# Quantum Stochastic Cable Equation Acting on Functionals of Discrete-Time Normal Martingales

Authors: Yuling Tang; Caishi Wang; Suling Ren; Jinshu Chen
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

LetM be a discrete-time normal martingale satisfying some mild conditions. Then Gel’fand triple S(M)⊂L2(M)⊂S⁎(M) can be constructed of functionals of M, where elements of S(M) are called testing functionals of M, while elements of S⁎(M) are called generalized functionals of M. In this paper, we consider a quantum stochastic cable equation in terms of operators from S(M) to S⁎(M). Mainly with the 2D-Fock transform as the tool, we establish the existence and uniqueness of a solution to the equation. We also examine the continuity of the solution and its continuous dependence on initial values.


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## Body

## 1. Introduction

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. It was originally proposed for developing mathematical models of signal decay in submarine (underwater) telegraphic cables. However, its utility for neuroscience was not recognized until many years later, after nerve axons came to be regarded as “core conductors.” Neurons are the basic unit in the brain; it transmits electrical signals through axons and receives electrical signals through dendrites. Axons and dendrites can be described as cables with special properties. Thus, the cable equation has been useful to explain different phenomena in dendrites and axons (see, e.g., [1–3]).The potential of the responses of nerve cells may evolve through a combination of diffusion and random fluctuation. To describe the change of such potential, Walsh [4, 5] introduced a stochastic cable equation, which is a second-order, parabolic, partial differential equation with a certain stochastic term. In 1998, Applebaum [6] proposed a quantum stochastic cable equation in the framework of Fock space, which can be viewed as a quantum version of Walsh’s stochastic cable equation and was intended to describe the microtubular potential as an observable that evolves in space and time through diffusion and quantum effects.Hida’s white noise analysis is essentially a theory of infinite dimensional calculus on generalized functionals of Brownian motion (also known as Gaussian white noise generalized functionals). In 2000, a quantum stochastic cable equation was considered in the framework of white noise analysis [7]. The main features of the equation considered in [7] lie in that the role of quantum noises was played by a family of generalized operators, which are continuous operators defined on Gaussian white noise testing functionals and valued in Gaussian white noise generalized functionals, and can be regarded as quantum Gaussian noises.Functionals of discrete-time normal martingales have attracted much attention in recent years (see, e.g., [8–12]). Let M be a discrete-time normal martingale satisfying some mild conditions. Then, by using a specific orthonormal basis for the space L2(M) of square integrable functionals of M, Gel’fand triple (1)SM⊂L2M⊂S∗Mcan be constructed [11], where elements of S(M) are called testing functionals of M, while elements of S∗(M) are called generalized functionals of M. Recently, a transform, called 2D-Fock transform, has been introduced [13] for operators from S(M) to S∗(M). It has been shown [13] that continuous linear operators from S(M) to S∗(M) are completely determined by a growth condition on their 2D-Fock transforms. Let L=L(S(M),S∗(M)) denote the set of all continuous linear operators from S(M) to S∗(M). Then one can define the convolution ∗ in L by means of the 2D-Fock transform, which makes (L,*) form a commutative algebra.In this paper, we consider the following quantum stochastic cable equation in terms of operators fromS(M) to S∗(M):(2)∂Vx,t∂t=∂2Vx,t∂x2-Vx,t+ΦVx,t∗Kx,t,0<x<l,t>0,∂V∂x0,t=∂V∂xl,t=0,t≥0,Vx,0=Jx,0≤x≤l,where l>0 is a given constant, ∗ stands for the convolution of operators, Φ:L→L is a given map, {J(x)∣x∈[0,l]} and {K(x,t)∣(x,t)∈[0,l]×R+} are L-valued quantum stochastic processes, and the solution {V(x,t)∣(x,t)∈[0,l]×R+}, if it exists, will be a L-valued quantum stochastic process. Compared to the equation considered in [7], (2) might be used to describe quantum stochastic evolution in space and time of a quantum system in the presence of quantum Bernoulli noises.The rest of the paper is organized as follows. In Section2, we briefly recall some notions and results on continuous linear operators from S(M) to S∗(M), which will be used in our later discussion. Sections 3 and 4 are our main work. Here, by using the 2D-Fock transform, we establish the existence and uniqueness of a solution to (2). We also examine the continuity and continuous dependence on initial values of the solution to (2).Throughout this paper,N designates the set of all nonnegative integers and Γ the finite power set of N, namely, (3)Γ=σ∣σ⊂Nand#σ<∞,where #(σ) means the cardinality of σ as a set.


