About nonuniqueness of solutions of the Showalter-Sidorov problem for one mathematical model of nerve impulse spread in membrane

An important part of the development of modern biophysics is the study of mathematical models of processes in living nature. Processes such as blood clotting, nerve impulse spreading, cardiac muscle contraction can be modelled using the Fitz Hugh-Nagumo system of equations [1,2]:

( e i v t = a i V ss + в 1 w — к i v,

[ e2 Wt = a2 Wss + в2 w — к2 v - w3, where parameters a 1, a2, в 1, к 1 G R+, в2, к2 G R, e 1, e2 > 0. System (1). on the one hand, is the development of the classical Kolmogorov-Petrovsky-Piskunov model, and on the other hand, some simplified version of the Hodgkin-Huxley model, which plays a significant role in the theory of nerve conduction. However, the majority of researchers considered the system of equations (1) under the assumption of e 1, e2 = 0 [3,4]. At the same time, cases of degenerate systems (e 1 = 0 о re 2 = 0) remained poorly understood, the necessity of studying of which is connected with the fact that the rate of change of one of the components of system (1) can significantly exceed another one. In case of e 1 = 0, the phase space of the system is a simple Banach C^-manifold, therefore, the problem has a unique solution. The question of the solvability of the Showalter-Sidorov-Dirichlet problem for Fitz Hugh-Nagumo system (1) in the case e 1 = 0 was considered in papers [5,6], it was also studied their optimal control, start control and final observations for this system. In this article, we will be interested in case of e2 = 0. In this case, the phase space of system of equations (1) contains singularity of Whitney fold type [7], which leads to nonuniqueness of solutions.

Consider degenerate system of equations (1) in case e2 = 0 in cylinder Q = Q x R+ , where Q C R ⁿ is a bounded doni;mi with boundary d Q of class C ^x

( vt = a i Vss + в 1 w - кi v, v = v (s,t) ,w = w (s,t),2

[ 0 = a2 Wss + в2 w - к2 v - w3, with boundary value conditions

v(s,t) = 0, w(s,t) = 0, (s,t) G dQ x R+,(3)

and initial value condition v (0) = v о.(4)

Problem (2) - (4) can be investigated within the framework of abstract Showalter-Sidorov problem

L(u(0) - uо) = 0(5)

for semilinear Sobolev type equation

Lui = Mu + N(u), ker L = {0}(6)

in specially constructed function spaces. Here L G L (U , F) , M G Cl (U; F), N is nonlinear operator, U , F are Banach spaces. By the phase space of equation (6) we mean the closure of the set of all admissible initial values, for which there is a local solution to problem (5), (6) [8]. So, based on the theory of ( L,p )-bounded operators or ( L,p )-sectorial operators, G.A. Sviridyuk, and later his adherers [9,10], found the conditions for the unique solvability of problem (5), (6). Namely, when the operator M is ( L,p )-sectorial (bounded) and the phase space of equation (6) is a simple Banach C^ -manifold, there is a single quasistationary the (semi)trajectory of problem (5), (6) passing through point u 0, which lies pointwise in phase space [11]. Recall that Banach C^ -manifold is called simple if any of its atlas is equivalent to an atlas containing a single chart. In particular, if operator M is ( L, 0)-sectorial (bounded), then any solution (5), (6) will be a quasistationary (semi)trajectory. The main method for studying problem (2) - (4) is the phase space method. Following it, we construct set M = {u G U : ( I — Q )( Mu + N ( u )) = 0 }, then all solutions of problem (1), (3) lie in set M as trajectories, where Q is spectral projector [11].

Back in 1987, G.A. Sviridyuk suggested that the solution to problem (5), (6) may not be unique if phase space of equation (6) is not simple Banach C^ -manifold. In review [12] it was shown that initial value condition (5) for (6) can have several solutions in cases where phase space of (6) lies on smooth Banach manifold having singularities such as Whitney folds. For example, the Showalter-Sidorov problem for the Korpusov-Pletner-Sveshnikov equation may have two different solutions [13], and for the system of Plotnikov equations - three [14]. In work [7] it was shown that in degenerate case (for e2 = 0) phase space of (2) contains singularity such as Whitney folds, therefore, it can have one or more solutions or the solution may not exist. In the course of this study, we will identify the conditions for the existence and uniqueness or multiplicity of solutions of Showalter-Sidorov problem (4) for Fitz Hugh-Nagumo system (2) depending on the parameters of the system.

• 1.    The Morphology of Phase Space

Let О C R n be a bounded domahi with boundary д 0 of class C^ю. In cyUnder Q = О x R+ we consider system of equations (2) with boundary value conditions (3) and initial value condition (4). We set H i = W ₂1(0) , i = 1 , 2 and define space

H = Hi x H = W1(0) x W ₂1(0) .

