# On the Existence of Variational Principles for a Class of the Evolutionary  Differential-Difference Equations

Authors: I. A. Kolesnikova; V. M. Savchin
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

Necessary and sufficient conditions for the existence of variational principles for a given wide class of evolutionary differential-difference operator equation are obtained. The theoretic results are illustrated by two examples.


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## Body

## 1. Introduction

We consider the equation:(1.1)N(u)=0,u∈D(N),whereD(N) is a domain of the definition of the operator N:D(N)⊆U→V, and U,V are normed linear spaces over the field of real numbers ℝ.Later, we will assume that at every pointu∈D(N), there exists the Gâteaux derivative Nu′ of N defined by the formula (d/dϵ)N(u+ϵh)|ϵ=0=δN(u,h)=Nu′h.The operatorN:D(N)→V is said to be potential [1] on the set D(N) relative to a given bilinear form Φ(·,·):V×U→R, if there exists a functional FN:D(FN)=D(N)→R such that(1.2)δFN[u,h]=Φ(N(u),h),∀u∈D(N),∀h∈D(Nu′).In that case, we also say that the given equation admits the direct variational formulation.A problem of the construction of the functionalFN upon the given operator N is known as the classical inverse problem of the calculus of variations [1]. Note that practically no one has been solving inverse problem of the calculus of variations for partial differential-difference operators until recently [2–4]. Let us also note that for a wide classes of partial differential equations, there has been developped the problem of recongnition of variationality upon the structure of corresponding operators [5, 6]. There is a theoretical and practical interst in the extention of these results on partial differential-difference equations [7–9].For what follows, we suppose thatD(N) is a convex set, and we also need the following potentiality criterion [1]:
(1.3)Φ(Nu′h,g)=Φ(Nu′g,h),∀u∈D(N),∀h,g∈D(Nu′).
Under this condition, the potential FN is given by(1.4)FN[u]=∫01Φ(N(u0+λ(u-u0)),u-u0)dλ+const,whereu0 is a fixed element of D(N).The functionalFN is called the potential of the operator N, and in turn the operator N is called the gradient of the functional FN.


## 2. Statement of the Problem

Let us consider the following differential-difference operator equation:(2.1)N(u)≡∑λ=-11Pλ(t)ut(t+λτ)+Q(t,u(t+λτ))=0,u∈D(N),t∈[t0,t1]⊂R.Here,Pλ:U1→V1(λ=-1,0,1) are linear operators depending on t;Q:[t0-τ,t1+τ]×U13→V1 is in general a nonlinear operator;N:D(N)⊆U→V;U=C1([t0-τ,t1+τ];U1);V=C([t0-τ,t1+τ];V1), where U1,V1 are real normed linear spaces, U1⊆V1.The domain of definitionD(N) is given by the equality:
(2.2)D(N)={u∈U:u(t)=φ1(t),t∈[t0-τ,t0],u(t)=φ2(t),t∈[t1,t1+τ]},
where φi,(i=1,2) are given functions.Under the solution of (2.1), we mean a function u∈D(N) satisfying the identity:
(2.3)N(u)≡∑λ=-11Pλ(t)ut(t+λτ)+Q(t,u(t+λτ))=0,∀t∈[t0,t1]⊂R.Let us give the following bilinear form:(2.4)Φ(⋅,⋅)≡∫t0t1〈⋅,⋅〉dt:V×U⟶R,
where the bilinear mapping Φ1≡〈·,·〉 satisfies the following conditions:
(2.5)〈v1,v2〉=〈v2,v1〉,∀v1,v2∈V1,Dt〈v,g〉=〈Dtv,g〉+〈v,Dtg〉,∀v,g∈V.Our aim is to define the structure of operatorsPλ(λ=-1,0,1) and Q under which (2.1) allows the solution of the inverse problem of the calculus of variations relative to the bilinear form (2.4) such that Dt=d/dt is the antisymmetric operator on D(Nu′), that is,
(2.6)Φ(Dth1,h2)=-Φ(Dth2,h1),∀h1,h2∈D(Nu′).


