UNSTABLE PERIODIC WAVE SOLUTIONS OF NERVE AXlON DIFFUSION EQUATIONS

BSTRAG’. Unstable periodic solutions of systems of parabolic equations are studied. Special attention is given to the existence and stability of solutions.


IHTRDDUTION.
Diffusion systems of partial differential equations are of great importance in biosciences.In this paper, unstable periodic solutions of systems of the form u t u + F(u,w), Equations of this type arise in neurophysiology in the study of nerve impulses on nerve axon, see [1,2].
Other classes of diffusion equations are also involved in biology, see for example [3-9].

ZISTZNCE OF SOLUTIONS
It is known that for G(u,w) u, if > 0 is sufficiently small, equation (I.I) has two types of wave solutions, namely, pulse travelling wave solutions and periodic travelling wave solutions.
In [I0], Evans showed that equation (1.1) has two pulse travelling solutions with different propagation speeds c and c 2. On the existence of periodic travelling wave solutions, Hastings [II] showed that equation (2.1) with G(u,w) eu has a non- constant periodic solution if e > 0 is sufficiently small and the speed c is limited to a certain range.
Rinzel and Keller [12] studied the case in which F(u,w) is a function of u only given by u for u < a, F(u,w) where 0 < a <I/2 Under this assumption, equation (2.1) has a non-constant periodic solution if c is limited in the range c < c < c 2 and the period p(c) is a smooth function of c.
They demonstrated the behavior of the function p(c) under the two cases when a is not very small and when a is very small.Dai [13] proved the existence and uniqueness of solutions for a general case and studied stability of the solution.
Stability of periodic travelling wave solutions is related to the eigenvalues of a matrix in the following theorem.Let A(z;%,c) be the matrix where F. and G. denote the partial derivatives as usual, and let X (z;,c) be a F(0,0) 0, (b) G(0,0) 0 and (c) the matrix X (p(c);/,c) has an eigenvalue of modulus I, for some complex number k with Re k > 0, then a periodic travelling wave solution [#(z;c), (z;c)] is unstable.
PROOF.With the change of variables, The linearized perturbation equation of the above system with respect to the solution where (Yl' Y2 satisfies the following system of linear ordinary differential equations Ay d2y where #(z;c) and (z;c).Note that if equation (3.3) has a solution which is bounded for all z in (-,) for a number A with Re(A) > 0, then equation (3.2) has a solution [U(z,t), W(z,t)] which grows exponentially, and hence, the travelling wave solution [#(z;c), (z;c)] is unstable.Using Floquet's theory, we can show that equation ( 3.3) has a bounded non-trivial solution if and only if one of the eigenvalues of X( p( c) A ,c) is a modulus I. Equa- tion (3.3) can be rewritten as and the matrix A is as defined before.Now, since the coefficient matrix A (z;k,c) is a p(c)-periodic function of z, Floquet's theory yields that equation (3.3) has a bounded non-trivial solution if and oaly if one of the eigenvalues of the matrix X(p(c) ;k ,c) defined before is of modulus I.The proof is now complete.
In the following lemma, it is shown that under the special case 0, one eigenvalue of X(p(c);0,c) is unity and the product of the other two eigenvalues is greater than one.But det {X (p(c);O,c)} I (0,c) 2 (0,c) 3 (0 ,c) and I (0,c) I, hence 2 (0,c) 3 (0,c) > I.
Note that under the assumptions of Lemma 3.1, either [2 (%,c)[ > or [12 3 (k,c[ > for k sufficiently small.In the next theorem, we will see that if L(c) is decreasing, i.e.L'(c) < 0, then B1 (l,c) is increasing at O, i.e.
PROOF: We claim that the following equality (), ,c) [k=O -p' (c) Tf-a actually holds.
Differentiation of the above equation with respect to c leads to w (p(c);c) p'(c) + w (p(c);c) w (0;c).
On the other hand, under certain conditions, two eigenvalues have modulus less than one and one has modulus greater than one.
The vectors qi (X'c) are qi(k,c) s.
F and G in equation (I.I) staisfy(a) and so can be represented by the matrix differential equation d d---v A(z;A,c) __v,