# Existence of Exponentialp-Stability Nonconstant Equilibrium of Markovian Jumping Nonlinear Diffusion Equations via Ekeland Variational Principle

Authors: Ruofeng Rao; Shouming Zhong
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

The authors obtained a delay-dependent exponentialp-stability criterion for a class of Markovian jumping nonlinear diffusion equations by employing the Lyapunov stability theory and some variational methods. As far as we know, it is the first time to apply Ekeland variational principle to obtain the existence of exponential stability equilibrium of p-Laplacian dynamic system so that some methods used in this paper are different from those methods of many previous related literatures. In addition, the obtained existence criterion is only involved in the activation functions so that the criterion is simpler and easier than other existence criteria to be verified in practical application. Moreover, a numerical example shows the effectiveness of the proposed methods owing to the large allowable variation range of time-delay.


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## Body

## 1. Introduction

Nonlinear diffusion equations have been investigated extensively by many authors owing to their physics and biological engineering backgrounds, population dynamics, and so on (see [1–9] and references therein). In addition, Markovian jumping systems have attracted rapidly growing interest due to the fact that Markovian jumping parameters are useful in modeling abrupt phenomena, such as random failures, operating in different points of a nonlinear plant, and changing in the interconnections of subsystems (see [8, 10–12] and references therein). On the other hand, fuzzy logic theory has shown to be an appealing and efficient approach to deal with the analysis and synthesis problems for complex nonlinear systems. Among various kinds of fuzzy methods, Takagi-Sugeno (T-S) fuzzy models provide a successful method to describe certain complex nonlinear system using some local linear subsystems (see [13–15] and references therein). As pointed out in the above related literature, the Markovian jumping T-S fuzzy mathematical models have always found their extensive applications in the real world. However, almost all the applications are greatly dependent on the stability of systems, which can often come down to the stability of the equilibrium solution for the corresponding mathematical models. So in this paper, we may consider the stability of the nonlinear p-Laplace (p>1) diffusion fuzzy equations with Markovian jumping. Note that when p=2, the so-called reaction-diffusion equations have been widely investigated (see [16–20] and references therein). For example, under Dirichlet boundary conditions, the existence result of μ-stability equilibrium solution in the sense of L2 norm for a class of time-delay reaction-diffusion equations was obtained in [16]. Motivated by the abovementioned literature, in this paper, we will synthetically employ Ekeland variational principle, Sobolev imbedding theorem, and the Lyapunov functional method to study the existence of exponential p-stability nonconstant equilibrium solution in the sense of Lp norm (p>1) for delayed Markovian jumping fuzzy equations with nonlinear p-Laplace diffusion (p>1).Let{r(t):[0,+∞)→S} be a right-continuous Markov process on the complete probability space (S,Ϝ,P) with a natural filtration {Ϝt}t⩾0 and take values in the finite space S≜{1,2,…,N} with generator ∏={πik} given by (1)Prt+δ=k∣rt=i=πikδ+oδ,k≠i,1+πikδ+oδ,k=i,where πik⩾0 is transition probability rate from i to k  (k≠i) and πii=-∑k∈S,k≠iπik, δ>0, and limδ→0o(δ)/δ=0.Consider the following fuzzy T-S Markovian jumpingp-Laplace partial dynamic equations.Fuzzy rulej: IF ω1 is νj1 and … ωκ is νjκ THEN (2)duit,xdt=Didiv⁡∇uip-2∇ui-biui+cijrtfiuit,x+dijrtgiuit-τit,x+Ii,i∈N,t>0,x∈Ω,uit,x=0,i∈N,t⩾0,x∈∂Ω,uis,x=ϕis,x,-τi0⩽s⩽0,0⩽τit⩽τi,i∈N,t⩾0,x∈Ω,where ωk(t)  (k=1,2,…,κ) is the premise variable and νjk  (j=1,2,…,J;k=1,2,…,κ) is the fuzzy set that is characterized by membership function. J is the number of the IF-THEN rules and κ is the number of the premise variables. Denote the premise variable vector ω(t)=[ω1(t),ω2(t),…,ωκ(t)] and ρj(ω(t))=wj(ω(t))/∑k=1Jwk(ω(t)), where wjωt:Rκ→0,1  (j=1,2,…,J) is the membership function of the system with respect to the fuzzy rule j.ρj can be regarded as the normalized weight of each IF-THEN rule, satisfying (3)ρjωt⩾0,∑j=1Jρjωt=1.Using a singleton fuzzifier, product fuzzy inference, and weighted average defuzzifier, system (2) is inferred as follows: (4)duit,xdt=Didiv⁡∇uip-2∇ui-biui+∑j=1Jρjωtcijrtfiuit,x+dijrtgiuit-τit,x+Ii,i∈N,t>0,x∈Ω,uit,x=0,i∈N,t⩾0,x∈∂Ωwith the initial condition4′uis,x=ϕis,x,-τi0⩽s⩽0,0⩽τit⩽τi,i∈N,t⩾0,x∈Ω,where Ω is a bounded subset in Rm with smooth boundary ∂Ω. p>1 is a scalar, and N={1,2,…,n} is a finite set. The divergence div(|∇ui(t,x)|p-2∇ui(t,x))=Δpui(t,x) is the p-Laplacian (see [21] for details). In mode r(t)=r, we denote cij(r(t))=cijr and dij(r(t))=dijr, which imply the connection strengths of the ith neuron on the jth neuron at the system mode r(t)=r, respectively. For any given i∈N, Di>0 is a corresponding constant, dependent on i.Remark 1.
Our methods employed in this paper are different from those of previous related literature. For example, homomorphic mapping theory was employed to obtain the existence of equilibrium of ordinary differential equations in [22]; topological degree theory was used to obtain the existence of equilibrium for fuzzy ordinary differential equations in [23] and of equilibrium for reaction-diffusion partial differential equations in [17]. In this paper, Ekeland variational principle is originally proposed to solve the existence of nonconstant equilibrium for nonlinear diffusion equations. Note that the abovementioned constant equilibrium point u∗=(u1∗,…,un∗)T can actually be regarded as the special case of our nonconstant equilibrium point u=u∗(x)=(u1∗(x),…,un∗(x))T with ui∗(x) ≡ constant for i=1,2,…,n. In addition, our criterion about existence is only involved in the activation functions while other more parameters need be considered in the proof of the existence of constant equilibrium point in those previous literatures (see Remark 8 below for details).


