s∇xκ∂tvt,∇xκvs−∇xκvt2ds+−1κσt∫0tϖjsds∇xκ∂tvt,∇xκvt2=−1κ+12σt∫0tϖjt−s∂t∇xκvs−∇xκvt22ds+−1κ2σt∫0tϖjsds∂t∇xκv22=−1κ+12∂tσt∫0tϖjt−s∇xκvs−∇xκvt22ds+−1κ2∂tσt∫0tϖjs∇xκv22ds+−1κ2σt∫0t∂tϖjt−s∇xκvs−∇xκvt22ds+−1κ+12σtϖjt∇xκv22+−1κ2σ′t∫0tϖjt−s∇xκvs−∇xκvt22ds+−1κ+12σ′t∫0tϖjsds∇xκv22.
Using the notation (36), we obtain the desired result.


## 3. Main Results

Before stating our main theorems, we define the weak solution for the problem (1)–(4).Definition 7.
The pairw,z is said to be weak solution to (1)–(4) on 0,T0 if it satisfies,
(40)∂tw,φ2+∇xwqx−2∇xw,∇xφ2+∂tz,ψ2+∇xzqx−2∇xz,∇xψ2,+∇x∂tw−σt∫0tϖ1t−swds,∇xφ2−f1w,z,φ2,+∇x∂tz−σt∫0tϖ2t−szds,∇xψ2−f2w,z,ψ2=0,for all test functions φ,ψ∈W01,q·Ω,t∈0,T0.
Here, we present without proof, the first known result concerning the local existence (in time) for the problem (1)–(4).Theorem 8.
Assume that (10), (12), and (14) hold. Then, the problem (1)–(4) has a unique local solution w,z satisfying
(41)w,z∈C0,T0;W01,q·Ω2,∂tw,∂tz∈C0,T0,L2Ω∩L20,T0;H01Ω2,for T0>0 depending on ∥w0∥1,q·,∥z0∥1,q·.To prove the previous theorem, we can adopt the Faedo-Galerkin method which is the same procedure used in [4].We introduce the main result concerned with the finite time blowup of solutions of the problem (1)–(4).We define(42)λ1=1q+−1q−σ∞∫0∞ϖ1sds,λ2=1q+−1q−σ∞∫0∞ϖ2sds,and find
(43)1p−<λ1,λ2<1q+<12,by (10) and (14).Theorem 9.
Assume that (10)–(14) hold. Given w0,z0∈W01,q·Ω satisfying
(44)E0<E1+E2<0,(45)∫Ω∇xw0qdx,∫Ω∇xz0qdx>α1λ1,α2λ2.
Then, any solution of (1) with (3) and (4) blows up in finite time T∗<∞, where
(46)T∗=L01−q−/2q−/2−1Γ,forL0,Γ>0,and E1 and E2 are given in (66).
In order to prove Theorem9, we need to exploit some lemmas. First, we define the modified energy functional Et associated to the problem (1)–(4) by
(47)Et=∫Ω∇xwx,tqxqxdx+∫Ω∇xzx,tqxqxdx+12σtϖ1∘∇xwt+ϖ2∘∇xzt−∫ΩFw,zdx−12σt∫0tϖ1sds∥∇xw∥22+∫0tϖ2sds∥∇xz∥22.Lemma 10.
Letw,z be the solution of (1)–(4) with (10)–(14). Then, the energy functional satisfies
(48)2∂tEt=−2∂tw22−2∇x∂tw22−2∂tz22−2∇x∂tz22+σt∂tϖ1∘∇xwt−ϖ1t∇xw22+σt∂tϖ2∘∇xzt−ϖ2t∇xz22+σ′tϖ1∘∇xwt−∫0tϖ1sds∇xw22+σ′tϖ2∘∇xzt−∫0tϖ2sds∇xz22≤0.Proof.
Multiplying (1)1 by ∂tw and (1)2 by ∂tz, integrating by parts over Ω, summing and using (14) and Lemma 6. Then, it is clear that f1w,z and f2w,z are bounded, so the potential Fw,z exists and is given by (85) as follows
(49)Fw,z=1pxw+zpx+2wzpx/2.
