invariant regions and asymptotic bounds for a hyperbolic version of the nerve equation

Nonhrreor Analysrs, Theory, Methods & App/icofiom, Vol. 16, No. 1 I. pp. 1035-1052. 1991. 0362-546X/91 $3.00+ .OO

Printed in Great Britain. 0 1991 Pergamon Press plc

INVARIANT REGIONS AND ASYMPTOTIC BOUNDS FOR A HYPERBOLIC VERSION OF THE NERVE EQUATION

MARTA VAL&NCIA*

Departament de Matematica Aplicada I, Universitat Politkcnica de Catahmya, Diagonal 647, 08028 Barcelona, Spain

(Received 1 March 1990; received in revised form 27 September 1990; received for publication 6 November 1990)

Key words and phrases: FitzHugh-Nagumo, contractiveness-property, partial differential equations.

INTRODUCTION: THE EQUATION

IN 1952, HODGKIN and Huxley [3] constructed a mathematical model describing the propaga- tion of electrical impulses on nerve fibers, composed of a system of ordinary differential equations coupled to a diffusion equation. In 1961, FitzHugh [2], and in 1962, Nagumo et al. [5] proposed the simpler model specified below, named the FitzHugh-Nagumo equations, which seems to describe the same qualitative behaviour,

(

U, = V,, + f(V) - 24 (x, t) E R x R+ (0.1)

24, = ou - yu

where cr and y are positive constants and f has the shape indicated in Fig. 1 (for example, f(u) = u - z?).

To deduce (O.l), the equations they used to relate the membrane current i = i(x, t) along the nerve fiber to the membrane voltage v = u(x, t) were

l

V* = - ri

i, = - GUI - j(u)

where x is the distance along the nerve fiber, r and c are constants denoting membrane resistance and membrane capacitance respectively, and -j(u) has the shape indicated in Fig. 1.

Fig. 1.

* Partially supported by DGICYT, project No. PB86-0306.

1035

1036 M. VALBNCIA

Eliminating i and resealing the time, one obtains the second order differential equation parabolic type

where f(v) = --I--(U). Then, system (0.1) was deduced, augmenting this equation with auxiliary variable u = U(X, t) (see [5]).

of

an

On the other hand, in 1967, Lieberstein [4] suggested to alter the diffusion equation in the original system of Hodgkin and Huxley in order to include the effects of line inductance along the nerve fiber. For that purpose, he used the following more general equations relating the membrane current i to the voltage current u,

l

v, = - ri - Ii,

ix = - cv, - j(v)

where I is a constant denoting membrane inductance (f will be small but not zero). Similarly to the parabolic case, eliminating i and recalling the time, one obtains the second order differential equation of hyperbolic type

EV,, + (1 - sf’(Wt = 0,X + f(v)

where E = I(r’c)-’ is a small parameter and f(v) = - Q(v) (more details can be found in [7]). In the present paper, we propose to put together both the simplification of FitzHugh and

Nagumo and the modification of Lieberstein, thus obtaining a new system which is a coupling of a semilinear wave equation with nonlinear damping and a parabolic equation (although the case 6 = 0 is also allowed), namely

EV,, + (1 - &f’(V))& = u,, + f(u) - u

24, = 62.4, + ou - yu (x, t) E R x R+

where E > 0, G > 0, y > 0 and 6 r 0 are constants and f E C2(R) has the shape indicated either in Fig. 1 or in Figs 2, 3 and 4, with h~_i+“mff’(u) < -(a/y) (for example, f(v) = v - v3).

Fig. 2. 0 5 Is,1 5 sj. Fig. 3. Is,1 2 sj 2 0. Fig. 4. s, I sz 5 0.

Note that the case of only one equilibrium point (for example in Fig. 5), which seems the more important case physically, is also considered.

Furthermore, we consider E so small that 1 - &f’(v) > 0, for all v E R (recall that E = l(r2c)-‘). Note that if we take formally E = 0 and 6 = 0 in (0.2) we obtain the FitzHugh-Nagumo equations (0.1). Therefore, we say that (0.2) is a hyperbolic version of the nerve equation.

Invariant regions of the nerve equation 1037

Fig. 5. s := s, = s, = sj.

The aim of this paper is to propose this hyperbolic version of the nerve equation, to give some results on the asymptotic behaviour of its solutions, and to compare them with the results obtained in the case E = 0. More specifically, this work is devoted to the study of the existence, the construction and the contractiveness property of positively invariant regions for the solutions of (0.2). We extend the studies due to Chueh et al. [I], Rauch and Smoller [6] and to ourselves [9] concerning the same problem for the case E = 0. As in these previous studies we do not impose boundary conditions, and our solutions will be defined for all x E R, though unlike those in [l] and those in [6], they will merely be restricted to be bounded. This last hypothesis has already been treated in [9].

For the case E = 0 of (0.2) Rauch and Smoller studied two families of positively invariant regions. The first family is formed by arbitrarily large positively invariant rectangles containing all constant steady state solutions and it allows to obtain a priori and asymptotic bounds on all the solutions, independently of its initial value. The second family is formed by small positively invariant rectangles containing the origin in their interior and it is used to prove that any solution having its initial values in one of these rectangles tends to zero.

In this paper, as in [9], we only deal with positively invariant regions for the solutions of (0.2) containing all its steady state solutions.

In what follows we focus on f such that

-p := -min(v E R:f(v) = -(a/y)u) I D := max(v E R:f(u) = -(a/y)vJ (0.3)

(that is the case of Fig. 1 and Fig. 2), since otherwise, analogous results would be obtained by symmetry.

1. DESCRIPTION OF THE MAIN RESULTS

Consider the system

V, = DAv + ~ MjV~, + G(V) j=l (x, t) E R” x R+, (1.1)

1038 M. VALBNCIA

where u E R”, D and M are diagonal-constant matrices with D L 0 and G is a smooth mapping from R” into R”. Assuming that this system has a local (in time) solution on some set X of smooth functions of x E R, we may consider the following.