## 2. Preliminaries

In this section, we briefly recall some notions and results on continuous linear operators fromS(M) to S∗(M). For details, see [10–14] and references therein.Let(Ω,F,P) be a given probability space and L2(Ω,F,P) the usual Hilbert space of square integrable complex-valued functions on (Ω,F,P) with inner product 〈·,·〉 and norm ·, respectively. By convention, 〈·,·〉 is conjugate-linear in its first argument and linear in its second argument.LetM=(Mn)n∈N be a discrete-time normal martingale on (Ω,F,P) that has the chaotic representation property. Denote by L2(M) the Hilbert space of square integrable functionals of M, which shares the same inner product and norm with L2(Ω,F,P), namely, 〈·,·〉 and ·. It is known that L2(M) has an orthonormal basis {Zσ∣σ∈Γ} defined by Z∅=1 and (4)Zσ=∏n∈σZn,σ∈Γ,σ≠∅, where Z=(Zn)n∈N is the discrete-time normal noise associated with M (see [11] for details).By using the orthonormal basis{Zσ∣σ∈Γ}, one can construct [11] a dense linear subspace S(M) of L2(M), which itself is a countable Hilbert nuclear space and continuously contained in L2(M). Let S∗(M) be the dual of S(M) endowed with the strong topology. Then, by identifying L2(M) with its dual, one comes to Gel’fand triple(5)SM⊂L2M⊂S∗M,which is the framework where we will work. Elements of S(M) are called testing functionals of M, while elements of S∗(M) are called generalized functionals of M. As mentioned above, we denote by L≡L(S(M),S∗(M)) the set of all continuous linear operators from S(M) to S∗(M).Lemma 1 (see [11]).
Letσ↦λσ be the N-valued function on Γ given by(6)λσ=∏k∈σk+1,σ≠∅,σ∈Γ;1,σ=∅,σ∈Γ.Then, for p>1, the positive term series ∑σ∈Γλσ-p converges and moreover(7)∑σ∈Γλσ-p≤exp⁡∑k=1∞k-p<∞Definition 2 (see [13]).
For an operatorT∈L, its 2D-Fock transform is the function T^ on Γ×Γ defined as(8)T^σ,τ=TZσ,Zτ,σ,τ∈Γ×Γ,where 〈〈·,·〉〉 is the canonical bilinear form on S∗(M)×S(M).Much like generalized functionals ofM, continuous linear operators in L are also completely determined by their 2D-Fock transforms.Lemma 3 (see [13]).
LetT1 and T2:S(M)→S∗(M) be continuous linear operators. Then T1=T2 if and only if T1^=T2^.The following lemma is known as the characterization theorem of operators inL through their 2D-Fock transforms.Lemma 4 (see [13]).
LetF be a function on Γ×Γ. Then F is the 2D-Fock transform of an element T in L if and only if it satisfies(9)Fσ,τ≤cλσpλτp,σ,τ∈Γ×Γfor some constants c≥0 and p≥0. In that case, for q>p+1/2, one has (10)Tq≤c∑σ∈Γλσ-2q-p,and in particular T takes values in Sq∗(M), where (11)Tp≡supξ≠0,ξ∈SMTξ-pξp,where p≥0.Definition 5 (see [13]).