Differentiating (49) with respect to time, it follows that
(50)∂tFw,z=f1w,z,∂tw2+f2w,z,∂tz2.
Hence, the proof is finished .Lemma 11.
Letw,z be a strong solution of (1)–(4) with (10)–(14). Then, we have
(51)Et≥λ1∫Ω∇xwqdx+λ2∫Ω∇xzqdx−∫ΩFw,zdx+12σtϖ1∘∇xwt+ϖ2∘∇xzt−q+−22q+σ∞Ω∫0∞ϖ1sds+∫0∞ϖ2sds.Proof.
Using Young’s inequality, we obtain(52)∇xw2≤2q∇xwq+q−2q≤2q−∇xwq+q+−2q+,∇xz2≤2q∇xzq+q−2q≤2q−∇xzq+q+−2q+,which proves the lemma.Lemma 12.
Letw,z be a strong solution of (1)–(4) with (10)–(14). Then, we have
(53)Et≥λ1∫Ω∇xwqdx+λ2∫Ω∇xzqdx−c1p−∫Ωwpdx−c1p−∫Ωzpdx+12σtϖ1∘∇xwt+ϖ2∘∇xzt−q+−22q+σ∞Ω∫0∞ϖ1sds+∫0∞ϖ2sds.Proof.
By using both equivalences (8) and (10), we deduce
(54)−∫ΩFw,z≥−c1px∫Ωwpxdx+∫Ωzpxdx≥−c1p−∫Ωwpdx+∫Ωzpdx.
The proof is completed by direct use of (54).In condition (10), Ω∞=∅ holds, which yields
(55)Qpxw=∫Ωwpxdx.We derive a shaper estimate than that in Lemma13.Lemma 13.
Letw,z be a strong solution of (1)–(4) with (10)–(14). Then, we have
(56)Et≥λ1∫Ω∇xwqdx+λ2∫Ω∇xzqdx−CSc1p−∫Ω∇xwqdxp+/q−+∫Ω∇xzqdxp+/q−+12σtϖ1∘∇xwt+ϖ2∘∇xzt−2c1p−CS+1−q+−22q+σ∞Ω∫0∞ϖ1sds+∫0∞ϖ2sds,where CS is an embedding constant defined in Lemma 1.Proof.
First of all, note that, fora≥0,
(57)maxap−,ap+=ap+ifa>1ap−ifa≤1≤ap++1.
Then, owing to Lemmas1 and 4, we have
(58)∫Ωwpdx=Qpw≤max∥w∥pp−,∥w∥pp+≤∥w∥pp++1≤CS∥∇xw∥qp++1,∫Ωzpdx=Qpz≤max∥z∥pp−,∥z∥pp+≤∥z∥pp++1≤CS∥∇xz∥qp++1.
Similarly, fora≥0, we have
(59)a=minaq−,aq+1/q−ifa>1,minaq−,aq+1/q+ifa≤1.
Hence, by Lemmas3 and 4, (58) is estimated as
(60)∫Ωwpdx≤CS∥∇xw∥qp++1≤CSQp∇xwp+/q−+1+1,∫Ωzpdx≤CS∥∇xz∥qp++1≤CSQp∇xzp+/q−+1+1.
