Asymptotic effects of boundary perturbations in excitable systems
aa r X i v : . [ m a t h - ph ] A p r Asymptotic effects of boundary perturbations inexcitable systems
Monica De Angelis, P. Renno ∗ Abstract
A Neumann problem in the strip for the Fitzhugh Nagumo system is consid-ered. The transformation in a non linear integral equation permits to deducea priori estimates for the solution. A complete asymptotic analysis shows thatfor large t the effects of the initial data vanish while the effects of bound-ary disturbances ϕ ( t ) , ϕ ( t ) depend on the properties of the data. When ϕ , ϕ are convergent for large t , the solution is everywhere bounded; when˙ ϕ i ∈ L (0 , ∞ )( i = 1 ,
2) too, the effects are vanishing.
Aim of the paper is the asymptotic analysis of the solution of the Fitzhugh Nagumosystem (FHN) for a strip ploblem with Neumann conditions. Some applicationsare related to the theory of excitable systems; in particular the cases of pacemak-ers [11] and when two species reaction-diffusion systems is governed by flux bound-ary condition [16]. Moreover, Neumann conditions are applied also in the distibutedFHN system. [17].Several aspects concerning the FHN model are discussed in pre-vious paper [4, 9, 10]. Moreover, owing to the equivalence between the FHN modeland the equation of superconductivity, other applications have been analyzed. [3]- [7], [19, 20].The present paper analyzes a transformation of the FHN model in a suitablenon linear integral-equation (see 3.14) whose kernel is a Green function which hasnumerous basic properties typical of the diffusion equation. Those properties implya priori estimates and so theorems on behaviour of the solution for large t can beobtained. ∗ Univ. of Naples ”Federico II”, Faculty of Engineering, Dip. Mat. Appl. ”R.Caccioppoli”,Via Claudio n.21, 80125, Naples, Italy. [email protected] Statement of the problem
Let u ( x, t ) be a trasmembrane potential and let v ( x, t ) be a variable associatedwith the contributions to the membrane current from sodium , potassium and otherions. The well known FHN system [11, 12, 15, 16, 19, 20] is(2.1) ∂ u∂ t = ε ∂ u∂ x − v + f ( u ) ∂ v∂ t = b u − β v where ε > b and β are positive constants that characterize the model’s kinetic. Further(2.2) f ( u ) = u ( a − u ) ( u −
1) (0 < a < . Assuming T as an arbitrary positive value, a typical example of problems whichtakes into account either initial perturbations and boundary perturbations is definedin Ω ≡ { ( x, t ) : 0 ≤ x ≤ L ; 0 < t < T } by(2.3) u ( x,
0) = u ( x ) , v ( x,
0) = v ( x )with the Neumann conditions(2.4) u x (0 , t ) = ϕ ( t ) u x ( L, t ) = ϕ ( t ) . It can be easiy verified (see,f.i [4, 9]) that the problem can be analyzed by meansof an integral differential problem with a single unknown function u ( x, t ). In fact,if F denotes the function:(2.5) F ( x, t, u ) = u ( a + 1 − u ) − v e − β t
2y (2.1) - (2.2) one has(2.6) u t − εu xx + au + b Z t e − β ( t − τ ) u ( x, τ ) dτ = F ( x, t, u ) ( x, t ) ∈ Ωwith u that has to satisfy the initial - boundary conditions (2 . , (2.4).As soon as u ( x, t ) is determined, the v ( x, t ) component will be given by(2.7) v ( x, t ) = v e − β t + b Z t e − β ( t − τ ) u ( x, τ ) dτ, where v is defined in (2 . .When source term F in (2.6) is a prefixed function depending only on x and t, then the initial-boundary problem (2.6), (2 . , (2 .
4) is linear and it can be solvedexplicitly by means of the Laplace transform. Moreover, when F depends on theunknown u ( x, t ) too, then by (2.6) one obtains an integral equation useful to studythe differential problem. The fundamental solution K ( x, t ) of the parabolic operator defined by (2.6) hasbeen already determined explictly in [9] and is given by(3.8) K ( r, t ) = 12 √ πε (cid:20) e − r t − a t √ t − b Z t e − r y − a y √ t − y e − β ( t − y ) J (2 p by ( t − y ) ) dy (cid:21) where r = | x | / √ ε and J n ( z ) denotes the Bessel function of first kind and order n. Moreover, one has [9]:
Teorema 3.1.
For all t > , the Laplace transform of K ( r, t ) with respect to t converges absolutely in the half-plane ℜ e s > max ( − a, − β ) and it results: (3.9) ˆ K ( r, s ) = Z ∞ e − st K ( r, t ) dt = e − r σ √ ε σ with σ = s + a + bs + β . t :ˆ u ( x, s ) = Z ∞ e − st u ( x, t ) dt , ˆ F ( x, s ) = Z ∞ e − st F [ x, t, u ( x, t ) ] dt , and let ˆ ϕ ( s ) , ˆ ϕ ( s ) be the L transforms of the data ϕ i ( t ) ( i = 1 , . Then the Laplace transform of the problem (2.6), (2 . , (2 .