## 2. Preparation

Throughout this paper, we assume the following:(A1)
There exists a positive definition matrixB=diag(B1,B2,…,Bn) such that (5)infs∈R⁡bi′s⩾Bi>0,∀i∈N.(A2)
There exist positive definite matricesF=diag(F1,F2,…,Fn) and G=diag(G1,G2,…,Gn) such that (6)fis-fit⩽Fis-t,gis-git⩽Gis-t,s,t∈R,i∈N.Definition 2.
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T is called a nonconstant equilibrium solution of PDEs (4) if and only if u=u∗(x) satisfies (4). In addition, the nonconstant equilibrium solution u∗(x) of PDEs (4) is called stochastically global exponential p-stability about Lp norm, if there are constants M>0 and  δ>0 for every stochastic field solution u(t,x)=(u1(t,x),u2(t,x),…,un(t,x))T of PDEs (4) such that (7)Eut,x-u∗Lp⩽Me-δt-t0,where one denotes by uLp=(∑i=1nuiLpp)1/p the Lp norm and denotes uiLp=(∫Ω|ui|dx)1/p. Usually, t0=0.Definition 3.
LetX be a Banach space, φ∈C1(X,R), and c∈R. φ satisfies the (PS)c condition if any sequence {un}⊂X, such that (8)φun⟶c,φ′un⟶0,has a convergent subsequence. By the way, the above sequence {un} with φ(un)→c and φ′(un)→0 is called the (PS)c sequence of φ for a given c∈R.The following lemma originated from the famous Sobolev imbedding theorem.Lemma 4.
LetΩ be a bounded subset in Rm with smooth boundary ∂Ω. For 1<q<p, there exist the corresponding positive constants c1 and cq such that, for any ξ∈W01,p(Ω),(9)∫Ωξdx⩽c1ξ,∫Ωξqdx1/q⩽cqξ,where the Sobolev space W01,p(Ω) is the completion of C0∞(Ω) with respect to the norm ξ=(∫Ω|∇ξ|pdx)1/p.In 1979, Ekeland proposed the following famous Ekeland variational principle and its proof in [24]. As is well known, Ekeland variational principle has been the most important result in nonlinear analysis and has been applied to optimization theory, control theory, economic equilibrium theory, critical point theory, dynamic systems, and so forth. In this paper, we also need the following Ekeland variational principle.Lemma 5 (Ekeland variational principle [24, Theorem 1]).
LetX be a complete metric space and let φ:X→(-∞,+∞] be a lower semicontinuous function, bounded from below and not identical to +∞. Let ɛ>0 be given and let ξ∈X be such that (10)φξ⩽infX⁡φ+ɛ.
Then there existsv∈X such that (11)φv⩽φξ,dist⁡ξ,v⩽1,and, for each w≠v in X,(12)φw>φv-ɛdist⁡v,w.