Definition. A closed subset C c R” is a positively invariant region for the solutions of (1.1) if any solution v(x, t) having all its initial values in C satisfies that u(x, t) E C, for all x E R and t 2 0.

Observe that when C is bounded, its invariance property provides a priori sup-norm bounds on the solutions and, for example, this can be used to prove the global existence of solutions with initial values in C.

In order to use compactness arguments, Chueh et al. in [l] only considered state spaces X and regions C satisfying the so-called K-condition i.e. if u E X there exists a compact set K c R”, that may depend on v, such that if x $ K then V(X) E C. For example, a region E containing the origin in its interior and the space X = C,,(Rm) of uniformly continuous functions which tend to zero as 1x1 tends to + 00 satisfy the K-condition. If D has all coefficients distinct, it is proved in [l] that any positively invariant region must be of rectangular form with sides parallel to the axes, and that a sufficient condition for its invariance is that G(v) points strictly into C on ax, i.e. G(u) * n(u) < 0, for all u E 13x, where n(v) is the outward pointing normal vector of ax at u. Such rectangles are called contracting rectangles for the vector field G(u). It is clear that this condition is not necessary: note, for example, that the intersection of an arbitrary family of contracting rectangles for the vector field G(v) is also a positively invariant region for the solutions of (l.l), though it is not necessarily contracting for the vector field G(u). Nevertheless, condition G(u) * n(u) 5 0 is certainly necessary. As compensation, Rauch and Smoller proved that the contracting rectangles satisfy that any solution of (1.1) belonging to X and with initial condition in a contracting rectangle E, lie in fact in an homothetic smaller rectangle for t, L t > 0. Such property is called the contractivenessproperty.

In [9], we succeeded in avoiding the K-condition in the theorem of positively invariant rectangles and in the contractiveness property when X C BC(Rm), where BC(R”) denotes the space of uniformly continuous functions on R” without any other specification about its behaviour at infinity. Furthermore, we also proved that if we are able to construct a decreasing Lipschitz family of contracting rectangles for the vector field G(u), any solution of (1.1) belonging to X and with initial condition in one of these rectangles approaches the infimum of the family. Thanks to these results, we improved the study of the asymptotic behaviour of the solutions for the case E = 0 of (0.2) in two aspects. Firstly, global existence results and bounds were obtained for a wider family of solutions (X = BC(Rm)) than in [l] and in [6] (X = C,(Rm)). Secondly, we proved that all solutions approach the (optimum) critical region

C, = ((u, U) E R’: A, 5 u 5 B,, (o/y&4, 5 u 5 (a/y)&)

(where A, and B, are defined in Section 3 of the present paper) which is smaller than the critical region found in [6]. So, we obtained better asymptotic bounds, independently of the initial values.

Our aim, now, is to find analogous results when E # 0. The basic idea is to transform (0.2) into an equivalent first order system of type (1.1) so that we can try to apply the results obtained in [9]. Since standard changes of variable do not appear to be suitable in our case, it has been essential to introduce the following new variable,

W = 2&l+ - 2v5Jx + u - &f(U),


which allows us to transform system (0.2) in the following equivalent first order system of type

(l.l),

[;J = [-(Z-Z) + [;;I- v+EY) 2(f(u) - 24) f (2E)_ (1 - &f’(U))(U - &f(U) - w) )* (1.2)

Note that this transformation is only meaningful in the case of x being one-dimensional. In fact, we have not suceeded in finding an analogue in more dimensions.

According to the results of [l], if the diffusion matrix has two diagonal-coefficients equal, it is not necessary that the positively invariant regions are of rectangular form. Nevertheless, since we want to extend and compare the results of the case E = 0, we study the existence of positively invariant regions for the solutions of (1.2) being of rectangular form. Furthermore, it is shown in [l] that in this case, the results will be independent of 6. We will obtain apriori and asymptotic bounds on v, u and also on w, and we will prove that this is equivalent to obtaining bounds on v, u, v, and v,.

Below, we describe our results more precisely, together with the organization of the paper. In Section 2, we prove the solvability of the initial value problem associated to (1.2) in

X = K’(R) x BC(R) x BC(R). In Section 3 we give necessary and sufficient conditions for the existence of positively

invariant rectangles for the solutions of (1.2), containing all constant steady state solutions, so that we can define, for E small, two kinds of positively invariant rectangles. The first kind is formed by arbitrarily large contracting rectangles, symmetric with respect to the v-axis and containing all constant steady state solutions of (1.2). Hence, we derive a priori bounds and global existence results on all the solutions of (1.2). The second kind of rectangle, even though for fixed E it is not arbitrarily large, is formed by contracting rectangles containing all constant steady state solutions of (1.2) but are not symmetric with respect to the v-axis. This fact is essential to obtain optimum asymptotic bounds on the solutions because this family of rectangles is bounded from below by the following rectangle

C,, = [(v, w, U) E R3 :A, I v I B,, cc(B,) % w I &(B,), (a/y)A, I u 5 (a/y)B,]

which is smaller than the one obtained with the first family. In fact, Z,, is critical in the sense of being the smallest positively invariant rectangle for the solutions of (1.2) containing all constant steady state solutions of (1.2).

In Section 4 we prove that any solution of (1.2) approaches the critical region Z,, which agrees in the (v, u)-variables with the critical region C, for the case E = 0 of (0.2). For this purpose, we construct a decreasing Lipschitz family of contracting rectangles containing all the constant steady state solutions of (1.2) (see theorem 4.1) that approaches Z,, . The construction is done in two stages. In the first stage, we consider, for E small, a family of rectangles of the symmetric type. This family does not provide us with optimum asymptotic bounds for the solutions, but as it contains arbitrarily large rectangles, it allows us to prove that any solution enters in a finite time in a fixed rectangle Z?&(BEO), which does not depend on E. Then, in the second stage, we extend from below (and starting from &:s,(Bco)) the first family of rectangles with a second family of rectangles of the nonsymmetric type and with infimum C, . In this way, and using the contractiveness property, we prove that any solution approaches C,, asttendsto +co.