4) is formally given by:(3.10) ˆ u xx − σ ε ˆ u = − ε [ ˆ F ( x, s, ˆ u ( x, s )) + u ( x ) ]ˆ u x (0 , s ) = ˆ ϕ ( s ) ˆ u x ( L, s ) = ˆ ϕ ( s ) . If one introduces the following theta functionˆ θ ( y, σ ) = 12 √ ε σ (cid:26) e − y √ ε σ + ∞ X n =1 (cid:20) e − nL + y √ ε σ + e − nL − y √ ε σ (cid:21) (cid:27) = cosh [ σ/ √ ε ( L − y ) ]2 √ ε σ sinh ( σ/ √ ε L ) =(3.11)then, by (3.10) and (3.11) one deduces:ˆ u ( x, s ) = Z L [ ˆ θ ( | x − ξ | , s ) + ˆ θ ( | x + ξ | , s ) ] [ u ( ξ ) + ˆ F ( ξ, s, ˆ u ( x, s ) ] dξ − ε ˆ ϕ ( s ) ˆ θ ( x, s ) + 2 ε ˆ ϕ ( s ) ˆ θ ( L − x, s ) . (3.12)Owing to dependence of source term F on the unknown, obviously all this is purelyformal. However, if one puts(3.13) θ ( x, t ) = ∞ X n = −∞ K ( x + 2 nL, t ) G ( x, ξ, t ) = θ ( | x − ξ | , t ) + θ ( | x + ξ | , t )4y (3.12) one deduces [4]: u ( x, t ) = Z L G ( x, ξ, t ) u ( ξ ) d ξ − ε Z t θ ( x, t − τ ) ϕ ( τ ) dτ + 2 ε Z t θ ( L − x, t − τ ) ϕ ( τ ) dτ + Z t dτ Z L G ( x, ξ, t − τ ) F [ ξ, τ, u ( ξ, τ ) ] dξ. (3.14)which represents an integral equation for the unknown u ( x, t ) . K ( x, t ) and θ ( x, t ) The behaviour for large t of the terms depending on the initial data and the source F has been already analyzed in [4] [5]. Now the effects of the boundary perturbations ϕ , ϕ will be estimated. For this an appropriate analysis of the kernels K ( x, t )and θ ( x, t ) will be considered.As for K ( x, t ) , in [9] has been proved that(4.15) | K | ≤ e − r t √ πεt [ e − at + bt E ( t ) ]where(4.16) E ( t ) = e − βt − e − at a − β > . Further, it results too:(4.17) Z ℜ | K ( x − ξ, t ) | dξ ≤ e − at + √ b π t e − ω t (4.18) Z t dτ Z ℜ | K ( x − ξ, t ) | dξ ≤ β . ω = min ( a, β ) β = 1 a + π √ b a + β aβ ) / . Now, if Γ( x ) is the gamma function and ζ ( x ) the Riemann’s Zeta function, let(4.20) C = 12 √ ε ω + b Γ(3 / ω − / √ π ε | a − β | (cid:20) Cb | a − β | + 3 C ω (cid:21) with C = 2 ε ζ (2) / ( eL ) . Then, one has the following theorem:
Teorema 4.2.
The θ ( x, t ) function defined in (3 . satisfies the following inequal-ities: (4.21) Z L | θ ( | x − ξ | , t ) | dξ ≤ (1 + √ b π t ) e − ω t (4.22) Z t dτ Z L | θ ( | x − ξ | , t ) | dξ ≤ β . Furthermore, it results: (4.23) lim t →∞ θ ( x, t ) = 0; Z ∞ | θ ( x, τ ) | dτ ≤ C , and (4.24) lim t →∞ Z t θ ( x, τ ) dτ = 12 ε σ cosh σ ( x − L )sinh ( σ L ) where σ = s(cid:18) a + bβ (cid:19) ε . roof. : We observ that properties of K ( x, t ) imply that: Z L | θ ( | x − ξ | , t ) | dξ ≤ ∞ X n = −∞ Z L | K ( | x − ξ + 2 nL | t ) | dξ = ∞ X n = −∞ Z x +2 nLx +(2 n − L | K ( y, t ) | dy ≤ Z ℜ | K ( y, t ) | dy (4.25)and so (4.21) and (4.22) follow by (4.17) and (4.18).Moreover , it results(4.26) ∞ X n = −∞ e − ( x +2 nL )24 εt ≤ t εe L ζ (2)and (4.15)implies:(4.27) | θ ( x, t ) | = 1 + C t √ π ε t [ e − a t + bt E ( t ) ] , consequently one obtains (4 . while considered that(4.28) Z ∞ t µ e − ωt dt = Γ( µ + 1) ω µ +1 re ( µ ) > − re ( ω ) > Z ∞ e − at √ t dt = r πa a > , by means of (4.15), (4 . can be deduced.Further as:(4.29) lim t →∞ Z t θ ( x, τ ) dτ = lim s → ˆ θ ( x, s ) ℜ e s > max ( − a, − β ) , by (3.11), one achieves(4.24). 7 Asymptotic effects of the boundary data
In the following we will have to refer to a known theorem on asymptotic behaviourof convolutions. ( [1],p 66).