## 3. Main Result

Theorem 6.
Letp>1. Assume that there exists a positive scalar q=q1/q2 with 1<q<p such that (13)lims→+∞⁡bissq-1=lims→+∞⁡fissq-1=lims→+∞⁡gissq-1=0=bi0,∀i∈N,where q1 is an odd number and so is q2. Assume, in addition, |Ii|<M.
If there exist a sequence of positive scalarshi(r(t))  (r∈S,i∈N) such that (14)-A+eβτ1-KB<0,where τ=maxi∈Nτi, τi′(t)⩽K<1 for all i∈N, A=diag(a1,a2,…,an), B=diag(b1,b2,…,bn), ai=minr∈Sai(r), bi=maxr∈Sbi(r), and(15)air=hirtpBi-β-pFi∑j=1Jcijr-p-1Gi∑j=1Jdijr,bir=hirtGi∑j=1Jdijr,r∈S,then there exists a nonconstant equilibrium solution for PDEs (4), which is stochastically global exponential p-stability about Lp norm.Proof. The following proof may be divided into two big steps.Step 1.
Firstly, we need prove that there exists a nonconstant equilibrium solution for (4).
Consider the functional(16)φiη=∫Ω1pDi∇ηp+Biη-∑j=1JρjωtcijrFiη+dijrGiη-Iiηdx,where Bi(s)=∫0sbi(θ)dθ, Fi(s)=∫0sfi(θ)dθ, Gi(s)=∫0sgi(θ)dθ, and η=η(t,x)∈W01,p(Ω), where the Sobolev space W01,p(Ω) is the completion of C0∞(Ω) with respect to the norm η=(∫Ω|∇η|pdx)1/p.
It is obvious thatφi∈C1(W01,p(Ω),R), for all i∈N. If its critical point ui∗(x) exists, then u∗(x)=(u1∗(x),u2∗(x),…,un∗(xn))T must be a nonconstant equilibrium solution of (4). So we only need to prove the existence of the critical point of φi for all i∈N.
Next, it follows from (13) that there exists a large enough aM>0 such that (17)fis+gis⩽sq-1,s>aM.Furthermore, we can conclude by the continuity of fi and gi that (18)fis+gis⩽bM,∀s⩽aMand hence, (19)fis+gis⩽bM+sq-1,∀s∈R,where bM>0 is a constant.
Fors⩾0, we can derive by q=q1/q2 and the restrictive conditions on q1, q2(20)Fis⩽∫0sfiydy⩽bMs+1qsq=bMs+1qsq,where bM>0 is a constant.
On the other hand, fors<0, we can get by q=q1/q2 and the restrictive conditions on q1, q2(21)Fis⩽∫s0fiydy⩽∫s0bM+yq-1dy⩽bMs+1qsq.
Hence,(22)Fis⩽bMs+1qsq,∀s∈R.Similarly, we can also deduce that (23)Gis⩽bMs+1qsq,∀s∈R.
Denote|cij|=maxr∈S|cijr| and |dij|=maxr∈S|dijr|; then |cij|⩾0 and |dij|⩾0 are constants, independent of r. Similarly, we can prove that there exist ai1>0 and ai2>0 such that (24)∑j=1JρjωtcijrFis+dijrGis⩽∑j=1JcijFis+∑j=1JdijGis⩽ai1s+ai2sq,∀s∈R,where ai1=bM(∑j=1J|cij|+∑j=1J|dij|) and ai2=1/q(∑j=1J|cij|+∑j=1J|dij|) are the constants, independent of r.
Thereby, we have(25)∑j=1JρjωtcijrFis+dijrGis+Iis⩽ai1+Ms+ai2sq,∀s∈R.
Sincebi′(s)>0, we know that bi(s)>bi(0)=0 if s>0 and bi(s)<bi(0)=0 if s<0. And hence Bi(s)=∫0sbi(θ)dθ⩾0 for all s∈R. Besides, we know from the above analysis and the Sobolev imbedding theorem (Lemma 4) that there exist positive constants bi1=c1(ai1+M) and bi2=cqqai2, independent of r, such that (26)φiη⩾1pDiηp-ai1+M∫Ωηdx-ai2∫Ωηqdx⩾1pDiηp-bi2ηq-bi1η,i∈N.