The substitution of (60) in the estimate (53) ends the proof.Now, we can see that (56) takes the following form
(61)Et≥λ1∫Ω∇xwqdx+λ2∫Ω∇xzqdx+12σtϖ1∘∇xwt+ϖ2∘∇xzt−C1λ1∫Ω∇xwqdx+12σtϖ1∘∇xwtp+/q−−ρ1−C2λ2∫Ω∇xzqdx+12σtϖ2∘∇xztp+/q−−ρ2,where
(62)C1=CSc1p−1λ1p+/q−,C2=CSc1p−1λ2p+/q−,(63)ρ1=c1p−CS+1+q+−22q+σ∞Ω∫0∞ϖ1sds,(64)ρ2=c1p−CS+1+q+−22q+σ∞Ω∫0∞ϖ2sds.Letf·,· be a function defined by
(65)f:ℝ+×ℝ+⟶ℝψ1,ψ2↦ψ1−C1ψ1p+/q−−ρ1+ψ2−C2ψ2p+/q−−ρ2.Then,f·,· is increasing in 0,α1×0,α2 decreasing for ψ1,ψ2>α1,α2, such that fψ1,ψ2⟶−∞ as ψ1,ψ2⟶+∞,+∞, and we have
(66)fα1,α2=p+−q−p+α1−ρ1⏟≡E1+p+−q−p+α2−ρ2⏟≡E2,for
(67)α1,α2=q−C1p+q−/p+−q−,q−C2p+q−/p+−q−.Since, by the fact (43), i.e., 1/λ1,1/λ2>2 holds. Then, by (63)1, we can find
(68)minC1,C2>2p+/q−Csc1p−,for all
(69)maxα1,α2<12p+/p+−q−p−q−p+q−/p+−q−1CSc1q−/p+−q−.Lemma 14.
Under the assumptions of Lemma13. Then, the pair E1,E2 defined in (66) is negative.Proof.
Direct use of (63) and (66). Thus, we obtain
(70)E1+q+−22q+σ∞Ω∫0∞ϖ1sds=p+−q−p+α1−c1p−CS+1,E2+q+−22q+σ∞Ω∫0∞ϖ2sds=p+−q−p+α2−c1p−CS+1.
By considering (68), (69). Then, we use Young’s inequality to get
(71)E1+q+−22q+σ∞Ω∫0∞ϖ1sds<p+−q−p+12p+/p+−q−p−q−p+q−/p+−q−1CSc1q−/p+−q−−c1p−CS+1≤δ01CSc1q−/p+−q−+1δ0p+−q−p+12p+/p+−q−p−q−p+q−/p+−q−−c1p−CS+1<0,E2+q+−22q+σ∞Ω∫0∞ϖ2sds<p+−q−p+12p+/p+−q−p−q−p+q−/p+−q−1CSc1q−/p+−q−−c1p−CS+1≤δ11CSc1q−/p+−q−+1δ1p+−q−p+12p+/p+−q−p−q−p+q−/p+−q−−c1p−CS+1<0.
To provide this, we takec1 sufficiently large and δ0,δ1 small enough.Lemma 15.
Letw,z be a strong solution of (1)–(4) with (10)–(14) and initial condition satisfying (44) and (45). Then, there exists a constants β1>α1 and β2>α2 such that
(72)λ1∫Ω∇xwtqdx+12σtϖ1∘∇xwt,λ2∫Ω∇xztqdx+12σtϖ2∘∇xzt≥β1,β2.Proof.
Since(73)fλ1∫Ω∇xw0qdx,λ2∫Ω∇xz0qdx≤E0<E1+E2,holds by (61). Then, there exists (β1>α1 and β2>α2 such that fβ1,β2=E0. Thus, we have
(74)fλ1∫Ω∇xw0qdx,λ2∫Ω∇xz0qdx≤E0≡fβ1,β2,which implies that
(75)λ1∫Ω∇xw0qdx,λ2∫Ω∇xz0qdx≥β1,β2.
Now, to establish (72), we suppose by contradiction that
(76)λ1∫Ω∇xwt0qdx+12σt0ϖ1∘∇xwt0,λ2∫Ω∇xzt0qdx+12σt0ϖ2∘∇xzt0<β1,β2,for some t0≥0 and by continuity of f·,·, we can choose t0 such that
(77)λ1∫Ω∇xwt0qdx+12σt0ϖ1∘∇xwt0,λ2∫Ω∇xzt0qdx+12σt0ϖ2∘∇xzt0>α1,α2.
Again, the use of (61) leads to
(78)Et0≥fλ1∫Ω∇xwt0qdx+12σt0ϖ1∘∇xwt0,λ2∫Ω∇xwt0qdx+12σt0ϖ2∘∇xzt0>fβ1,β2=E0.
This is impossible, since(79)Et<E0,∀t∈0,T0.
Hence, (72) established.Lemma 16.