## 3. Conditions of Potentiality and the Structure of (2.1) in the Case of Its Variationality

We denote byK* the operator adjoint to K.Theorem 3.1.
IfDt*=-Dt on the set D(Nu′), then for the existence of the direct variational formulation for the operator (2.1) on the set D(N) relative to (2.4), it is necessary and sufficient that the following conditions hold on the set D(Nu′):
(3.1)P-λ+Pλ*|t→t-λτ=0,∂Pλ*∂t|t→t-λτ+Qu(t-λτ)′-Qu(t+λτ)′*|t→t-λτ=0,λ=-1,0,1,∀u∈D(N),∀t∈(t0,t1).Proof.
Taking into account formula (2.1), we get
(3.2)Nu′h=∑λ=-11Pλht(t+λτ)+∑λ=-11Qu(t+λτ)′h(t+λτ).
The criterion of potentiality takes the following form:
(3.3)∫t0t1〈∑λ=-11{Pλht(t+λτ)+Qu(t+λτ)′h(t+λτ)},g(t)〉dt=∫t0t1〈∑λ=-11{Pλgt(t+λτ)+Qu(t+λτ)′g(t+λτ)},h(t)〉dt,
or
(3.5)∫t0t1[〈∑λ=-11{Pλht(t+λτ)+Qu(t+λτ)′h(t+λτ)},g(t)〉-〈∑λ=-11(Pλgt(t+λτ)+Qu(t+λτ)′g(t+λτ)),h(t)〉]dt=0,∀u∈D(N),∀g,h∈D(Nu′).
Bearing into account the condition Dt*=-Dt on the set D(Nu′), from (3.5), we get
(3.4)∫t0t1[〈∑λ=-11(PλDt+Qu(t+λτ)′)h(t+λτ),g(t)〉-∑λ=-11〈[-Dt(Pλ*h(t))+Qu(t+λτ)′*h(t)],g(t+λτ)〉]dt=0,
or
(3.6)∫t0t1[〈∑λ=-11(P-λDt+Qu(t-λτ)′)h(t-λτ),g(t)〉-〈∑λ=-11-∂Pλ*∂t-Pλ*Dt+Qu′*|t→t-λτh(t-λτ),g(t)〉]dt=0.
Thus, condition (3.5) can be reduced to the following form:
(3.7)∫t0t1∑λ=-11〈(P-λDt+Qu(t-λτ)′)h(t-λτ)-(-∂Pλ*∂t-Pλ*Dt+Qu′*)|t→t-λτh(t-λτ),g(t)〉dt=0,∀u∈D(N),∀g,h∈D(Nu′).
This equality is fulfilled identically if and only if
(3.8)∑λ=-11[(P-λ+Pλ*|t→t-λτ)Dt+∂Pλ*∂t|t→t-λτ+Qu(t-λτ)′-Qu(t+λτ)′*|t→t-λτ]⋅h(t-λτ)=0,
for all u∈D(N). Thus, it is necessary and sufficient that conditions (3.1) hold.Theorem 3.2.
Conditions (3.1) are held if and only if (2.1) has the following form:
(3.9)N1(u)≡∑λ=-11(Rλ*|t→t-λτ-R-λ)ut(t-λτ)+(gradΦB[u]-∑λ=-11∂Rλ∂tu(t+λτ))=0,∀u∈D(N1),t∈[t0,t1]⊂R.
The operatorsRλ and B depend on Pλ(t) and Q(t,u(t+λτ)).Proof.
IfDt*=-Dt on the set D(N1u′) and conditions (3.1) are held, then according to Theorem 3.1, operator (2.1) is potential on the set D(N) relative to a given bilinear form (2.4).
Let us consider the following functional:(3.11)FN[u]=∫t0t1[〈∑λ=-11Rλ(t)u(t+λτ),ut(t)〉+B[u]]dt+FN[u0].
It is easy to check that
(3.10)δFN[u,h]=∫t0t1〈∑λ=-11Rλh(t+λτ),ut(t)〉+〈∑λ=-11Rλu(t+λτ),ht(t)〉+〈gradΦB[u],h(t)〉dt=∫t0t1〈∑λ=-11Rλ*|t→t-λτut(t-λτ),h(t)〉-〈∑λ=-11Dt(Rλu(t+λτ)),h(t)〉+〈gradΦB[u],h(t)〉dt=∫t0t1〈∑λ=-11Rλ*|t→t-λτut(t-λτ),h(t)〉-〈∑λ=-11∂Rλ∂tu(t+λτ),h(t)〉-〈∑λ=-11R-λut(t-λτ),h(t)〉+〈gradΦB[u],h〉dt=∫t0t1〈∑λ=-11(Rλ*|t→t-λτ-R-λ)ut(t-λτ)+(gradΦB[u]-∑λ=-11∂Rλ∂tu(t+λτ)),h(t)〉dt=∫t0t1〈N(u),h〉dt,∀u∈D(N),∀h∈D(Nu′).
Then functional (3.11) is a potential of evolutionary operator (2.1).
IfDt*=-Dt on D(Nu′), then(3.12)P-λ(t)=Rλ*|t→t-λτ-R-λ,Q(t,u(t+λτ))=-∑λ=-11∂Rλ∂tu(t+λτ)+gradΦB[u].
This theorem shows the structure of the given kind of differential-difference operator, which admits the solution of the inverse problem of the calculus of variations.