Let vector functions u = ( v,w ) , Z = ( C, n ) , consider HiIbert space H = L 2(0) x L 2(0) with scalar product

[u,Z ] h = (v,(j Hi + (w,nj H, arid space UN = L 4(0) x L 4(0). В у U = F we denote the space adjoint to H with respect to duality of [•, • ] in H. By virtue of the Sobolev embedding Theorems there are dense and continuous embeddings

H ^ U N - H ^ U N ^ U = F .                   (7)

Note that space H is identified with its adjoint. Construct linear operators L,M : U ^ F

[ Lu,Z ] = (v,(j, u,Z C U,

[ Mu,Z ] = -a 1 (v s i ,^ з , j - a 2 (w s i ,n s i j, u,Z C U, where dom M = H and nonlinear operator

[ N ( u ) , Z ] = (в 1 w — к 1 v, ^j + (в 2 w — к ₂ v — w ³ , nj, ^here dom N = U_N.

(Note that the Einstein agreement on summation over repeated indices is fulfilled everywhere.) By construction, operator L C L (U , F) , M C Cl (U; F) .

Denote by

U⁰ = ker L = { 0 }x W— 1(0) , U¹ = W ^- 1(0) x{ 0 },

F¹ = im L = W ^- 1(0) x { 0 }, F0 = M [U⁰ П dom M ] = { 0 } x W ^- 1(0) , when U = U⁰ Ф U¹ , F = F⁰ Ф F¹ . Set L 1 as the restrict!on of operator L 10 U¹ , then L- ¹ C L (F¹ , U¹) .

Lemma 1. For any a 1 , a ₂ C R₊ , в 1 , в 2 , к 1 , к₂ C R , n <  4

• (i)    the operator M is ( L, 0)-seetonal:

• (u) N c C 1(U N ; UN ).

Proof, (i) ( L, 0)-sectoriality operator M was shown in work [14].

• (ii)    We show that N C C ¹ (U _N ; U*_N ) , where U*_N is dual space of U _N with respect to duality of [ •, • ] . Indeed, due to Holder inequality, we have

I ^{[ N ( u} ) , ^{Z ]} I <  ^{( C} 1 H ull U _N + ^C 2 H ullU N ) IICIIU N ,

I [ N U Z 1 ,Z 2] I =

J ^{( в} 1 ^Z 1 ^Z 2 ^{— к} 1 ^Z 1 ^Z 2) ^ds + J ^{( в} 2 ⁿ 1 ⁿ 2 ^{— к} 2 ⁿ 1 ⁿ 2 <  ^{( C} 3 ^||u|| U _N + ^C 4) • ^IIZ 1 H U N • ^IIZ 2 H U N ,

— 3 w ² n 1 n ₂) ds

≤

where constants C i C R₊ , i = 1 , 4 , depend neither on u, nor on Z, Z 1 , Z ₂ • Here N U is the Frechet derivative of operator N at point u. The inclusion of N C C 1(U _N ; U*_N ) is proved.

□

Thus, we reduced problem (2), (3) to a semilinear equation of Sobolev type (6). Note that condition (4) takes form (5). We are interested in the solvability of problem (2) - (4) for any u о = ( v о ,w о) G H.

Let {v k } denote the sequence of eigenvalues of the following spectral problem:

— А ф = vф, s G fi , ф ( s) = 0 , s G d fi ,

where eigenvalues are numbered in nondecreasing order of their multiplicity. Denote by {ф_к} the corresponding eigenfunctions orthonormal in the sense of scalar product (•, •} in L 2(D) .

Definition 1. Vector-function u G C 1((0, т); U) ПC((0, т); UN), satisfying equation (2), is called the solution of the equation. Solution u = u(t) of equation (2) is called the solution of problem (2), (4) if t lim \\L(u(t) — u о) || f = 0.

Build

M = ^ u G H : —(v,n) = —~

β ₂ w

κ 2

+ K- w ³ ,n ^ +

α 2 w

κ 2

s i , η s i

and note that all solutions of system of equations (2) satisfying boundary value conditions (3) will he in this set.

Lemma 2. Let a ₂ , к ₂ G R₊ , в 2 G (0 , a ₂ v 1) , n <  4 , then for any vector v G H1 there exists 'unique vector w G H₂ such that u = col ( v,w ) G M .

Proof. Construct an auxiliary operator

(A ( w ) ,n) = (-— w + — w ³ ,n\ + ("w W s i , κ 2       κ 2               κ 2 ⁱ

η s i

, w,n G H2, dom A = H2.

Denote by H 2 the space conjugate to H₂ with respect to duality of (•, •}. Insofar as

|(A(w 1) ,w 2 )| < C 1(||w 1 11H2 + ||w 1 11H2 ) ||w 2 11H2 , where constant C1 G R+ depen ds on в 2 ,k 2 ,a 2 and embedding constants (7) and does not depend on w, thus the action of operator A : H2 ^ H2 is proved. Note that operator A : H2 ^ H2 Is coercive, he.

