# Properties of Weak Solutions for a Pseudoparabolic Equation with Logarithmic Nonlinearity of Variable Exponents

Authors: Rongting Pan; Yunzhu Gao; Qiu Meng
Journal: Journal of Mathematics
 (2023)

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## Abstract

In this paper, a new pseudoparabolic equation with logarithmic nonlinearity of variable exponents is investigated. By using the energy functional and the classical potential well, we obtain the global existence and blow-up results of weak solutions with variable exponents. These results extend and improve some recent results in which the blow-up results were showed involving the logarithmic nonlinearity with variable exponents.


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## Body

## 1. Introduction

Consider the following a pseudoparabolic equation problem with variable exponents and logarithmic nonlinear term(1)ut−Δut−div∇upx−2∇u=uqx−2ulnu,x,t∈QT,ux,t=0,x∈∂Ω,ux,0=u0x,x∈Ω,where Ω⊂ℝN is a bounded domain with smooth boundary ∂Ω, QT=Ω×0,T,T>0. The exponent functions px and qx are satisfying the following conditions:(H1)px,qx are measurable functions defined on Ω such that
(2)2≤p−≤px≤p+<q−≤qx≤q+<q−∗x,where p−≔essinfx∈Ωpx,p+≔esssupx∈Ωpx,q−∗x=Np−/N−p−,ifp−<N,+∞,ifp−≥N.(H2)∀z, ξ∈Ω,z−ξ<δ,
(3)pz−pξ+qz−qξ≤αlog1/z−ξ,where α>0 and 0<δ<1.Pseudoparabolic-type equations appear in many areas of applications, such as unidirectional propagation of nonlinear diffusive long waves, aggregation of populations, and nonstationary processes in semiconductor groups with input sources, which have been considered by different authors; see [1–8] and the references therein. They studied the global existence, uniqueness, asymptotic behavior, and blow-up of solutions with pseudoparabolic equations.Let us consider the following pseudoparabolic equations of the form(4)ut−∆ut−div∇up−2∇u=fu.In the case offu=uqx−2u with qx>1, they are usually called the pseudoequations with variable exponents; see [9–13]; they investigated the existence, nonexistence, and blow-up of weak solutions and also considered a long-time behavior of the weak solutions.In the case of logarithmic nonlinearityfu=up−2ulnu, the Dirichlet boundary value problem for equation (4), which is refereed to [14], has been studied by many authors. Moreover, such problems have been extensively investigated by using the method of potential wells and the logarithmic Sobolev inequality. Many results concerning global existence and nonexistence have been proved; see [15–23].In the recent monograph [24], Xiangyu Zhu, Bin Guo, and Menglan Liao studied the following problem
(5)ut−Δut−div∇umx−2∇u=upx−2u.They obtained the global existence and blow-up results of weak solutions with arbitrarily high initial energy.Motivated by these works, we consider the problem (1) with the logarithmic nonlinearity uqx−2ulnu. The aim of this paper is to study the existence and blow-up of weak solutions for a pseudoparabolic equation with the logarithmic nonlinearity of variable exponents. To the best of our knowledge, there are only a few works about the logarithmic nonlinearity with variable exponents; see [25, 26]. So, the study of problem (1) is a new and interesting topic. The main difficulties for treating the problem are caused by the existence of the gap between the norm and the module. In addition, owing to the term uqx−2ulnu leads to some difficulties in deploying the potential well method.The outline of this work is the following: In Section2, we shall introduce some definitions and the function spaces of Orlicz−Sobolev type; in Section 3, we obtain the global existence and blow-up results of weak solutions by analyzing the controlled relationship between the energy functional and the sources and using the method of potential wells.