Let h ( t ) and g ( t ) be two continuous functions on [0 , ∞ [ . If they satisfy the fol-lowing hypotheses (5.30) ∃ lim t →∞ h ( t ) = h ( ∞ ) ∃ lim t →∞ g ( t ) = g ( ∞ ) , (5.31) ˙ g ( t ) ∈ L [ 0 , ∞ ) , then, it results: (5.32) lim t →∞ Z to h ( t − τ ) ˙ g ( τ ) dτ = h ( ∞ ) [ g ( ∞ ) − g (0) ] . According to this, it is possible to state:
Teorema 5.3.
Let ϕ i ( i = 1 , be two continuous functions which converge for t → ∞ . In this case one has: (5.33) lim t →∞ Z t θ ( x, τ ) ϕ i ( t − τ ) d τ = ϕ i, ∞ ε σ cosh σ ( x − L )sinh σ L where σ = s(cid:18) a + bβ (cid:19) ε . Proof.
Let apply (5.32) with g = R t θ ( x, τ ) dτ and f = ϕ i ( i = 1 , . and (4.24). 8 eorema 5.4. When the data ϕ i ( i = 1 , verify conditions (5.30) (5.31), it re-sults: (5.34) lim t →∞ [ θ ( x, t ) ∗ ϕ i ( t ) ] = 0 ( i = 1 , Proof.
It sufficies to put h = θ ( x, t ) and g = ϕ i and to apply (4 . . Let us denote with f ∗ f the convolution f ( · , t ) ∗ f ( · , t ) = Z t f ( · , t ) f ( · , t − τ ) d τ and let N ( x, t ) be the following known function depending on the data ( u , v , ϕ , ϕ ) N ( x, t ) = − ε ϕ ( t ) ∗ θ ( x, t ) + 2 ε ϕ ( t ) ∗ θ ( L − x, t )+ Z L u ( ξ ) G ( x, ξ, t ) dξ − e − β t ∗ Z L v ( ξ ) G ( x, ξ, t ) dξ . (6.35)Owing to (2.5), (2.7) and (3.14),the solution related to the initial boundary FHNsystem 2.1-2.4 is given by [4]:(6.36) u ( x, t ) = Z L G ( x, ξ, t − τ ) ∗ { u ( ξ, τ )[ a + 1 − u ( ξ, τ ) ] } dξ + Nv ( x, t ) = v e − β t + b e − β t ∗ N ( x, t )+ b e − β t ∗ Z L G ( x, ξ, t − τ ) ∗ { u ( ξ, τ )[ a + 1 − u ( ξ, τ ) ] } dξ (6.37) 9hese formulae represent two integral equations for u and v . By means of theestimates deduced in sec.4 it is possible to apply the fixed point theorem in orderto obtain existence and uniqueness results [2, 4, 8]. When the Nagumo polinomial(2.5) is approximated by means of its linear part, then (6.36) (6.37) give the explicitsolution of the problem.As for the analysis and the stability of solutions of nonlinear binary reaction -diffusion systems of PDE’s, as well as the existence of global compact attractors,there exists a large bibliography . (see e. g. [10, 12–14, 18]. Moreover, as it is wellknown,the (FHN) system admits arbitrary large invariant rectangles Σ containing(0 ,
0) so that the solution ( u, v ), for all times t >
0, lies in the interior of Σ whenthe initial data ( u o , v o ) belong to Σ. [21]So, letting k F k = sup Ω T | u ( a + 1 − u ) | , k u k = sup Ω T | u ( x, ) | ; k v k = sup Ω T | v ( x ) | one has: Teorema 6.5.
For regular solution ( u, v ) of the (FHN) model, when the boundaryconditions are homogeneous, ( ϕ = ϕ = 0 ), the following estimates hold: (6.38) | u | ≤ k u k (1 + π √ b t ) e − ω t + k v k E ( t ) + β k F k ] | v | ≤ k v k e − β t + 2 (cid:2) b ( k u k + t k v k ) E ( t ) + baβ k F k (cid:3) For boundary data different from zero,the asymptotic behaviour of the solution ( u, v )of FHN system is established by theorems 5.3 and 5.4.
In conclution . When t tends to infinity, the effect due to the initial disturbances( u , v ) vanishes while the effect of the non linear source is bounded for all t. Moreover, also the effects determined by boundary disturbance ϕ , ϕ are vanishingin the hypotheses ( b ). Otherwise, they are always bounded. Acknowledgments
This work has been performed under the auspices of Programma F.A.R.O. (Fi-nanziamenti per l’ Avvio di Ricerche Originali, III tornata) “Controllo e stabilita’10i processi diffusivi nell’ambiente”, Polo delle Scienze e Tecnologie, Universita’ degliStudi di Napoli Federico II (2012).
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