1040

Finally, in Section invariant regions for obtaining bounds for

Let us consider the

i

wt

where E > 0, cr > 0, y in Fig. 1 or in Fig.

1 - &f’(v) > 0 for all

M. VAL~NCIA

5 we prove that, owing to the existence the solutions of (1.2), obtaining bounds v, v, and vx.

2. FUNCTIONAL SETTING

of arbitrarily large positively for v and w is equivalent to

hyperbolic version of the nerve equation proposed in (0.2),

+ (1 - &f’(U))& = v, + f(v) - U

Ut = 6L4, + (TV - yu (x, t) E R x R+

> 0 and 6 2 0 are constants and f E C’(R) has the shape indicated either 2, with f’lr;n,i:,ff ‘(v) < -(a/y). Assume also that E is so small that

v E R. As we said before, defining the new variable

w = 2&V, - 2V5v, + v - &f(U),

system (0.2) is equivalent to the first order system (1.2), namely

i,i = c_:;;YJ + [;I- v+Ef(v!) 2(f(u) - 2.4) + (2E)_ (1 - &f’(V))(V - &f(V) - w) j

which is of type (1.1) with

D= [! i ;I M= i”i-l + ;I

and

GE(b, w, u)

= ((2E)_‘(W - v + &f(V)), 2(f(v) - u) + (2&))‘(1 - &f’(V))(V - &f(V) - w), ov - p).

This form tacitly suggests a division of the second member into the linear and the nonlinear parts because it appears not suitable for applying the classical theory of existence of solutions.

Observe that defining the function v(x, t) = exp( - (2~))‘t)(~(x, t), the homogeneous wave equation eel,, = 01, is transformed into the damped wave equation (with linear damping), EV,, + VI = v,, - (4&)-lv. Then, using the variable w = 2~21~ - 2&vX + v as above, one obtains the equivalent first order system

This suggests that (1.2) be rewritten in the following form: consider in X = BC’(R) x

K’(R) x BC(R) the initial value problem, x’(t) = &x(t) + F,(x(t))

x(0) = xg (2.1)


where x(t) = (u(*, t), w(., t), u(+, t)), x0 = (u,, w,, II,), F,: X -+ Xis defined by

FE(u) w, u) = (23(v), 2(f(v) - U) + (2&)-‘(1 - .sJ’(v))(u - &f(v) - w) + (2e)-‘w, CTU - yu)

and A, is the matricial operator

A, =

@-I-$ - (2&)_‘1 (2&))9 0

0 -(Q-l& - (2&)_‘Z 0

0 0 g%

Now and in what follows, BC’(R) denotes the space of functions u such that I~‘u/&T~ is a bounded uniformly continuous function on R, 0 I j I i, for i = 0, 1,2.

Going back to the aforementioned variable 01, one sees that the linear part of (2.1) defines a semigroup S(t) that, using the D’Alembert formula, takes the form

s(t): x -+ X

u(x) (Ii exp( - (2e))lt) (

x+(vw’t v(x + (v5)‘t) + (2&)-l i w(s) ds

x-(vF-‘t >

w(x) + exp( - (2e)-‘t)w(x - (@-It)

u(x) (27r)’ +m .1

(v&-l exp( - (4&)-7x - s)~)u(s) ds -co I

and D(A) = BC2(R) x BC’(R) x MT’(R) (except for the case 6 = 0 where D(A) = BC2(R) x B&(R) x BC(R)).

With respect to the nonlinear part of (2. l), since f E C’(R) it follows that F, : X + X is well defined, Lipschitz on bounded sets of X and continuously (Frechet) differentiable on X. So, using the classical results of Segal [8], we have the following result.

THEOREM 2.1 (existence and uniqueness of solutions). (i) For each x0 E X, there exists a unique continuous function

X(‘, x0): IO, Mxo)) --) X

defined on some maximal right interval of existence [0, 0(x0)), such that

tb) x(t, x0) = exp[At]xo + explA(t - WW~, x0)) do 0

(this equation is called the integral equation associated to (2.1)); (ii) if 0(x0) < + 00, then ((x(t, x0)1(X --, + CO for t + a(~,,); (iii) if x0 E D(A), then for each t E [0, w(xo)) we have that

(a) x(t, x0) E D(A), (b) x(t, xo> E CL(IO, Wo)), Xl, (c) x(t, x0) satisfies (2.1);

(iv) x(t, x0) is jointly continuous on t and x0.

(2.2)

1042 M. VAL~NCIA

As we see, every solution of (2.1) is a solution of (2.2) but the integral equation (2.2) has solutions that do not verify (2.1). These are usually called mild solutions. For simplicity in the exposition, in the rest of the paper we focus on strict solutions of (2.1), but, by a continuity argument applied on the results, they will also hold for the solutions of integral equation (2.2).

3. POSITIVELY INVARIANT REGIONS

In this section, we look for the existence of positively invariant regions for the solutions of (1.2) belonging to X, containing all constant steady state solutions of (1.2).

Since (1.2) is of type (1.1) and since we want to extend the results concerning the case E = 0 contained in [9], we consider rectangle-shaped regions with sides parallel to the axis. Then, in order to obtain sufficient conditions for its invariance and to apply the contractiveness property we restrict ourselves to considering contracting rectangles for the vector field GE(u) W, U) in the following sense.

Definition. A bounded rectangle C c R3 is called a contracting rectangle for the vector field GE(u, W, U) if and only if for all (u, W, U) E Z, GE(u) w, u) * n(u, w, U) < 0, where n(v, w, U) is the outward pointing normal vector of LJZ at (u, w, u).

In this way, consider

~=((~,w,~)ER~:A%uIB,C(WID,E~~~F).

Imposing that the vector field GE(u) w, U) points strictly into C on X5, we obtain the following conditions for the bounds of C,

A - &f(A) < c,

B - &f(B) > D,

4&f(u) + (1 - ef’(u))(u - &f(u) - C) > 4&F, u E [A,Bl,

4&f(u) + (1 - eJ’(u))(u - &f(u) - D) < 4&E, u E [A,Bl,

aA > yE,

aB < yF.