## 2. Preliminaries

In this section, we introduce some notations and basic properties of the generalized Lebesgue spaceLqxΩ and Lebesgue-Sobolev space Wk,pxΩ.Letpx,qx satisfy the condition (2); we give the definition of variable exponent Lebesgue space Lq·Ω that consists of all measurable functions such that
(6)Aq·u=Δ∫Ωuqxdx<∞,equipped with the Luxemburgnorm(7)uq·=infλ>0:Aq·uλ⩽1.The variable exponent Lebesgue space is a special case of Orlicz-Musielak spaces studied by Tsutsumi in [27].It is obvious thatLq·Ω is a Banach space in [28]. It follows directly from the definition that
(8)minuq·,Ωq−,uq·,Ωq+≤∫Ωuxqxdx≤maxuq·,Ωq−,uq·,Ωq+.By Corollary 3.3.4 [28], we have Lq++1Ω↪Lqx+1Ω, and we also give some lemmas needed in our main results as the following.Lemma 1.
LetΩ be a bounded domain of RN and q· satisfies H1 and H2, and then for all σ>0, the following result hold
(9)minuq·+σ,Ωq−+σ,uq·+σ,Ωq++σ≤∫Ωuxqx+σdx≤maxuq·+σ,Ωq−+σ,uq·+σ,Ωq++σProof.
Sethx=qx+σ, and then we clearly have
(10)uh·,Ωh−=uq·+σ,Ωq−+σ,uh·,Ωh+=uq·+σ,Ωq++σ.
In view of (8), we easily obtain
(11)minuh·,Ωh−,uh·,Ωh+≤∫Ωuxhxdx≤maxuh·,Ωh−,uh·,Ωh+.
Substituting (10) into (11), we get immediately (9).Lemma 2 (see [28, 29]).
LetΩ be a bounded domain of RN andp· satisfies H1 and H2, then
(12)up·≤C∇up·,forallu∈W01,p·Ω,where C is the optimal embedding positive constant which depends on p· and Ω. In particular, the space W01,p·Ω has an equivalent norm given by uW1,p·=∇up·.Lemma 3 (see [28, 29]).
Ifp·∈CΩ¯, q·:Ω⟶1,∞ is a measurable function and
(13)essinfx∈Ωp∗x−qx>0,withp∗x=Npxesssupx∈ΩN−px,ifpx<N,+∞,ifpx≥N.
Then the embeddingW01,p·Ω↪Lq·Ω is continuous and compact.