Denoteμ(s)=1/pDisp-bi2sq-bi1s. Owing to p>q>1, there exists a large enough constant s0>0 such that μ(s)>0 for all |s|>s0. And hence (27)μs⩾min⁡0,mins∈-s0,s0⁡μs,s∈R,which implies that φi is bounded below. And the infimum may be defined as ci=infXφi, where we denote X=W01,p(Ω) for convenience and denote by X∗ its dual space. Define the operators Ai,Bi:X→X∗ as follows: (28)Aiη,ξ=Di∫Ω∇ηp-2∇η∇ξdx,Biη,ξ=∫Ωbiη-∑j=1Jρjωtcijrfiη+dijrgiη-Iiξdx,where η,ξ∈X,〈·,·〉 denotes the adjoint pair for X and X∗. It follows by (A1) and (A2) that all bi, fi, gi are continuous. Similarly to (19), we can get (29)bis+fis+gis⩽cM+sq-1,∀s∈R,where cM is a constant. we can conclude from (13) and (29) that there exist positive constants ai3 and ai4, independent of r, such that (30)bis-∑j=1Jρjωtcijrfis+dijrgis-Ii⩽cM+sq-1+∑j=1Jcij·fis+∑j=1Jdijr·gis+M⩽ai3+ai4sq-1,s∈R.Here, ai3=M+cM+cM(∑j=1J|cij|+∑j=1J|dij|) and ai4=1+∑j=1J|cij|+∑j=1J|dij|.
So we know from [25, 26] that both Ai and Ai-1 are continuous, and Bi is compact.
Next, we claim that, for eachi∈N, the functional φi must satisfy the (PS)c condition if only every (PS)c sequence of φi is bounded.
Indeed, let{ηn} be the (PS)c sequence of φi for any given c. It is obvious that (31)φi′ηn,ξ=Aiηn-Biηn,ξ,∀ξ∈X.Owing to φi′(ηn)→0, we have (Ai(ηn)-Bi(ηn))→0. If {ηn} is bounded, we know from the reflexivity of the Banach space W01,p(Ω) that there exists a weakly convergent subsequence (say, {ηn}). Since Bi is compact, {Bi(ηn)} must own a convergent subsequence, which implies that Ai(ηn) must own a convergent subsequence. And furthermore, the continuity of Ai-1 yields that {ηn} owns a convergent subsequence.
Below, we only need to prove that every(PS)ci sequence (say, {ηn}) of φi is bounded.
Indeed, similarly to (26), we have (32)φiηn⩾1pDiηnp-bi2ηnq-bi1ηn.
Owing to the boundedness of{φi(ηn)}, it is not difficult to prove by the application of reduction to absurdity that {ηn} must be bounded in W01,p(Ω).
Now we may define the metric for the spaceX as follows: (33)dist⁡η,ξ=η-ξ.Then X is a complete metric space with the above metric. From the continuity of φi and the above analysis, we know that φ:W01,p(Ω)→(-∞,+∞] is a lower semicontinuous function and bounded from below. Owing to 0∈W01,p(Ω), we can compute and deduce that |φi(0)|⩽∑j=1J(cijFi(0)+dijGi(0)), which implies φi≢+∞, ∀i∈N.
According to Ekeland variational principle, for givenɛ=1/n, there exists ηn∈X such that (34)φiη>φiηn-1nη-ηn,∀η≠ηn,(35)φiηn<ci+1n.So we can deduce from (34) (36)φi′ηn=supξ=1⁡dφiηn,ξ⩽1n.Here, dφi(ηn,ξ) is Gâteaux derivative of φi at ηn, and φi′(ηn) is Fréchet derivative of φi at ηn. Besides, (35) yields φi(ηn)→ci. Then we can conclude from the PSci condition that there exist a convergent subsequence {ηnj}⊂{ηn} and ui∗∈W01,p(Ω) such that ηnj→ui∗. Moreover, the continuity of φi yields φi(ui∗)=ci. Hence, φi has a critical point ui∗(x) in X for all i∈N. That is, there exists an equilibrium u∗=(u1∗,u2∗,…,un∗)T for (4).Step 2.