(3.1)

That is equivalent to choosing (if we know A, B, C and D) E and F among those constants satisfying

t

4&f(u) + (1 - ~f’(u))(u - &f(u) - C) > 4&F > 4e(a/y)B

4.$(u) + (1 - ef’(u))(u/- &f(u) - D) < 4&E < cle(a/y)A (3.2)

for all u E [A, B], and so to choosing (if we only know A and B) C and D among those constants satisfying

- &A) < C < u - &f(u) + 4~ f(u) - (o/Y)B

1 - &f’(U) (3.3)

B - &f(B) > D > u - &f(u) + 4~~(Li)_;:;~

for all u E [A, B].


s, = (% (1 - s(o/y))s3 9 Wrb3)

where Is,, s2, s3) = {v E R:.f(v) = (a/y)u) are such that s1 5 s, I sj (i.e. the constant steady state solutions of (1.2), which are not necessarily different) and let

(u,, v2, v,) = (v E R:f(v) = 0)

be such that vi < v2 < vj (Fig. 6).

Fig. 6.

Taking v = A and v = B respectively in first and second equation of (3.3), it follows that necessarily the bounds of C must satisfy that

1

(o/y)B < f(A)

(o/y)A > f(B). (3.4)

Then, since A < B it must hold that

t

(o/y)A < f(A)

(o/y)B > f(B). (3.5)

Thus, as we are interested in rectangles containing all the constant steady state solutions of (1.2), we restrict ourselves to considering A < s1 and B > s3. This implies that A < 0 < B and thus, from (3. l), (3.4) and the shape off, it follows that B, F andf(A) must be strictly positive and that A, E andf(B) must be strictly negative. So, in order to obtain contracting rectangles for the vector field GE(u, W, U) containing all constant steady state solutions of (1.2), it is necessary that B > v3 and that A < vl. In this case and due to the shape off, (3.4) is equivalent to

(o/y)B < f(Wo)f(B)) (3.6)

with suitable A < v1 satisfying (3.4). The following lemma holds.

LEMMA 3.1. Defining the function T(B) = f((y/a)f(B)) - (a/y)B for B 2 v3 it holds that there is a unique B, > v3 such that T(B,) = 0. Furthermore, it holds that for B > B,, 7’(B) is strictly increasing and positive and that for B E [u3, B,., T(B) is strictly negative.

1044 M. VALBNCIA

(The proof is given in [9, lemma 4.11. In fact we have that uj < B, I u, where u, = max(u E R:f(u) = - (a/y)u).)

Note that lemma 3.1 together with (3.6) imply that if B E [u,, B,[ and A < ur then GE(u, w, U) * n(u, w, U) > 0 for some (u, w, u) E aYZ. Thus, there do not exist positively invariant rectangles with B E [u, , B,[ and A < ur . Furthermore, the values B = B, and A = A, (where A, = (y/o)f(B,)) define a lower bound on the u-axis for the positively invariant rectangles for the solutions of (1.2) containing all the constant steady state solutions (note that from lemma 3.1 it follows that A, < ul). In fact, we will prove that these bounds are critical in the sense of defining (with suitable C, D, E and F) the smallest positively invariant rectangle for the solutions of (1.2) containing all constant steady state solutions.

So, the problem of the existence of contracting rectangles for the vector field GE(u) w, u) containing all constant steady state solutions of (1.2) is reduced to proving the existence of B > B, and A < A, such that

4c(o/y)B A - &f(A) + 1 _ Ef ‘(u) < ZJ - &f(U) +

4&f(V) < B - &f(B) +

4e(a/y)A

1 - &f’(U) 1 - &f’(U) (3.7)

for all u E [A, B] and to choosing suitable C, D, E and F among those constants satisfying (3.2) and (3.3).

As we have said in Section 1, in a first stage we look for rectangles which are symmetric with respect to the u-axis. So, we have the following theorem.

THEOREM 3.1. For E small (E E (0, EJ), there exist arbitrarily large contracting rectangles for the vector field GE(u) w, U) containing the steady state solutions of (1.2), of the form

C = ((u, w, u) E R3: -B_=u~B,Crw~D,Esu~F).

Proof of theorem 3.1. Consider A = - B and recall E > 0 small enough so that 1 - &f’(u) > 0 for all u E R. Defining the functions

H,(u) = u - &f(U) + 4&f(U)

1 - &f’(U)

which is C’(R) with respect to the variable u, and

4&(O/Y)Y &(u9 Y) = Y - &f(Y) - I _ Ef ,(u)

which is C’(R) with respect to the variable u and C2(R) with respect to the variable y, the problem is reduced (see (3.7)) to proving the existence of B > B, such that

forall UE [-B,B]. Ke(u, -B) < Hz(u) < K(u, B)

Let a be the inflection point off. Due to the shape off one can prove that

if f(u) *f”(v) > 0 then H,’ (u) > 0,

H;(a) # 0 so CI is not an extremum of H,, (3.8)


and that if y > 0 then Ke(a, y) < KE(u, y) for all v # a,

if y < 0 then KE(a, y) c Ke(u, y) for all u # a,

for E small (0 < E < E,), K,(u,y) is strictly

increasing with respect to y and for all u we have

thatK,(u,y)++oo asy-, +co.

(3.9)

Define now u, = maxlu E R:f(v) = -(a/y)u) and note from (0.3) that

-v, < min(u E R:f(u) = -(a/y)u).

Consider M, = max(lH,(u)l: u E [-uc, u,.). Thus, since K,(a,y) -+ foe as y-’ foe there exists B > 0, B depending on E, such that M, 5 KE(a, B) and -Me 2 KE(a, -B). Further- more, since KE(a, u,) < KE(uCr u,) = IYE < Me, it holds that, for E small, B > u, 2 B, and thus, that A = -B < - u, < ul. Then, by using (3.8) and (3.9) we have that

(i) if u E [-u,, u,] and u # a, then KE(u, -B) < K,(a, -B) I -Me I HE(u) _( M, I

KE(a, B) < Kdu, B). (ii) If u = a, since H,‘(a) # 0, then Ke(a, -B) I -Me < H,(a) < M, I K,(a, B).