## 3. Main Results

In this section, the existence and blow-up of weak solutions will be studied. For this purpose, we give the definition of weak solutions as follow.Definition 4.
A functionux,t∈QT is called a weak solution of problem (1) if ux,t∈L∞0,T;W01,pxΩ∩H01Ω∩C0,T,L2Ω with ut∈L20,T;H01Ω,ux,0=u0x, and for all v∈W01,pxΩ∩H01Ω, the following equality holds
(14)∫Ωutv+∇ut∇v+∇upx−2∇u∇vdx=∫Ωuqx−2ulnuvdx,a.e.t∈0,T.Take into account of (2), we get W01,pxΩ∩H01Ω=W01,pxΩ, and then, for any u∈W01,pxΩ, define the functional Ju and Nehari functional Iu as follows:
(15)Ju=∫Ω1px∇upxdx−∫Ω1qxuqxlnudx+∫Ω1p2xupxdx,Iu=∫Ω∇upxdx−∫Ωuqxlnudx,we also define energy functional
(16)Et≔Eux,t=∫Ω1px∇ux,tpxdx−∫Ω1qxux,tqxlnux,tdx+∫Ω1p2xux,tpxdx.Let(17)d=infu∈NJu,where N=u∈W01,pxΩ\0Iu=0. Define
(18)N+=u∈W01,pxΩ\0Iu>0,N−=u∈W01,pxΩ\0Iu<0.In the following, in order to consider the weak solutions of the problem (1), we give some sets and functionals.
(19)Jη=u∈W01,pxΩJu≤η,Nη=N∩Jη=u∈NJu≤η,∀η>d.Furthermore, for anyη>d, we define γη=infuH012u∈Nη, and Γη=supuH012u∈Nη. Obviously, γηΓη is nonincreasing (nondecreasing) with respect to η.Thus, we haveLemma 5 (see [10]).
Ifu is a weak solution to problem (1), the energy functional Et is nonincreasing with respect to t as long as t belongs to the existence interval of weak solutions and
(20)Et=E0−∫0tuτH012dτ.Lemma 6.
The potential depthd is positive.Proof.
For fixedu∈N, from Lemma 1-Lemma 3, and by simple calculations, for each q>0,0≤s−qlns≤1/eq,s∈1,+∞, we have
(21)∫Ω∇upxdx=∫Ωuqxlnudx=∫Ω1uqxlnudx+∫Ω2uqxlnudx≤∫Ω2uqxlnudx≤1eσ∫Ω2uqx+σdx≤1eσ∫Ωuqx+σdx≤1eσmaxuqx+σq++σ,uqx+σq−+σ≤1eσmaxC1q++σmax∫Ω∇upxdxq++σ/p+,∫Ω∇upxdxq++σ/p−,C1q−+σmax∫Ω∇upxdxq−+σ/p+,∫Ω∇upxdxq−+σ/p−,where C1=maxC,1 and C>0 is the optimal embedding constant
(22)Ω1=uu∈W01,pxΩ,0<u<1,Ω2=uu∈W01,pxΩ,u≥1.
Therefore, from (21), we have
(23)∫Ω∇upxdx≥mineσp−/q−+σ−p−C1p−q++σ/p−−q−+σ,eσp+/q−+σ−p+C1p+q−+σ/p+−q−+σ=eσp+/q−+σ−p+C1p+q−+σ/p+−q−+σ>0.
It follows from the definitions ofJu, Iu, and (23) that
(24)Ju≥1p+∫Ω∇upxdx−1q−∫Ωuqxlnudx+1q+2∫Ωuqxdx=1q−∫Ω∇upxdx−∫Ωuqxlnudx+1p+−1q−∫Ω∇upxdx+1q+2∫Ωuqxdx=1q−Iu+q−−p+p+q−∫Ω∇upxdx+1q+2∫Ωuqxdx≥q−−p+p+q−∫Ω∇upxdx.
As a result, from (23), (24), and the definition of d, we obtain
(25)d≥q−−p+p+q−eσp+/q−+σ−p+C1p+q−+σ/p+−q−+σ>0.
Next, we give the properties ofγη and Γη.Lemma 7.
For anyη>d, γη and Γη satisfy 0<γη≤Γη<+∞.Proof.
Consider the following problem of elliptic equation(26)Δw+λw=0,inΩ,w=0,on∂Ω.
Poincaré’s inequality shows ∇u22≥λ1u22; here λ1 is the first eigenvalue of problem (26). Then we get
(27)∇u22=λ11+λ1∇u22+11+λ1∇u22≥λ11+λ1uH012.
On the other hand, forpx≥2, the continuous embedding LpxΩ↪L2Ω implies that
(28)∇u22≤1+Ω2∇upx2.
Combining (27) with (28), we have
(29)uH012≤1+λ1λ11+Ω2max∫Ω∇upxdx2/p+,∫Ω∇upxdx2/p−.
For anyu∈Nη, recalling (19) and (24), we have
(30)∫Ω∇upxdx≤ηp+q−q−−p+.
(29) and (30) yield
(31)uH012≤1+λ1λ11+Ω2maxηp+q−q−−p+2/p+,ηp+q−q−−p+2/p−,which implies Γη<+∞.
For anyu∈Nη, in view of the Sobolev embedding W01,pxΩ↪LqxΩ, as well as the definition of Iu and d, we have
(32)d=infu∈NJu≤Ju≤1p−∫Ω∇upxdx−1q+∫Ωuqxlnudx+1q−2∫Ωuqxdx=1q+∫Ω∇upxdx−∫Ωuqxlnudx+1p−−1q+∫Ω∇upxdx+1q−2∫Ωuqxdx=1q+Iu+q+−p−p−q+∫Ω∇upxdx+1q−2∫Ωuqxdx≤q+−p−p−q+∫Ω∇upxdx+1q−2maxuqxq+,uqxq−≤q+−p−p−q+∫Ω∇upxdx+1q−2maxC2q+max∫Ω∇upxdxq+/p+,∫Ω∇upxdxq+/p−,C2q−max∫Ω∇upxdxq−/p+,∫Ω∇upxdxq−/p−,where C2=maxC∗,1, C∗>0 is the optimal embedding constant.