Letw,z be a strong solution of (1)–(4) with (10)–(14) and (44). Then, we have
(80)∫Ωwpdx+∫Ωzpdx≥λ1p−c1∫Ω∇xwqdx+λ2p−c1∫Ω∇xzqdx+2CS+1+p−c1C1β1p+/q−−β1+C2β2p+/q−−β2.(81)∫Ωwpdx+∫Ωzpdx≥2CS+1+p−c1C1β1p+/q−+C2β2p+/q−,for all t∈0,T0.Proof.
To prove (80), we exploit Lemmas 10 and 13 to obtain
(82)c1p−∫Ωwpdx+∫Ωzpdx=It≥λ1∫Ω∇xwqdx+λ2∫Ω∇xzqdx−E0+12σtϖ1∘∇xwt+ϖ2∘∇xzt−q+−22q+σ∞Ωmin∫0∞ϖ1sds,∫0∞ϖ2sds.
Then, by the definitions ofρ1,ρ2, f, and E0. So, inequality (82) can be estimated as follows
(83)It≥λ1∫Ω∇xwqdx+λ2∫Ω∇xzqdx−fβ1,β2−q+−22q+σ∞Ω∫0∞ϖ1sds+∫0∞ϖ2sds≥λ1∫Ω∇xwqdx−q+−22q+σ∞Ω∫0∞ϖ1sds−β1−C1β1p+q−−ρ1+λ2∫Ω∇xzqdx−q+−22q+σ∞Ω∫0∞ϖ2sds−β2−C2β2p+/q−−ρ2≥λ1∫Ω∇xwqdx+λ2∫Ω∇xzqdx+2c1p−CS+1+C1β1p+/q−−β2+C2β2p+/q−−β2.
From the previous estimate, we deduce(84)It≥β1−q+−22q+σ∞Ω∫0∞ϖ1sds−β1−C1β1p+/q−−ρ1+β2−q+−22q+σ∞Ω∫0∞ϖ2sds−β2−C2β2p+/q−−ρ2≥2c1p−CS+1+C1β1p+/q−+C2β2p+/q−.
Thanks to (72) and (84) and the definition of different constants, then the desired results are proved.


## 4. Proof of Theorem9

We are now ready to prove Theorem9.Proof.
Let the auxiliary function defined as(85)Lt=12wt22+zt22+∇xwt22+∇xzt22.
By differentiating the functionalL, we get
(86)∂tLt=∫Ωw∂twdx+∫Ωz∂tzdx+∫Ω∇xw∇x∂twdx+∫Ω∇xz∇x∂tzdx=∫Ωwdiv∇xwqx−2∇xw+Δ∂tw−σt∫0tϖ1t−sΔxwsds+f1w,zdx+∫Ω∇xw∇x∂twdx+∫Ωzdiv∇xzqx−2∇xz+Δ∂tz−σt∫0tϖ2t−sΔxzsds+f2w,zdx+∫Ω∇xz∇x∂tzdx=−∫Ω∇xwtqdx−∫Ω∇xztqdx+∫Ωwt+ztp+2wzp/2dx⏞=∫ΩpxFw,zdx+∫Ωσt∫0tϖ1t−s∇xws·∇xwtdsdx+∫Ωσt∫0tϖ2t−s∇xzs·∇xztdsdx.
By using (8), we get
(87)∂tLt≥−∫Ω∇xwtqdx−∫Ω∇xztqdx+c0∫Ωwtpdx+∫Ωztpdx+∫Ωσt∫0tϖ1t−s∇xws·∇xwtdsdx+∫Ωσt∫0tϖ2t−s∇xzs·∇xztdsdx.
Thanks to Cauchy Schwarz’s inequality, we get(88)∫Ω∇xws·∇xwtdx≥2C02−12C02∇xwt22−C022∇xwt−∇xws22,(89)∫Ω∇xzs·∇xwtdx≥2C02−12C02∇xzt22−C022∇xzt−∇xzs22,holds for some positive constant C0>0 to be determined later.
Substituting (89) in (86) to get
(90)∂tLt≥−∫Ω∇xwtqdx−∫Ω∇xztqdx+c0∫Ωwtpdx+∫Ωztpdx+2C02−12C02σt∫0tϖ1sds∇xwt22+∫0tϖ2sds∇xzt22−C022σtϖ1∘∇xwt+ϖ2∘∇xzt.