(iii) If u E L-B, - u,[, since./(u) > - (a/v)u (note (0.3)) and H,‘(u) > 0, then Ke(u, -B) I Kc(u) u) < H,(u) < H,(- II,) I M, I KE(a, B) < Kc(u) B).

(iv) If u E Iv,, B], since f(u) < -(a/y)v and H,‘(u) > 0, then KE(u, -B) < KE(a, -B) s -Me 5 H,(v,) < He(u) < Ke(u, u) < Kc(u) B).

Putting these results together, it follows that KE(u, -B) < H,(v) < KE(u, B) holds for all u E [-B, B]. So, any rectangle determined by B > 0 such that M, s KE(a, B) and -M, 2 Ke(a, -B), A = B and suitable C, D, E and F satisfying (3.3) and (3.4) is contracting for the vector field GE(u) W, u).

In order to prove that these rectangles contain all the constant steady state solutions of (1.2), note that since B > u, it follows that B > s3 and A < s, . Thus, taking u = sr and u = s3 in the first and second equations of (3.3), it holds that C < (1 - e(a/y))s, and D > (1 - E((T/Y))s~. Finally, from (3.1) we have that E < (a/y)s, and F > (o/y)s,, and so S, , S2 and S3 belong to these rectangles.

Noting that B can be arbitrarily large and that - C, D, -E and F tend to + CL) as B does, we finish the proof of theorem 3.1. n

Remark 3.1. Consider E E (0, cl) and define the following functions

C,(B) = min u

and

- &j-(u) + 4~~(~) - (a’y)B. u E [ - B, B] 1 - &f’(U) . 1

D,(B) = max u - &f(u) + 4.~‘(;)~F:4~~~~: u E f-B, B]].

Noting that - C,(B) and D,(B) are increasing functions for B > B, and using (3.3) and (3.4) it follows that those rectangles considered in the proof of theorem 3.1 are bounded from below by the rectangle

EE = ((u, W, U) E R3: -B, 5 u 5 B,, C,(B,) I w 5 D,(B,), - (o/y)B, 5 u h (a/y)B,)

1046 M. VALkNClA

where B, = max{B E R: M, = KE(a, B) or M, = - &(a, -B)). Furthermore, XE is positively invariant for the solutions of (1.2) because it is the intersection of positively invariant rectangles for the solutions of (1.2) but it is not contracting for the vector field GE(u, w, u). Nevertheless, the dependence of C, on E will be a problem in the following section. In order to avoid this problem consider A& = M(E) as a continuous function of E on [0, q]. So, there exists co E [0, q] such that k&s,) = max{M(e): E E [0, EJ]. Thus, it follows that for all E E [0, EJ, B,,, r B, and C, c C,,. So, each B z Be0 defines contracting rectangles for the vector field GE(u, w, U) of the type considered in theorem 3.1, for any E E (0, .si). If we restrict ourselves to consider B r B,,, it follows that these rectangles are bounded from below by C,,, which is a positively invariant rectangle for the solutions of (1.2) for any E E (0, EJ and does not depend on E.

Theorem 3.1 gives us a priori bounds on the solutions of (1.2) and this allows us to conclude, for example, the existence for all t 1 0 of any solution of (1.2). The symmetry on the u-axis of the rectangles considered on it appears to be very restrictive to obtain asymptotic bounds on the solutions as sharp as possible. This is the reason why, in a second stage, other types of rectangles are to be considered. We have the following theorem.

THEOREM 3.2. Choose 6 > 0 and B, > B, + 26. There exists s2 = .cZ(BO) > 0 such that for E E (0, Q), each B E [B, + 26, B,] defines a contracting rectangle for the vector field GE(u, w, U) containing all constant steady state solutions of (1.2), of the form

C = l(v, w, U) E R3: (y/a)f(B - 26,) 5 u 5 B, C 5 w s D, E I u 5 F)

where 6, is chosen in the range 0 < 26, < min(d, (y/a)T(B, + 6)) and T(e) is the function defined in lemma 3.1.

Proof of theorem 3.2. Consider A(B) = (y/a)f(B - 26i). From (3.7), we must prove that each B E [B, + 26, B,] satisfies

A(B) - &f(A(B)> + 4c(o/y)B

1 - &f’(U) < u - &f(U) +

4&f(U)

1 - &j-‘(U) < B - &f(B) +

4da/y)A(B)

1 - &f’(U)

for all u E [A(B), B] and recall also that we consider E so small that 1 - &f’(u) > 0 for all u. Furthermore, for B E [B, + 26, B,] it holds that [A(B), B] c [A(B,), B,] with [A(B,), B,] independently of E. Then, considering the function

Mu9 Y) = Y - &f(Y) + 4&f(Y)

1 - &f’(U) ’

which is C’([A(B,), B,]) with respect to the variable u and C2([A(Bo), B,]) with respect to the variable y, it holds that there exists E 2 = c2(BO) such that if E E (0, e2) then L,(u,y) is an increasing function with respect to y, for all u E [A(B,), B,]. So, if u E [A(B), B] (B E [B, + 26, B,]) and E E (0, e2) we have that

(i) u - &f(u) + 4&f(U) 4&f W 4&B - 2d1)

1 - &J’(U) 5 B - &f(B) +

1 - &f’(U) < B - &f(B) +

1 - &f’(U)

= B - &f(B) + 4Ha/y)A(B)

1 - &f’(U) .


(ii) Since B - 26r > (B, + 26) - 26r > B, + 6 > B,, from lemma 3.1 and the range of 6, it follows that T(B - 26r) > T(B, + 6) > 2(o/y)d,, and thus that f@(B)) > (a/y)B. So,

v - &f(V) + 4&f(V) 4c.M (B)) 4.$a/ y)B

1 - &f’(V) 2 A(B) - &I(B)) +

1 - &f’(V) > A(B) - &M(B)) +

1 - &f’(V) .