Case 1. For∫Ω∇upxdx>1, on the basis of the discussion above, we can easily obtain
(33)d≤q+−p−p−q+∫Ω∇upxdxq+/p−+1q−2C2q++C2q−∫Ω∇upxdxq+/p−=q+−p−p−q++1q−2C2q++C2q−∫Ω∇upxdxq+/p−,which implies that
(34)∫Ω∇upxdx≥q+−p−p−q++1q−2C2q++C2q−−p−/q+dp−/q+.
Case 2. For0<∫Ω∇upxdx<1, according to discussion above, we can easily get
(35)d≤q+−p−p−q+∫Ω∇upxdx+1q−2C2q++C2q−∫Ω∇upxdx=q+−p−p−q++1q−2C2q++C2q−∫Ω∇upxdx,which implies that
(36)∫Ω∇upxdx≥q+−p−p−q++1q−2C2q++C2q−−1d.
Set(37)s=minq+−p−p−q++1q−2C2q++C2q−−p−/q+dp−/q+,q+−p−p−q++1q−2C2q++C2q−−1d.
From (34) to (36), we get
(38)∫Ω∇upxdx≥s.
The Gagliardo-Nirenberg inequality and the continuous embeddingLq++σΩ↪Lqx+σ imply that there exists a positive constant C3 such that
(39)uqx+σ≤Ω+1uq++σ≤C3∇upxθu21−θ≤C3∇upxθuH011−θ,where θ=1/2−1/q++σ1/N−1/p−+1/2−1∈0,1 as a result of (2). Furthermore, noticing that u∈N, then we can obtain
(40)∫Ω∇upxdx=∫Ωuqxlnudx=∫Ω1uqxlnudx+∫Ω2uqxlnudx≤∫Ω2uqxlnudx≤1eσ∫Ω2uqx+σdx≤1eσ∫Ωuqx+σdx=1eσuqx+σqx+σ.
Moreover, we have(41)uqx+σ≥mineσ∫Ω∇upxdx1/q++σ,eσ∫Ω∇upxdx1/q−+σ.
In addition, we combine (39) with (41) to obtain
(42)uH012≥mina+C3∫Ω∇upxdx1/q++σ−θ/p−2/1−θ,a−C3∫Ω∇upx1/q−+σ−θ/p+2/1−θ,where a+=eσ1/q++σ, a−=eσ1/q−+σ.
Obviously, the right-hand side of the above inequality remains bounded away from0 no matter what the sign of 1/q++σ−θ/p− and 1/q−+σ−θ/p+ are due to (30) and (38). Therefore, γη>0.In fact, we can get a lower bound ofγη that is independent of η.Proposition 8.
If2≤p−≤px≤p+<q−≤qx≤q+<2∗p−, then for any η>d, γη≥M>0, here
(43)2∗p−=Np−N−2,if2<N,+∞,if2≥N.
M is given in (48).Proof.
In the following, we will divide the proof of this part into two cases.
Case 1. For2<q−≤qx≤q+≤2N/N−2, the Sobolev embedding H01Ω↪Lqx+σΩ (see [30]) and (9) yield
(44)s≤∫Ω∇upxdx=∫Ωuqxlnudx=∫Ω1uqxlnudx+∫Ω2uqxlnudx≤∫Ω2uqxlnudx≤1eσ∫Ω2uqx+σdx≤1eσ∫Ωuqx+σdx≤1eσmaxuqx+σq++σ,uqx+σq−+σ≤1eσmaxC¯q++σuH01q++σ,C¯q−+σuH01q−+σ,which implies
(45)γη≥1C¯2mina+2s2/q++σ,a−2s2/q−+σ>0,where C¯ is the optimal embedding constant for H01Ω↪Lqx+σΩ.
Case 2. For2N/N−2<q−≤qx≤q+≤Np−/N−2, the Gagliardo-Nirenberg inequality and the embedding inequality Lq++σΩ↪Lqx+σΩ indicate that there exists a positive constant C4 such that
(46)uqx+σ≤Ω+1uq++σ≤C4∇upxθu2∗1−θ≤C4∇upxθuH011−θ,where 1/q++σ=θ/p−∗+1−θ/2∗, p−∗=Np−/N−p−, and 2∗=2N/N−2. A simple computation shows that θq++σ/p−=q++σ−2∗p−∗/p−∗−2∗p−<1. By the same argument of Lemma 7, we easily obtain
(47)γη≥mina+C4s1/q++σ−θ/p−2/1−θ,a−C4s1/q−+σ−θ/p+2/1−θ>0.
By (45) and (47), we get γη≥M, here
(48)M=1C¯2mina+2s2/q++σ,a−2s2/q−+σ,2<q+≤2NN−2,1C42/1−θmina+sp−−q++σθ/q++σp−2/1−θ,a−sp+−q−+σθ/q−+σp+2/1−θ,2NN−2<q+<Np−N−2.Theorem 9.
Assume thatd<Ju0=E0≤η with η∈d,+∞, we have
(1)
ifu0∈N+ and u0H012≤γJu0, the weak solution u of problem (1) exists globally and ut⟶0 as t⟶+∞(2)
ifu0∈N− and u0H012≥ΓJu0, the weak solution u of problem (1) blows up in finite timeProof.
Denote byTu0 the maximal existence time of weak solutions for problem (1). If there exists a global solution, i.e., Tu0=∞, we denote by wu0=∩t≥0us:s≥t¯ the w-limit of u0.
(1)