Below, we will prove the exponentialp-stability for the equilibrium point u∗.
Consider the Lyapunov-Krasovskii functional as(37)Vit,rt=V1it,rt+V2it,rti∈N,where (38)V1it,rt=eβt∫Ωhirtuit,x-ui∗xpdx,V2it,rt=bi1-Keβτ∫Ω∫t-τitteβsuis,x-ui∗xpdsdx.Let L be the weak infinitesimal operator; then for any given mode r(t)=r∈S, taking the derivative of V1i(t,r) with respect to t along the trajectory of (4) yields (39)LV1it,r=limδ→0+⁡1δE∫Ωeβt+δhirt+δuit+δ,x-ui∗pdx∣rt=r-eβt∫Ωhirtuit,x-ui∗xpdx=eβt∫Ωhirtpui-ui∗p-2ui-ui∗·Di∑k=1m∂∂xk∇uip-2∂ui∂xk-∂∂xk∇ui∗p-2∂ui∗∂xk-biui-biui∗+∑j=1Jρjωtcijrfiui-fiui∗+∑j=1Jρjωtdijrgiuit-τit,x-giui∗dx+∫Ω∑k∈Sπikhikuit,x-ui∗pdx+βhirt∫Ωui-ui∗pdx.Next, we claim that (40)∑k=1m∫ΩDiui-ui∗p-2ui-ui∗∂∂xk∇uip-2∂ui∂xk-∂∂xk∇ui∗p-2∂ui∗∂xkdx⩽0.To verify (40), we have to prove firstly the following proposition by the Yang inequality.Proposition 7.
Forη,ψ∈W01,p(Ω), one has (41)∇ηp-2∇η-∇ψp-2∇ψ∇η-∇ψ⩾0.Proof.
In fact, the Yang inequality yields(42)∇ηp-2∇η∇ψ⩽∇ηp-1∇ψ⩽p-1p∇ηp-1p/p-1+∇ψpp,∇ψp-2∇ψ∇η⩽p-1p∇ψp-1p/p-1+∇ηpp.Synthesizing the above two inequalities results in (41).
So we can get by Gauss formula, the Dirichlet zero-boundary value, and Proposition7(43)∑k=1m∫ΩDiui-ui∗p-2ui-ui∗·∂∂xk∇uip-2∂ui∂xk-∂∂xk∇ui∗p-2∂ui∗∂xkdx=-∑k=1m∫ΩDi∇uip-2∂ui∂xk-∇ui∗p-2∂ui∗∂xk·∂∂xkui-ui∗p-2ui-ui∗dx=-∫ΩDip-1ui-ui∗p-2∇uip-2∇ui-∇ui∗p-2∇ui∗·∇ui-∇ui∗dx⩽0,which proves (40).In addition, we get by (A1)(44)∫Ωhirtpui-ui∗p-2ui-ui∗·biui-biui∗dx⩾hirt·pBi∫Ωui-ui∗pdx.Further, we can derive by (A2) (45)∫Ωhirtpui-ui∗p-2ui-ui∗·∑j=1Jρjωtcijrfiui-fiui∗dx⩽hirt·pFi∑j=1Jcijr∫Ωui-ui∗pdx.In addition, (A2) and the Yang inequality yield(46)∫Ωhirtpui-ui∗p-2ui-ui∗·∑j=1Jρjωtdijrgiuit-τit,x-giui∗⩽hirtGi∑j=1Jdijrp-1∫Ωui-ui∗pdx+∫Ωuit-τit,x-ui∗pdx.Combining the above analyses results in (47)LV1it,r⩽eβt-air∫Ωui-ui∗pdx+bir∫Ωuit-τit,x-ui∗pdx,∀r∈S.On the other hand, we have(48)LV2i⩽bi1-Keβτ∫Ωeβtuit,x-ui∗xpdx-bi∫Ωeβtuit-τit,x-ui∗xpdx.Thus, we get by (14) (49)LVit,rt⩽eβt-ai+bi1-Keβτ∫Ωui-ui∗pdx⩽0,∀t⩾0.So we can obtain by the Dynkin formula(50)EVt,rt-EV0,r0=E∫0tLVs,rsds⩽0,t⩾0.Hence, we have (51)minr∈S⁡hirtEeβt∫Ωuit,x-ui∗xpdx⩽EVt,rt⩽EV0,r0⩽Emaxr∈S⁡hirt∫Ωui0,x-ui∗xpdx+bi1-Keβτ∫Ω∫-τ0eβsuis,x-ui∗xpdsdx⩽maxr∈S⁡hirt+bi1-Keβτ·sup-τ⩽s⩽0⁡Eϕis-ui∗Lpp,which implies (52)E∫Ωuit,x-ui∗xpdx⩽hie-βtsup-τ⩽s⩽0⁡Eϕis-ui∗Lpp,i∈N,where hi=maxr∈Shi(r(t))+bi/1-Keβτ/minr∈Shi(r(t))>0, ∀i∈N. Then we have (53)Eu-u∗Lpp=∑i∈NE∫Ωuit,x-ui∗xpdx⩽maxi∈N⁡hie-βtEϕ-u∗τp,where we denote Eϕ-u∗τp=∑i∈Nsup-τ⩽s⩽0(Eϕi(s)-ui∗Lpp).Now, we can conclude from Definition2 that the nonconstant equilibrium solution of (4) is stochastically exponentially p-stable about Lp norm. And that completes the proof of Theorem 6.Remark 8.