LetT1 and T2∈L and then their convolution T1∗T2 is defined by(12)T1∗T2^σ,τ=T1^σ,τT2^σ,τ,σ,τ∈Γ×Γ.It can be verified that(L,*) forms a commutative algebra with a unit. The next definition describes convergence of sequences in L.Definition 6 (see [14]).
A sequence(Tn)n≥1 in L is called to converge strongly to T∈L, if for any ξ∈S(M), we have Tn(ξ)→T(ξ) in the strong topology of S∗(M).Lemma 7 (see [14]).
Let(Tn)n≥1 be a sequence of operators in L and T an operator in L. Then the sequence (Tn)n≥1 converges strongly to T if and only if:
(1)Tn^(σ,τ)→T^(σ,τ) for all σ,τ∈Γ;
(2) there exist constantsα≥0 and p≥0, such that (13)supn≥1Tn^σ,τ≤αλσpλτp,σ,τ∈Γ×Γ.Lemma 8 (see [14]).
Let(E,ε,ν) be a measure space and T(·):E→L be a L-valued function satisfying the following conditions:
(1) for anyσ,τ∈Γ, the function T(·)^(σ,τ):E→C is measurable;
(2) there existp≥0 and a positive function c(ω)∈L1(E,ν), such that for ν-a.e., ω∈E it holds that (14)Tω^σ,τ≤cωλσpλτpσ,τ∈Γ.Then T is Bochner integrable with respect to ν on E and its Bochner integral satisfies the following norm inequality: (15)∫ETωdνωq≤∑σ∈Γλσ-2q-p∫Ecωdνω.where q>p+1/2. In particular, ∫ET(ω)dν(ω)∈L.Definition 9 (see [14]).
LetW be a compact metric space. A map T(·):W→L is said to be continuous, if for each sequence (ωn)n≥1 in W that converges to ω0∈W, one has T(ωn) converges to T(ω0).Lemma 10 (see [14]).
LetW be a compact metric space and T(·):W→L is a L-valued function. Then T(·) is continuous if and only if
(1)T(·)^(σ,τ):W→C is continuous for any σ,τ∈Γ;
(2) there existp≥0 and α≥0, such that(16)Tω^σ,τ≤αλσpλτp.for each σ,τ∈Γ and ω∈W.


## 3. Existence and Uniqueness of Solution

In the present section, we establish the existence and uniqueness of a solution to quantum (2). Recall that L≡L(S(M),S∗(M)) is the space of all continuous linear operators from S(M) to S∗(M), which forms a commutative algebra with the convolution ∗.LetG:[0,l]×[0,l]×R+→R+ be the Green’s function associated with the classical cable equation. Then G has following properties (see[5] for details):(1)G is continuous;(2)∫0lGs(x,y)Gt(x,y)dy=Gs+t(x,y);(3)Gt(x,y)=Gt(y,x);(4) for each0<r<3, there exists kr>0, such that(17)∫0lGtx,yrdy≤kre-rtt1-r/2,x∈0,l,t>0.And we put (18)ψx,t≡∫0lGtx,ydy,x∈0,l,t>0.Then, obviously, 0≤ψ(x,t)≤k1, x∈[0,l], t>0.LetV:[0,l]×R+→L be a L-valued function. Then, V is said to be a solution to Equation(2) if it satisfies the following integral equation: (19)Vx,t=∫0lGtx,yJydy+∫0t∫0lGt-sx,yΦVy,s∗Ky,sdyds,where (x,t)∈[0,l]×R+, the integral is the Bochner integral of operator valued function in L.Definition 11.
LetΦ:L→L be a map. If there exists a constant θ>0, such that
(1) for anyX,Y∈L and σ,τ∈Γ, we have (20)ΦX^σ,τ-ΦY^σ,τ≤θX^σ,τ-Y^σ,τ;
(2) for anyX∈L,σ,τ∈Γ, we have ΦX^σ,τ≤θ(1+X^σ,τ).ThenΦ is said to satisfy Lipschitz and linear growth conditions by means of 2D-Fock transforms.Theorem 12.