Ifu0∈N+ and u0H012≤γJu0, we claim that u∈N+ for all t∈0,Tu0. By contradiction, there exists a t0∈0,Tu0 such that u∈N+ for t∈0,t0 and ut0∈N. Choosing v=u in (14), we obtain(49)ddtu0H012=−2Iu.
Therefore, it follows from (49) that ∫0tuτH012dτ≠0 for Ω×0,t0. Further, (20) yield Jut0=Et0<Ju0, which implies ut0∈JJu0. Therefore, ut0∈NJu0. By the definition of γJu0, we obtain
(50)ut0H012≥γJu0.
Noticing thatIu>0 for t∈0,t0, according to (49), we have ut0H012<u0H012≤γJu0, which contradicts (50). Therefore, u∈N+ for all t∈0,Tu0. Further, we get for all t∈0,Tu0, ut∈JJu0∩N+. Moreover, it follows from (24) that
(51)Ju0≥Ju≥1q−Iu+q−+p+p+q−∫Ω∇upxdx+1q+2∫Ωuqxdx>q−+p+p+q−∫Ω∇upxdx,which indicates that the boundedness of uW01,px is independent of t, Tu0=+∞.
For anyw∈wu0, then wH012<u0H012≤γJu0, Jw≤Ju0 by (49) and (20), respectively. The second inequality implies w∈JJu0. Noticing that the definition of γJu0 and the first inequality, we derive w∉NJu0; further, w∉N. Thus, wu0∩N=∅, which indicates wu0=0. Therefore, the weak solution u of problem (1) exists globally and ut⟶0 as t⟶+∞.
(2)
Ifu0∈N− and u0H012≥ΓJu0, we claim that u∈N− for all t∈0,Tu0. By contradiction, there exists a t0∈0,Tu0 such that u∈N− for t∈0,t0 and ut0∈N. Similar to case (1), we get Jut0<Ju0, which implies ut0∈JJu0. Therefore, ut0∈NJu0. By the definition of ΓJu0, we obtain(52)ut0H012≤ΓJu0.
Noticing thatIut<0 for t∈0,t0, it follows from (49) that ut0H012>u0H012≥ΓJu0, which contradicts (52). Therefore, u∈N− for all t∈0,Tu0. Further, we get for all t∈0,Tu0, ut∈JJu0∩N−. Suppose Tu0=∞, then for any w∈wu0, then wH012>u0H012≥ΓJu0, Jw≤Ju0 by (49) and (20), respectively. The second inequality implies w∈JJu0. Noticing that the definition of ΓJu0 and the first inequality, we derive w∉NJu0; further, w∉N. Thus, wu0∩N=∅, which indicates wu0=0. This result contradicts d which is dist0,N−=infu∈N−uW01,px>0 that is obtained by the same argument of (23). Therefore, wu0=∅, Tu0<∞; that is, the weak solution u of problem (1) blows up in finite time.Forpx=p, in particular, we have the following Proposition 10 that is easily proved based on Theorem 9(2), here we omit the proof. Meanwhile, we give the following Theorem 11 to illustrate there exists u0 such that Ju0 is arbitrarily large, and the corresponding solution ux,t to problem (1) with u0 as initial datum blows up in finite time as well.Proposition 10.
IfJu0 satisfies d<Ju0<q−−p/pq−1+λ1/λ11+Ω2−p/2uH01p, then u0∈N− and the weak solution u of problem (1) blows up in finite time.Theorem 11.
For anyM>d, there exists initial value uM∈N− such that JuM=M, and the weak solution u of problem (1) blows up in finite time.Proof.
Assume thatM>d and Ω3, Ω4 are two arbitrary disjoint open subdomains of Ω. Furthermore, we assume that ν∈W01,pΩ3 is an arbitrary nonzero function. Recalling (2), then we take ζ>0 large enough such that
(53)Jζν=∫Ωζpp∇νp−ζqxqxνqxlnζν+ζqxq2xνqxdx≤0,M<q−−ppq−1+λ1λ11+Ω2−p/2ζνH01p.
We fix such a numberζ>0 and choose a function μ∈W01,pΩ4 satisfying M=Jμ+Jζν. Extend ν and μ to be 0 in Ω\Ω3 and Ω\Ω4. Set uM=ζν+μ, then M=Jζν+μ=JuM, and it follows that
(54)q−−ppq−1+λ1λ11+Ω2−p/2uMH01p≥q−−ppq−1+λ1λ11+Ω2−p/2ζνH01p>JuM.
By Proposition10, uM∈N− and the weak solution u of problem (1) blows up in finite time.



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