In [27], existence theorems of stochastic differential equations on t∈[t0,T) were given under some conditions on activation functions, where T>0 is a constant. And in [28, 29], existence theorems of stochastic differential equations were presented under some conditions on function V∈C1,2([t0-τ,T)×Rn;R+). Motivated by [27], we proposed some conditions on activation functions to set up existence criterion for the equilibrium solution of system (4). In [22, 23], the constant equilibrium solution u=u∗ for all t∈[t0,+∞) was obtained by homomorphic mapping theory and matrix theory, or matrix theory and homotopy invariance theorem, where u∗=(u1∗,u2∗,…,ui∗,…,un∗), and each ui∗ is a constant. In this paper, we also need to consider the equilibrium solution of (4) defined on [t0,+∞). Different from [22, 23], we consider the nonconstant equilibrium solution u=u∗(x)=(u1∗(x),u2∗(x),…,ui∗(x),…,un∗(x)) for all t∈[t0,+∞). This equilibrium solution is a solution for a nonlinear p-Laplacian elliptic partial differential equation whose space frame may be considered as infinite dimension function space W01,p(Ω). And variational method is always a powerful tool to solve the problem. Although the variational method is more complicated than homomorphic mapping method, M-matrix method, or homotopy invariance theorem, our criterion about existence is only involved in the activation functions (remark: condition (14) is not used in the proof of existence) and hence is simpler and more effective than other criteria, such as M-matrix criteria and LMI-based criteria, because LMI-based criteria or M-matrix criteria always involve the computer MATLAB programming in practical application while our condition (13) is easy to verify. So our existence criterion is actually simpler and more effective than LMI-based criteria and other criteria, which is the main contribution in this paper.Remark 9.
LMI-based stability criteria orM-matrix stability criteria are always proposed in many literatures related to the mean square stability (see, e.g., [30–33] and references therein). However, when p>1 and p≠2, p-stability criteria always involve more complicated mathematical method and mathematical deduction. For example, the stability criteria in [34] are not simpler than our stability criterion in Theorem 6. Similar phenomena exposed in many literatures related to p-stability (see [15, 34–38]). Besides, the nonlinear p-Laplacian (p>1) operator produces great difficulties in p-stability proof. However, our condition (14) is still a LMI condition, which can be computed and verified by computer MATLAB LMI Toolbox in practical application.