^{lim (A ( w} ) ^,w) H w H h 11 = _IIH^lim f f ^{( —} Ks ^{w 2 —} a2 ⁽ w s i ^{)2 +} k c ^{w 4) d}A • H₂ → + ∞                   ²     || w || H₂ → + ∞ _Ω ^{κ 2        κ 2            κ 2}

- ¹

• f ( ws. + w ²) ds^    = + to.

In addition, operator A is strictly monotone, that is,

(A(w 1) — A(w2) ,w 1 — w2) = f ( — K| (w 1 — w2)2 — 02 (w 1 si — w2si )2 + + K12 • (w 1 — w2)2(w2 + w 1 w2 + w2))ds > 0 Vw 1, w2 G H2, as soon as w 1 = w2. Finally, we show the smoothness of operator A. Indeed,

I AK,n)l = I [( -^в2Щ— -4..U + -3 w ² пе ) *>1 <  ( C 1 + C 2 llwll H) llnll H2 I K II H ■ κ ₂       κ ₂          κ ₂

Ω where constants C1, C2 depend only on a2,в2,к2 and the nesting constants. Hence, by virtue of the Vishik-Minty-Browder Theorem [15], equation A(w) = —v has a unique solution.

□

Consider the case of в ₂ = a ₂ v 1 , put

H ^ = {v ¹ G Hi : ^± ,^) = 0 }■ H ^ = {w ¹ G H2 : ¹ ■ф) = 0 }.

Let v 1 be a single root and ф be an eigenfunction of problem (8), corresponding to the eigenvalue of v 1 , normalized in sense L ₂(O) . If v G H1 ai id w G H₂ be represented as v = v ¹ + гф and w = w ¹ + qф, where r,q G R , then set M takes the following form:

{ f — v ¹ = — —w ¹ — " 2 A w ¹ + ¹ f ( w ¹ + qф )³ ф*з, ) κ 2           κ 2              κ 2

u G H :                 ,                           ^Q                         .        (9)

—к ₂ r = ( w ¹ + qф )³ фаз.

Ω

Lemma 3. Let a2, к2 G R+, в2 = a2v 1, n < 4, then for any vector v1 G H1 there exists unique vector w1 G H1 such that v1 = — w1 + — A w1

κ 2        κ 2

( w ¹ + qф )³ ф*з. κ 2

Ω

The proof of this lemma is carried out similarly to the proof of Lemma 2, if we consider the following operator as an auxiliary operator:

A ( w ¹ )

-β 2 w

-

κ 2

^a A w1 + 1

κ 2          κ 2

j ( w¹ + qф )³ ф*з.

Ω

By Lemma 3 by v 0- aiid r ₀. we coiistruct w ¹

w ¹ + q ₀ ф, then u ₀ = ( v ₀ , w ₀) G M .

and q 0 . Put v 0 = v 0- + r 0 ф and w 0 =

Theorem 1. Let a 2 , к 2 G R+ , n <  4 , в 2 G (0 , a 2 v 1) , or в 2 = a 2 v 1 , q ² ||ф||L4 + 2 q J w¹ф ³ *з + f(w¹ )² ф ² = 0 , then the set M at the point u ₀ is a simple Banach C^ж-manifold.

The second equation of system (9) can be represented as:

q ³ φ ⁴ L ₄ (Ω)

+ 3 q ²

I w¹ф ³ d> + 3 q

Ω

J ( w¹ )² ф ² d> +

Ω

У ф ( w ¹ )³ d> + к ₂ r = 0 .

Ω

The equation (10) is a cubic equation of general form aq3 + bq2 + cq2 + d = 0 with respect to q. According to Cardano formulas, any cubic equation of general form with the help of replacement q = y — 3a can be reduced 10 canonical form y3 + py + e = 0 with coefficients a = 11 ф11L 4 (Q)

b =3

/

Ω

w ¹ ф ³ d>, c = 3

j ( w¹ )² ф ² d>, d =

Ω

У ф ( w ¹ )³ d> — к ₂ r,

Ω

3 ac — b ²

p =^0^

■e =2(

2 b ³      bc d

27 a ³   3 a ^{2 +} a) ¹ Q

-

= p ³ + e ² .

By virtue of the already mentioned Cardano formulas, Theorem 1 and Theorem on the existence of a solution of problem (5), (6) [8,10] is valid.

Theorem 2. For any u0 = (v0, w0) G H, n < 4, a2,k2 G R+ and (г) в2 G (0, a2v 1) there exists a unique solution to problem (2) - (4); (H) в2 = a2v 1, Q > 0 there exists a unique solution to problem (2) - (4); (Hi) в2 = a2v 1, Q = 0 and following condition is fulfilled q2 ll^llL 4(q) + 2 q I

Ω

w ^± Ф ³ ds +

I ^{( w± )2} ф ² = ⁰

Ω

there exists two solutions to problem (2) - (4);

(iv) в2 = a2v 1 , Q <  0 there exists three solutions to problem (2) - (4).

The authors would like to thank Professor G.A. Sviridyuk for the support and given opportunities.

The work was supported by Act 211 Government of the Russian Federation, contract no. 02.A03.21.0011.