LetK:[0,l]×R+→L and J:[0,l]→L satisfy
(1) for anyσ,τ∈Γ, the functions K(·,·)^(σ,τ):[0,l]×R+→C and J(·)^(σ,τ):[0,l]→C are continuous;
(2) there exist constantsp≥0,A≥1 associated with p, and α>0, such that (21)Ky,s^σ,τ≤A,σ,τ∈Γ,y,s∈0,l×R+,Jx^σ,τ≤αλσpλτp,σ,τ∈Γ,x∈0,l.In addition, let Φ:L→L satisfy Lipschitz and linear growth conditions. Then there exists a sequence of L-valued functions Vn:[0,l]×R+→L,n≥0, satisfying
(1) for anyn≥0 and σ,τ∈Γ, the function Vn(·,·)^(σ,τ):[0,l]×R+→C is continuous;
(2) forn≥1,(x,t)∈[0,l]×R+,σ,τ∈Γ, it holds that (22)Vnx,t^σ,τ≤∑i=1nk1iθitiAii!+αψx,tλσpλτp∑i=0nθitiAii!;
(3) for(x,t)∈[0,l]×R+,n≥1, we have (23)V0x,t=∫0lGtx,yJydy,Vnx,t=V0x,t+∫0t∫0lGt-sx,yΦVn-1y,s∗Ky,sdyds.Proof.
By properties of the Green’s function and the conditions of Theorem12, the integral ∫0lGt(x,y)J(y)dy exists and belongs to L for (x,t)∈[0,l]×R+. Put (24)V0x,t≡∫0lGtx,yJydy,x,t∈0,l×R+.Then, for any σ,τ∈Γ, the function V0(·,·)^(σ,τ):[0,l]×R+→C is continuous. For x∈[0,l], t>0, let (25)H1y,s=Gt-sx,yΦV0y,s∗Ky,s,y,s∈0,l×0,t.Then, we have (26)H1y,s^σ,τ=Gt-sx,yΦV0y,s^σ,τKy,s^σ,τ,(y,s)∈[0,l]×[0,t),σ,τ∈Γ, which implies that the function (27)H1·,·^σ,τ:0,l×0,t→Cis continuous. Furthermore, we have (28)H1y,s^σ,τ≤θGt-sx,yA1+V0y,s^σ,τ≤θGt-sx,yA1+αψy,sλσpλτp≤θ1+αk1Gt-sx,yAλσpλτp,where σ,τ∈Γ and (x,t)∈[0,l]×R+. By Lemma 8, the integral (29)∫0t∫0lH1y,sdyds=∫0t∫0lGt-sx,yΦV0y,s∗Ky,sdydsexists and belongs to L. Now we define V1:[0,l]×R+→L as (30)V1x,t≡V0x,t+∫0t∫0lGt-sx,yΦV0y,s∗Ky,sdyds.Then, by 2D-Fock transform, we have (31)V1x,t^σ,τ=V0x,t^σ,τ+∫0t∫0lGt-sx,yΦV0y,s^σ,τKy,s^σ,τdyds,σ,τ∈Γ,(x,t)∈[0,l]×R+, which implies that the function V1(·,·)^(σ,τ):[0,l]×R+→C is continuous and satisfies (32)V1x,t^σ,τ≤V0x,t^σ,τ+∫0t∫0lGt-sx,yΦV0y,s^σ,τKy,s^σ,τdyds≤V0x,t^σ,τ+θA∫0t∫0lGt-sx,ydyds+θA∫0t∫0lGt-sx,yV0y,s^σ,τdyds≤αψx,tλσpλτp+k1θAt+αθAλσpλτp∫0t∫0lGt-sx,yψy,sdyds=∑i=11k1iθiAitii!+αψx,tλσpλτp∑i=01θitiAii!.