Putting these results together, we have that for E E (0, Q), (3.7) holds withA = A(B). So, for E E (0, Ed), the rectangles determined by B E [B, + 26, B,], A = A(B) and suitable C, D, E and F satisfying (3.2) and (3.3) are contracting for the vector field GE(v) w, u).

In order to prove that these rectangles contain all the constant steady state solutions of (1.2), note that since B > B, it follows that B > s3 and then A(B) = (y/o)f(B - 26,) < (y/a)f(B,) = A, < s1. Thus, as in the proof of theorem 3.1, it follows that C < (1 - s(a/y))s, ,

D > (1 - s(a/y))s,, E < (a/y)s, and F > (a/y)s, and so, S,, S2 and S3 belong to these rectangles. n

Remark 3.2. Consider E E (0, E*) and define the following functions

Cc(B) = min v - &J‘(v) + 4s f(v) - (a/@

1 - &j-‘(V) : v E [(Y/@-VO, Bl

1 and

D,(B) = max v - &f(v) + 4~ f(v) - f(B)

1 - &f’(V) : v E [OJ/WW), Bl .

1

Noting that - cc(B), 6,(B) and -f(B) are increasing functions for B 2 B, it follows that the rectangles studied in the proof of theorem 3.2 are bounded from below (see (3.2) and (3.3)) by the rectangle

c, = ((v, W, U) E R3: (yh)f(~,) I v I B,, Ce(~,) 5 w 5 D,(B,)J(B,) I 2.4 I (dy)~,)

where B, is the value defined in lemma 3.1. Similar to C, in remark 3.1, we have that Eh is positively invariant for the solutions of (1.2) but it is not contracting for the vector field GE(v, w, u). Furthermore, remember that we proved at the beginning of this section that B = B, and A = (y/a)f(BC) define a lower bound on the v-axis for the positively invariant rectangles for the solutions of (1.2) containing all constant steady state solutions, so that, Xh is critical in the sense of being the smallest positively invariant rectangle for the solutions of (1.2) containing all the constant steady state solutions. In particular, if E is chosen in the range 0 < E < min(e, , Q) it follows that Ch c C,.

Finally, note that C, is independent of E in the (v, u)-variables and that Eh agrees in the (v, u)- variables with the critical region, C,, found in [9] for the case E = 0.

4. ASYMPTOTIC BOUNDS

From Section 3, we have that the rectangle C,, is critical in the sense of being the smallest positively invariant rectangle for the solutions of (1.2), containing all the constant steady state solutions. Our aim now is to prove that all solutions of (1.2) approach arbitrarily E,, (theorem 4.2) so in that way we obtain the best asymptotic bounds for the solutions that could be obtained by the method of positively invariant rectangles.

We state now the contractiveness property, that was proved in [9], which will allow us to obtain the wanted result, namely the following theorem.

1048 M. VALBNCIA

THEOREM 4.1. Let C(B), for B E [B, , B,], be an increasing one-parameter family, of parameter B, of contracting rectangles for the vector field GE(u, w, U) of the type

C(B) = ((u, w, U) E R3: A(B) I u I B, C(B) I w 5 D(B), E(B) I u s F(B)]

where A(B), C(B), D(B), E(B) and F(B) are (locally) Lipschitz functions of B. Then, any solution of (1.2) belonging to X and having its initial values in one of these rectangles, will eventually enter in Z(B,).

As we have said in Section I, in a first stage we will construct, for E small, a family of rectangles of the symmetric type satisfying the hypothesis of theorem 4.1. For that, take B, = B, in theorem 4.1 where BE0 is defined in remark 3.1 and B, > BcO. Consider the functions C,(B) and D,(B) defined too in remark 3.1 and note that they are (locally) Lipschitz functions of B. Since C,(B) + cf( - B) + B and B - &f(B) - D,(B) are continuous and strictly positive functions in [BcO, B,] (see the proof of theorem 3.1), we can choose 6, in the range 0 < 6, < min(C,(B) + cf( - B) + B, B - &f(B) - D,(B) : B E [BE,, B,] and define 8, such that 8~8~ = &(I - &f’(a)), where a is the inflection point off. Then, considering

&O(B) = ((v, w, U) E R3: -B i u I B, C,(B) - 6, s w 5 D,(B) + 6,,

-(a/y)B - 6, i u YS (a/y)B + 6,]

we have the following lemma.

LEMMA 4.1. For E > 0 small (E E (0, aI)) it follows that X,,,(B), B E [BcO, B,], is a family of rectangles of the type considered in theorem 4.1.

Proof of lemma 4.1. Fix E E (0, cl) and consider B E [BeO, B,]. Define A(B) = -B, C(B) = C,(B) - do, D(B) = D,(B) + 6,) E(B) = -(a/y)B - 3, and F(B) = (a/y)B + 8,. Then:

(1) on the one hand, since 6, < C,(B) + &f( - B) + B we have that

A(B) - &4(B)) = - cf( - B) - B < C,(B) - 6, = C(B)

and on the other hand, since 6, < B - &f(B) - D,(B) we have that

D(B) = D,(B) + 6, < B - &f(B).

(2) By the definition of C,(B), for u E [-B, B] it holds that

C,(B) + (d/2) < u - &f(u) + 4.5 f(u) - (a/y)B + 6

1 - &f’(U) O’

or equivalently that (1 - &f’(u))(S/2) < (u - &f(v) - C(B))(l - &f’(v)) + 4.$f(u) - (a/y)B). But since a is the maximum off ‘, for u E [-B, B] it holds that

(a/2)(1 - &f’(a)) + 4r(a/y)B < (v - &f(u) - C(B))(l - &f’(u)) + 4&f(u).

so, 4&F(B) = 4~((a/y)B + 8,) < (u - &f(u) - C(B))(l - &f’(v)) + 4&f(v)

holds for v E [-B, B]. Similarly, we can prove that for u E [-B, B] it holds that

(u - &f(u) - D(B))(l - &f’(u)) + 4&f(u) < -4~((o/y)B + 8,) = 4&E(B).