## 4. Numerical Example

Example 1.
Consider the 5-Laplace fuzzy T-S dynamical equations as follows.
Fuzzy Ruler 1. IF ω1(t) is 1/e-2ω1(t), THEN(54a)du1t,xdt=D1div⁡∇u13∇u1-b1u1+c11rf1u1t,x+d11rg1u1t-τ1t,x+I1,t>0,x∈Ω,du2t,xdt=D2div⁡∇u23∇u2-b2u2+c21rf2u2t,x+d21rg2u2t-τ2t,x+I2,t>0,x∈Ω,uit,x=0,i∈N,t⩾0,x∈∂Ω.
Fuzzy Ruler 2. IF ω2(t) is 1-1/e-2ω1(t), THEN (54b)du1t,xdt=D1div⁡∇u13∇u1-b1u1+c12rf1u1t,x+d12rg1u1t-τ1t,x+I1,t>0,x∈Ω,du2t,xdt=D2div⁡∇u23∇u2-b2u2+c22rf2u2t,x+d22rg2u2t-τ2t,x+I2,t>0,x∈Ω,uit,x=0,i∈N,t⩾0,x∈∂Ωequipped with the initial value (55)ϕs,x=0.2cos⁡2πx1-x2sin⁡πs20.25sin⁡6x1-x2cos⁡3πs2,-83≤s≤0,where x∈Ω≜{(x1,x2)T∈R2:|xi|<1,i∈N}, N={1,2}, r∈S≜{1,2},  c111=c211=0.1,  c112=c212=0.11, c121=c221=0.15, c122=0.12=c222,  d111=d211=0.1, d112=d212=0.11, d121=d221=0.15, d122=0.12=d222, b1(u1)=u13+1/2u1, b2(u2)=u23+1/3u2, and B1=1/2, and B2=1/3. Consider f1u1=0.2u1sin⁡u1=g1(u1), f2u2=0.3u2+0.2cos⁡u2=g2(u2), and F1=G1=0.4, F2=G2=0.5; let τi(t)≡83,  i∈N, and τ=83,  K=0; denote hi(r)=hir, r∈S. Let h11=1,  h21=2,  h12=3,  h22=4, and β=0.01. Denote ai(r)=air and bi(r)=bir; then we can compute by computer MATLAB that a11=1.5900, a21=4.9860, a12=1.0633, a22=2.4867, b11=0.1000, b21=0.2760, b12=0.2500, and b22=0.4600, and hence a1=1.5900, a2=1.0633, b1=0.2760, b2=0.4600, and (56)-a1+b11-Keβτ=-0.9570<0,-a2+b21-Keβτ=-0.0084<0,which imply that condition (14) is satisfied. In addition, condition (13) is obviously satisfied. Therefore, there exists a nonconstant equilibrium solution for PDEs (54a)-(54b), which is stochastically global exponential p-stability about Lp norm (see Figures 1 and 2).Figure 1
The state variableu1(t,x).Figure 2
The state variableu2(t,x).


## 5. Conclusions

The nonlinearp-Laplace (p>1,p≠2) brings great difficulties to the proof of the existence of the p-stability nonconstant equilibrium solution, for p-Laplace (p>1,p≠2) partial differential equations (PDEs) always need be considered in Banach space W1,p(Ω) while the common linear Laplace (p=2) PDEs can be studied in the setting of the special Hilbert space H1 that can be orthogonally decomposed into the direct sum of the eigenfunction spaces. However, by applying Ekeland variational principle and the Yang inequality and constructing the suitable Lyapunov functional, we overcome those difficulties to obtain the existence of exponential p-stability nonconstant equilibrium solution for Markovian jumping p-Laplace (p>1) partial differential equations (PDEs) (4) under Dirichlet boundary condition. As far as we know, it is the first time to apply the Ekeland variational principle to solve the above problem so that some methods used in this paper are different from those of many previous related literatures (see Remark 1). In addition, the obtained existence criterion is only involved in the activation functions so that the criterion is simpler and easier than other existence criteria to be verified in practical application. Moreover, a numerical example shows the effectiveness of the proposed methods owing to the large allowable variation range of time-delay.



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