Assume that, for somen≥1, Vn:[0,l]×R+→L has been defined and the function Vn(·,·)^(σ,τ) is continuous and satisfies (33)Vnx,t^σ,τ≤∑i=1nk1iθiAitii!+αψx,tλσpλτp∑i=0nθitiAii!.For x∈[0,l],t>0, define (34)Hny,s=Gt-sx,yΦVny,s∗Ky,s,y,s∈0,l×0,t.Similarly, for any x∈[0,l],t>0,σ,τ∈Γ, the function Hn(·,·)^(σ,τ) is continuous, and by computation we have (35)Hny,s^σ,τ≤βGt-sx,yλσpλτp,where β=θA(1+αk1)(1+θt+k1θt) and (y,s)∈[0,l]×[0,t),σ,τ∈Γ. Hence, the integral (36)∫0t∫0lHny,sdyds=∫0t∫0lGt-sx,yΦVny,s∗Ky,sdydsexists and belongs to L. So we can define Vn+1:[0,l]×R+→L as (37)Vn+1x,t≡V0x,t+∫0t∫0lGt-sx,yΦVny,s∗Ky,sdyds,(x,t)∈[0,l]×R+. For σ,τ∈Γ, the function Vn+1(·,·)^(σ,τ) is continuous and satisfies(38)Vn+1x,t^σ,τ≤V0x,t^σ,τ+θA∫0t∫0lGt-sx,ydyds+θA∫0t∫0lGt-sx,yVny,s^σ,τdyds≤αψx,tλσpλτp+k1θtA+θA∫0t∫0lGt-sx,yVny,s^σ,τdyds.Note that (39)θA∫0t∫0lGt-sx,yVny,s^σ,τdyds≤θA∫0t∫0l∑i=1nk1iθiAisii!Gt-sx,ydyds+θAλσpλτp∫0t∫0l∑i=0nθisiAii!Gt-sx,yψy,sdyds≤∑i=2n+1k1iθiAitii!+αψx,tλσpλτp∑i=1n+1θiAitii!.Thus from (38), we get (40)Vn+1x,t^σ,τ≤∑i=1n+1k1iθiAitii!+αψx,tλσpλτp∑i=0n+1θiAitii!,(x,t)∈[0,l]×R+,σ,τ∈Γ. Hence, by induction we come to our conclusion.Theorem 13.
Let the noise process{K(x,t)∣(x,t)∈[0,l]×R+}, initial value process {J(x)}, and the map Φ in (2) satisfy the conditions in Theorem 12. Then there exists a unique continuous L-valued function V:[0,l]×R+→L which satisfies (2). Moreover, the solution V has following estimate:(41)Vx,t^σ,τ≤1+αk1λσpλτpexp⁡θ1+k1At.Proof (existence).
Consider the sequence ofL-valued functions {Vn}n≥0 given in Theorem 12. We assert that, for any n≥1, (42)Vnx,t^σ,τ-Vn-1x,t^σ,τ≤1n!1+αλσpλτpk1nθnAntn,σ,τ∈Γ,(x,t)∈[0,l]×R+. In fact, this is obviously true for n=1. Now, we assume that the inequality is satisfied for some n>1, then, for n+1, we have (43)Vn+1x,t^σ,τ-Vnx,t^σ,τ=∫0t∫0lGt-sx,yΦVny,s^σ,τ-ΦVn-1y,s^σ,τKy,s^σ,τdyds≤θA∫0t∫0lGt-sx,yVny,s^σ,τ-Vn-1y,s^σ,τdyds≤1n!1+αλσpλτpk1nθn+1An+1∫0t∫0lGt-sx,ysndyds≤1n+1!1+αλσpλτpk1n+1θn+1An+1tn+1,σ,τ∈Γ,(x,t)∈[0,l]×R+. By induction, we come to the assertion.