## 4. Examples

Example 4.1.
Let us consider the evolutionary differential-difference equation with partial derivatives in the following form:(4.1)N1(u)≡∑λ=-11(aλ(x,t)∂u∂t(x,t+λτ)-bλij(x,t)∂2u∂xi∂xj(x,t+λτ))=0,
where (x,t)∈Q=Ω×(t0,t1),t1-t0>2τ,i,j=1,n¯,aλ(x,t)∈Cx,t0,1(Q¯),bλij(x,t)∈Cx,t2,0(Q¯).
Ω is a bounded domain in ℝn with piecewise smooth boundary ∂Ω.
The domain of definitionD(N1) is given by the equality:
(3.13)D(N1)={∂μu∂nxμ|∂Γτ=ψμ(x,t),u∈U=Cx,t2,1(Ω¯×[t0-τ,t1+τ]):u(x,t)=φ1(x,t),(x,t)∈E1=Ω¯×[t0-τ,t0],u(x,t)=φ2(x,t),(x,t)∈E2=Ω¯×[t1,t1+τ],∂μu∂nxμ|∂Γτ=ψμ(x,t),μ=0,1},
where Γτ=∂Ω×[t0-τ,t1+τ], φi∈C(Ei),(i=0,1),ψμ∈C(Γτ),(μ=0,1) are given functions.
We investigate the existence of variational principle for (4.1) relative to a given bilinear form (2.4).For (4.1), we getN1u′h=∑λ=-11(aλ(x,t)ht(x,t+λτ)-bλij(x,t)hxixj(x,t+λτ)).
We investigate the existence of variational principle for (4.1) relative to a given bilinear form (2.4).
For (4.1), we get
(4.2)N1u′h=∑λ=-11(aλ(x,t)ht(x,t+λτ)-bλij(x,t)hxixj(x,t+λτ)).
Necessary and sufficient conditions of potentiality take the form:(4.3)Φ(N1u′h,g)-Φ(N1u′g,h)≡-∫t0t1∫Ω{(∑λ=-11Dt(aλ(x,t)g(x,t))∑λ=-11|t→t-λτ+(∑λ=-11Dxixjbλij(x,t)g(x,t))|t→t-λτh(x,t)}dxdt-∫t0t1∫Ω∑λ=-11{aλ(x,t)gt(x,t+λτ)-bλij(x,t)gxixj(x,t+λτ)}⋅h(x,t)dxdt=0,∀u∈D(N1),∀g,h∈D(N1u′).
From that, we come  to the following:(4.4)-∑λ=-11(Dt(aλg(x,t))-Dxixj(bλijg(x,t)))|t→t-λτ=∑λ=-11{aλgt(x,t+λτ)-bλijgxixj(x,t+λτ)},∀g∈D(N1u′).
That is true if and only if(4.5)-a-λ(x)=aλ(x),λ=1,-1,a0(x)=0,bλij(t)|t→t-λτ=b-λij(t),λ=-1,0,1,i,j=1,n¯,∀(x,t)∈Q.
Under the fulfilment of that conditions, the corresponding functional is given by(4.6)FN1[u]=12∫t0t1∫Ω{a1(x)ut(x,t-τ)u(x,t)-a1(x)ut(x,t+τ)u(x,t)+b1ij(t-τ)uxi(x,t-τ)⋅uxj(x,t)+b0ij(t)uxi(x,t)uxj(x,t)+b-1ij(t+τ)uxi(x,t+τ)uxj(x,t)}dxdt+const.