Using Lemma12 to estimate the terms ϖ1∘∇xw and ϖ2∘∇xz, we get
(91)∂tLt≥C02λ1−1Qq∇xw+C02λ2−1Qq∇xz+c0−c1C02p−Qpw+Qpz+2C02−12C02σt∫0tϖ1sds∇xwt22+∫0tϖ2sds∇xzt22+C02−E1−q+−22q+σ∞Ω∫0∞ϖ1sds+C02−E2−q+−22q+σ∞Ω∫0∞ϖ2sds.
Owing to (43), we can choose C0 as
(92)2<1λ1,1λ2<C02<p−,where c0 is sufficiently large such that c0>C02c1/p−, by Lemma 14, we have
(93)∂tLt>C02λ1−1Qq∇xw+C02λ2−1Qq∇xz+c0−c1C02p−Qpw+Qpz>C02λmin−1Qq∇xw+Qq∇xz+c0−c1C02p−Qpw+Qpz>0,where λmin=minλ1,λ2.
We derive the differential inequality from (93). First, owing to Poincaré’s inequality, we have
(94)w2≤CP∇xw2,z2≤CP∇xz2,holds. For some constant CP>0, we have
(95)Lt≤1+CP22∇xwt22+∇xzt22≤21+CP212q/q−22∇xwtq2+∇xztq2.
By using the following Minkowski’s inequality(96)a+bp≤2p−1ap+bp,we can obtain
(97)∫Ω∇xwqdx+∫Ω∇xzqdx≥21−q−/2∇xwtq2+∇xztq2q−/2≥21−q−/2Lt21+CP212q/q−22q−/2.
Then, by Lemma5, we have L0≥0 and
(98)∂tLt>c0−c1C02p−2CS+1+p−c1C1β1p+/q−+C2β2p+/q−>0,by (82) and (93), we can assume that Lt is sufficiently large for t∈0,T0. Hence, thanks to (95), ∇xwtq>1 and ∇xztq>1 hold for t∈0,T0. Then, using Lemma 4 to get
(99)∇xwtqq−=min∇xwtqq−,∇xwtqq+≤Qp∇xw,∇xztqq−=min∇xztqq−,∇xztqq+≤Qp∇xz,which gives
(100)∂tLt>ΓLtq−/2forallt>0,where
(101)Γ=C02λmin−122−q−/q−21+CP212q/q−2−2q−/2>0.
A direct integration of (100) over 0,t then
(102)Lt1−q−/2<1−q−2Γt+L01−q−/2,yields, which implies that
(103)Lt>1L01−q−/2−q−/2−1Γt2/q−−2,along with 1−q−/2<0. Finally, we have
(104)Lt⟶+∞whent⟶T∗−,where
(105)T∗=L01−q−/2q−/2−1Γ.
Then, the proof is now completed.


## 5. Conclusion

In the present paper, we examined the global nonexistence of solutions for a class of coupled pseudoparabolic equations with weak memories. In fact, the influence of the memory terms are unable to guarantee the stability of our problem. More precisely, we derived some conditions on the functionspx,qx,ϖ1t, and ϖ2t that could occurs blowing up solutions under conditions (10) and (45) with negative initial data. The novelty of our work is to outlined the effects of weak-memory terms, i.e., it lies primarily in the use of a new relation between the relaxation functions and the exponent of the nonlinear sources to get some necessary conditions, which make the problem very interesting from the application point of view, in particular related to the energy systems (including heat systems).The damping terms in this paper are composed with weak-viscoelastic terms(106)σt∫0tϖjt−susds,and strong damping, which let our problem dissipative named pseudoparabolic. The functions fj:ℝ2⟶ℝ,j=1,2 are given in more complicated nonlinearities. We used a large class of Laplacian operator, with variable exponents.After restriction on initial data, we found that when the sources dominate, we get a blow-up finite time, even though the existence of the damping terms which well known to ensure the stability of solutions.



---

*Source: 5573959-2021-02-28.xml*