For(x,t)∈[0,l]×R+, consider the sequence {Vn(x,t)}⊂L; we have (44)Vm+nx,t^σ,τ-Vnx,t^σ,τ≤∑i=n+1m+nVix,t^σ,τ-Vi-1x,t^σ,τ≤∑i=n+1m+n1i!1+αλσpλτpk1iθiAiti,for any σ,τ∈Γ. Hence, the sequence {Vn(x,t)^} converges in C. On the other hand, from the Theorem 12, we have (45)Vnx,t^σ,τ≤∑i=1n1i!k1iθitiAi+αψx,tλσpλτp∑i=0n1i!θitiAi≤1+αk1λσpλτpexp⁡θ1+k1At,σ,τ∈Γ, which together with Lemma 7 implies that {Vn(x,t)} converges in L.
LetV(x,t)=limn→∞Vn(x,t),(x,t)∈[0,l]×R+; then, it is easy to see that {V(x,t)∣(x,t)∈[0,l]×R+}⊂L and (46)Vx,t^σ,τ≤1+αk1λσpλτpexp⁡θ1+k1At.Thus, for any (x,t)∈[0,l]×R+, the integral (47)∫0t∫0lGt-sx,yΦVy,s∗Ky,sdyds, exists and belongs to L.
Moreover, by the dominated convergence theorem, for eachσ,τ∈Γ,(x,t)∈[0,l]×R+, (48)∫0t∫0lGt-sx,yΦVny,s^σ,τKy,s^σ,τdyds→∫0t∫0lGt-sx,yΦVy,s^σ,τKy,s^σ,τdyds.On the other hand, we have (49)Vnx,t^σ,τ=V0x,t^σ,τ+∫0t∫0lGt-sx,yΦVn-1y,s^σ,τKy,s^σ,τdyds,
n ≥ 1 , σ , τ ∈ Γ , ( x , t ) ∈ [ 0 , l ] × R +, so (50)Vx,t^σ,τ=V0x,t^σ,τ+∫0t∫0lGt-sx,yΦVy,s^σ,τKy,s^σ,τdyds,σ,τ∈Γ,(x,t)∈[0,l]×R+, which implies that {V(x,t)∣(x,t)∈[0,l]×R+} is a solution to (2).
For the continuity ofV:∈[0,l]×R+→L, we should only show that for any L>0,V is continuous on [0,l]×[0,L]. In fact, for any σ,τ∈Γ, we have (51)Vnx,t^σ,τ-Vn-1x,t^σ,τ≤1n!1+αλσpλτpk1nθntnAn≤1n!1+αλσpλτpk1nθnTnAn. So (52)sup⁡Vnx,t^σ,τ-Vx,t^σ,τ∣x,t∈0,l×0,T→0n→∞, which means that the function V(·,·)^(σ,τ) is continuous. In addition, we have (53)Vx,t^σ,τ≤1+αk1λσpλτpexp⁡θ1+k1At≤1+αk1λσpλτpexp⁡θ1+k1AT,σ,τ∈Γ,(x,t)∈[0,l]×[0,T]. Then by Lemma 10, the function V:[0,l]×[0,T]→L is continuous.
Finally, we prove the uniqueness of the solution. Assume that{U(x,t)∣(x,t)∈[0,l]×R+} is a another continuous solution to (2). Then for any σ,τ∈Γ, we have (54)Ux,t^σ,τ-Vx,t^σ,τ=∫0t∫0lGt-sx,yΦUy,s^σ,τ-ΦVy,s^σ,τKy,s^σ,τdyds≤θA∫0t∫0lGt-sx,yUy,s^σ,τ-Vy,s^σ,τdyds≤θA∫0t∫0lGt-sx,y2dyds1/2∫0t∫0lUy,s^σ,τ-Vy,s^σ,τ2dyds1/2≤θAδ1/2∫0t∫0lUy,s^σ,τ-Vy,s^σ,τ2dyds1/2,where δ=k2∫0+∞e-2ss-1/2ds<+∞. Hence, (55)∫0lUx,t^σ,τ-Vx,t^σ,τ2dx≤lθ2A2δ∫0t∫0lUy,s^σ,τ-Vy,s^σ,τ2dyds. By Gronwall inequality, we obtain that (56)∫0lUx,t^σ,τ-Vx,t^σ,τ2dx=0,t∈R+.By Lemma 3, U(x,t)=V(x,t) and (x,t)∈[0,l]×R+; namely, the solution is unique.