Let us consider an example when this criterion of potantiality fails.Example 4.2.
Consider the equation:(4.7)N2(u)=ut(t,x)-2u(t,x+2τ)ux(t,x+2τ)-2ux(t,x)u(t,x-2τ)-2u(t,x)ux(t,x-2τ)+12uxxx(t,x+2τ)+12uxxx(t,x-2τ)=0,(t,x)∈QT=(0,T)×(-∞,+∞).
Let us note that this equation is a Korteweg-de Vries’ equation τ=0.
We denote(4.8)D(N2)={u∈U:Ct,x1,3((0,T)×(-∞,+∞)):u|t=0=u0(x),u|t=T=u1(x),lim⁡|x|→+∞Dxnu=0(n=0,3¯)}.
It is easy to be convinced that operator (4.7) is not potential on set (4.8) relative to the bilinear form:
(4.9)Φ(v,g)=∫0T∫-∞+∞v(t,x)⋅g(t,x)dxdt,v∈V,g∈U.
Here, V={v∈Ct,x0,3((0,T)×(-∞,+∞)):lim|x|→+∞Dxnv=0,(n=0,3¯)}.
We define the integrating operatorM as Mv(t,x)=∫-∞xv(t,y)dy.
Then, the operator,(4.10)Ñ(u)≡MN2(u)=∫-∞xut(t,y)dy-u2(t,x+2τ)-2u(t,x)u(t,x-2τ)+12uxx(t,x+2τ)+12uxx(t,x-2τ)dy
is potential on set (4.8) relative to bilinear form (4.9). The corresponding functional FN2[u] has the form:
(4.11)FN2[u]=12∫0T∫-∞+∞{u(t,x)∫-∞xut(t,y)dy-ux(t,x-τ)ux(t,x+τ)-2u2(t,x+τ)u(t,x-τ)dy}dxdt.
Indeed, using (4.11), we find that
(4.12)δFN2[u,h]=∫0T∫-∞+∞{∫-∞xut(t,y)dy+12uxx(t,x+2τ)+12uxx(t,x-2τ)-2u(t,x)u(t,x-2τ)-u2(t,x+2τ)dy∫-∞+∞{∫-∞xut(t,y)dy+12uxx(t,x+2τ)+12uxx(t,x-2τ)-2u(t,x)u(t,x-2τ)}h(t,x)dxdt,∀u∈D(N),∀h∈D(Nu′).
From the conditionδFN[u,h]=0,u∈D(N), for all h∈D(Nu′), we obtain
(4.13)Ñ(u)≡MN2(u)≡∫-∞xut(t,y)dy-u2(t,x+2τ)-2u(t,x)u(t,x-2τ)+12uxx(t,x+2τ)+12uxx(t,x-2τ)dy=0,u∈D(N),
This equation is equivalent to (4.7).
Let us note that the formulaI[u]=∫-∞+∞(1/2)ux(t,x-τ)ux(t,x+τ)+u2(t,x+τ)u(t,x-τ)dx defines the first integral I[u]=const of (4.7).



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*Source: 780382-2012-01-04.xml*