(3) We have that aA = - aB > - y((a/y)B + 8,) = YE(B) and that

CJB < y((a/y)B + 8,) = yF(B).

Putting these results together, we have that (3.1) holds and thus, C,0(B), B E [B,,,, B,], is a contracting rectangle for the vector field G,(v, w, u). Note also that it is symmetric with respect to the v-axis and that B > BEO, so from theorem 3.1 it follows that Z&,,(B) contains all the constant steady state solutions of (1.2).

Finally, note that if B E [B,,, B,], we have that -A(B), - C(B), D(B), -E(B) and F(B) are increasing functions of B and since f E C2(R), that they are (locally) Lipschitz functions of B. n

For the second family of rectangles let us consider, for simplicity, S, the same as that used in the first stage. Consider B, = B, + 2&, in theorem 4.1 and consider B, > B, + 26, where B, is defined in lemma 3.1 and 6r chosen in the range 0 < 26r < min(d,, (y/a)T(B, + 6,)). Consider the functions cc(B) and DE(B) defined in remark 3.2 and note that they are (locally) Lipschitz functions of B. Since cc(B) + cf((y/a)f(B - 26,)) - (y/df(B - 26,) and B - &f(B) - d,(B) are continuous and strictly positive functions in [B, + 26,) B,] (see the proof of theorem 3.2) we can choose a2 in the range 0 < a2 < min{ce(B) + .sf((y/g)f(B - 26,)) - (y/a)f(B - 26,), B - &f(B) - B,(B): B E [B, + 26,, B2]] and define 3, such that S&S2 = 6,(1 - &f’(a)), where a is the inflection point off. Then considering

C,,(B) = {(u, W, U) E R3: (y/a)f(B - 26,) I u I B, cc(B) - 6, 5 w I D’,(B) + d2,

f(B - 26J - 5, I u 5 (a/y)B + &I,

we have the following lemma.

LEMMA 4.2. For E > 0 small (E E (0, c2(B2))) it follows that I&(B), B E [B, + 26,, B,], is a family of rectangles of the type considered in theorem 4.1.

Proof of lemma 4.2. Fix E E (0, s2(B2)) and consider B E [B, + 26,, B,]. Define A(B) =

(y/o)f(B - 26,), C(B) = cc’,(B) - 6,, D(B) = B,(B) + d2, E(B) = f(B - 26,) - 6, and F(B) = (o/y)B + 6,. Then,

(1) On the one hand, since a2 < ce((B) + &f((y/o)f(B - 26,)) - (y/cT)f(B - 26,) we have that A(B) - &f(A(B)) = (y/a)f(B - 26,) - &f((y/o)f(B - 26,)) < cc(B) - B2= C(B) and on the other hand, since 6, < B - &f(B) - D’,(B) we have that B - &f(B) > DE(B) + d2 = D(B).

(2) By the definition of cc(B), for u E [A(B), B] c [(o/y)f(B), B], it holds that

cc(B) + (6,/2) < v - &f(u) + 4ef(Y)_C(fq;;)B + d2,

or equivalently that

(6,/2)(1 - &f’(u)) < (u - cfv - C(B))(I - &f’(u)) + 4E(f(v) - (a/y)B).

Then, since a is the maximum off’, it holds that

(6,/2)(1 - &f’(a)) + 4c(a/y)B < (v - &f(u) - C(B))(l - &f’(v)) + 4&f(u),

for u E [A(B), B]. So, for v E [A(B), B] we have that

4&F(B) = 4.$(a/y)B + 3,) < (v - &f(u) - C(B))(l - &f’(v)) + 4&f(v).

1050 M. VALtiNCL4

Similarly, we can prove that for u E [A(B), B] we have that

4&E(B) > (u - &f(U) - D(B))(l - &J’(V)) + 4&f(U).

(3) We have that oA(B) = yf(B - 26,) > y(f(B - 26J - 6,) = y,?(B) and that

OB < y((a/y)B + 8,) = yF(B).

Putting these results together, we have that (3.1) holds and thus ChO(B), B E [B, + 2&, B,], is a contracting rectangle for the vector field GE(u, w, u). Note also that it is of the type considered in theorem 3.2 and that B > B,, so from theorem 3.2 it follows that e,,(B) contains all the constant steady state solutions of (1.2).

Finally, note that if B E [B, + 2&, B,], we have that -A(B), -C(B), D(B), -E(B) and F(B) are increasing functions of B and, as f E C’(R), they are (locally) Lipschitz functions of B. n

Finally, we are ready to obtain the asymptotic bounds for the solutions of (1.2) belonging to X. Note that taking 6,, small, we have that B,, > B, + 2& (where Be0 is defined in remark 3.1.1) and so, if E E (0, min(e, , c2 (B,,))) the following holds.

THEOREM 4.2. For E small, any solution of (1.2) belonging to X arbitrarily approaches the rectangle C, .

Proof of theorem 4.2. Fix E E (0, min(e,, e2(B,J)) and let (u(*, t), w(*, t), u(*, t)) be a solution of (1.2) belonging to X with initial data (q,, wO, u,) = (u( *, o), w(. , 0), u(. , 0)). Consider for B > B,, the auxiliary family of rectangles

Z,(B) = ((u, w, U) E R3: -B I u I B, C,(B) s w 5 D,(B), - (a/y)B 5 u I (a/y)B).

Note that for B > BcO, C,(B) is a family of positively invariant regions for the solutions of (1.2) because they are the intersection of contracting rectangles for the vector field GE(u, w, U) but that they are not contracting for the vector field GE(u, w, u). Furthermore, this family is increasing and it contains arbitrarily large rectangles, thus there exists B1 L B,, such that (u, 3 wo 9 ~0) E Co (4).