## 4. Stability of Solution

In this section, we will prove that the solution to (2) continuously depends on the initial values.LetC([0,l],L) be the set of continuous functions from [0,l] to L, namely, C([0,l],L)={J∣J:[0,l]→L}. For any σ,τ∈Γ, we define seminorm ·σ,τ on C([0,l],L) as (57)Jσ,τ=sup0≤x≤lJx^σ,τ and endow C([0,l],L) with the topology generated by the family of seminorms {·σ,τ∣σ,τ∈Γ}. Then C([0,l],L) becomes a Hausdorff topological linear space. In the following, we always assume that the map Φ:L→L satisfies Lipschitz and linear growth conditions in terms of 2D-Fock transform, and the noise process {K(x,t)} satisfies conditions:(1) For anyσ,τ∈Γ, the function K(·,·)^(σ,τ) is continuous;(2) there exist constantsp0≥0 and A0≥1 associated with p0, such that (58)Kx,t^σ,τ≤A0,σ,τ∈Γ,x,t∈0,l×R+.Theorem 14.
For eachJ∈C([0,l],L), there exists a unique continuous L-valued function V:[0,l]×R+→L satisfying (2); moreover, for each t∈R+,V(·,t)∈C([0,l],L).Proof.
By Theorem13, we should only prove that there exist constants α>0, p≥0, and A>1, such that(59)K·,·^σ,τ≤A,σ,τ∈Γ,x,t∈0,l×R+,and(60)Jx^σ,τ≤αλσpλτp,σ,τ∈Γ,x∈0,l.In fact, because J∈C([0,l],L), so by Lemma 10, there exist p1≥0,α>0, such that (61)Jx^σ,τ≤αλσp1λτp1,σ,τ∈Γ,x∈0,l.Set p=p0∨p1; then for any σ∈Γ, we have λσp0≤λσp,λσp1≤λσp; hence, (62)Kx,t^σ,τ≤A,σ,τ∈Γ,x,t∈0,l×R+,and (63)Jx^σ,τ≤αλσpλτp,σ,τ∈Γ,x∈0,l.ForJ∈C([0,l],L), we denote by VJ the solution to (2) with J as its initial value. Then, we have the following estimate:(64)VJ1·,t-VJ2·,tσ,τ≤k1eθk1tA0J1-J2σ,τ,t∈R+.In fact, because (65)VJ1x,t^σ,τ-VJ2x,t^σ,τ≤∫0lGtx,yJ1y^σ,τ-J2y^σ,τdy+∫0t∫0lGt-sx,yΦVJ1y,s^σ,τ-ΦVJ2y,s^σ,τKy,s^σ,τdyds≤k1J1-J2σ,τ+θA0∫0t∫0lGt-sx,yVJ1y,s^σ,τ-VJ2y,s^σ,τdyds≤k1J1-J2σ,τ+θA0∫0tk1VJ1·,s-VJ2·,sσ,τds,(x,t)∈[0,l]×R+. Therefore, (66)VJ1·,t-VJ2·,tσ,τ≤k1J1-J2σ,τ+θk1A0∫0tVJ1·,s-VJ2·,sσ,τds,t∈R+. Then, by Gronwall inequatily, we come to the estimate (64).Using estimate (64) and 2D-Fock transform and together with the theory of classical cable equation, we obtain the next theorem, which shows that the solution to (2) depends continuously on the initial value.Theorem 15.
For eacht∈R+, define a map Ψt:C([0,l],L)→C([0,l],L) as Ψt(J)=VJ(·,t),J∈C([0,l],L); then {Ψt∣t∈R+} is a family of continuous injective maps and Ψ0(J)≡J.



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*Source: 9382079-2019-01-01.xml*