Consider now the family E,,(B) , B E [B,, B,], described in lemma 4.1, and note that for any fixed 6, > 0 we have that C(B,) C C6,(B1). It follows that for E small we are in the hypothesis of theorem 4.1. So, there is a To(&, So) > 0 such that for t 2 TO(.s, a,), (u(*, t), w(-, t), u(*, t)) E

%, (4,). Take now B, = B, in lemma 4.2. Since Be0 > u, where u, is defined in the proof of

lemma 3.1, it follows that - (a/y)B,, > f(B,J. So, considering the family l&,(B), for B E [B, + 26,, B,] defined in lemma 4.2, it holds that &,(BEO) C &,(B,J and thus, for t 1 TO(&, So) we have that (u(*, I), w(*, t), u(*, t)) E&,(B,J. So, we are again in the hypothesis of theorem 4.1 and thus, there is a Ti (E, 6,) > 0 such that for t L Tl(c, a,),

(u(*, t), w(*, t), ~(‘9 t)) E %JB, + 260). Finally, note that Tl(e, 6,) and the bounds of &,,(Bc + 26,) are continuous functions of a0

and that if do tends to 0 then zaO(B, + 26,) tends to C, . So, by letting 6, tend to zero we prove that all the solutions of (1.2) belonging to X, approach arbitrarily Ch . n


5. COMMENTS

In this section we prove that owing to the existence of arbitrarily large positively invariant regions for the solutions of (1.2), bounding v and w is equivalent to bounding v, V, and v,.

Remember that system (1.2) is obtained introducing the new variable w in (0.2), but observe that if we introduce the variable

iit = 2&V* + 2&JX + v - &f(V)

instead of w then system (0.2) is transformed in the following first order equivalent system,

(i) = ($Y;) + (“‘z- u+Ef(? 2(f(v) - u) + (2&)_ (1 - &f’(U))(V - &f(U) - W) ). (5.1)

This system differs from system (1.2) in the matrix A4 which transformed into the matrix - A4 (in fact, introducing ii, instead of w is the same as making a symmetry with respect to the variable x). So, a rectangle of the type

contracting for the vector field G,(v, ti, u) is a positively invariant region for the solutions of (1.2) but also for the solutions of (5.1). Then, we have the following.

Let (v, u) be a solution of system (0.2) with initial values (v( a, 0), z+ (* , 0), u( *, 0)) and consider the initial values corresponding to (1.2) and (5.1), (uO, w,, uO) = (v(., 0), w(., 0), u(*, 0)) and (UC89 tiO, uo) = (v( * , 0), ti( *, 0), u( *, 0)) respectively. Suppose now that I: is a positively invariant rectangle for the solutions of (1.2) and (5.1) such that (vO, w,, uo) E Z and (uO , tie, uo) E iY (that is possible because there exist arbitrarily large positively invariant rectangles). It follows that

c 5 w(x, t) I D

c 5 iqx, t) 5 D (x, t) E R x R+.

Using now that A I v(x, t) -i B for all (x, t) E R x R+ and that f is continuous in R and by the definition of w and W, it holds that there exist constants G and H such that

I 2evt(x, t) - 2&(x, t) 5 G

5 2&24(x, t) + 24&(x, t) I G (x, t) E R x R+.

So, we have that

(H/2&) 5 v,(x, t) i (G/2&)

- (max(G, - H)/2fi) I v,(x, t) I (max(G, - H)/2fi) (x, t) E R x R+.

In this way we have shown how a priori and asymptotic bounds on v, w and iit imply a priori and asymptotic bounds on v, V, and v, . Note that the existence of positively invariant rectangles is a sufficient but not a necessary condition in order to obtain bounds on u, , v, . This means that if we know the bounds on u, w and ii, by another method, the same conclusion will hold.

Reciprocally, if there exist constants A, B, I, J, K and L such that

A I V(X, t) 5 B

II v,(x, t) I J (x, t) E R x R+

K I u,(x, t) I L

1052

then,

M. VALBNCIA

2&I - 26L I 2w,(x, t) - 2d&x, t) 5 2&J - 2fiK

2.d + 2fiK I 2q(x, t) + 2fivx(x, t) I 2&J + 2v5L (x, t) E R x R+.

Using now, that A 5 v(x, t) 5 B for all (x, t) E R x R+ and thatfis continuous in R, we have that there exist C, D, c and b such that

That is to say that apriori bounds on v, w and 13.

c 5 w(x, t) 5 D

c 5 iit(x, t) I D (x, t) E R x R+.

or asymptotic bounds on v, v, and v, imply a priori or asymptotic

Acknowledgements-The author wishes to thank J. Sol&Morales for helpful discussions and valuable suggestions during the development of this work.

1.

2.

3.

4. 5.

6. 1. 8. 9.

REFERENCES

CHUEH K. N., CONLEY C. C. & SMOLLER J. A., Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. math. J. 26, 313-392 (1977). FITZHUGH R., Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445-466 (1961). HODGKIN A. L. & HUXLEY A. F., A quantitative description of membrane current and its application to conduction and excitation in nerves, J. Physiol. 117, 500-504 (1952). LIEBERSTEIN H. M., On the Hodgkin-Huxley partial differential equation, Mafh. Biosci. 1, 45-69 (1967). NACUMO J., ARIMOTO S. & YOSHIZAWA S., An active pulse transmission line simulating nerve axon, Proc. IRE. 50, 2061-2070 (1962). RAUCH J. & SMOLLER J. A., Qualitative theory of the FitzHugh-Nagumo equations, Adv. Math. 27, 12-44 (1978). SCOTT A. C., The electrophysics of a nerve fiber, Rev. Mod. Phys. 47, 487-533 (1975). SECAL I., Nonlinear semigroups, Ann. Math. 78, 339-364 (1963). VALBNCIA M., On invariant regions and asymptotic bounds for semilinear partial differential equations, Nonlinear Analysis 14, 217-230 (1990).

invariant regions and asymptotic bounds for a hyperbolic version of the nerve equation

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