by integration with respect to t and putting f u(x, z)dr=w(x, t),

Transcription

1 76 [Vol. 39, 160. The Asymptotic Behaviour o f the Solution o f a Semi.linear Partial Differential Equation Related to an Active Pulse Transmission Line By Masaya YAMAGUTI (Comm. by Kinjiro KUNUGI, M.J.A., Dec. 1, 1963) 1. Introduction. J. Nagumo [1] proposed as active pulse transmission line simulating an animal nerve axon. The equation of propagation of his line is the following: (1) a ate u _ axat a3u -~(1-~]-EU) au at --G p>0, x>0 s>0, t>0 with the boundary data; (u(x,0)=0 (x> 0) () ut(x, 0) = 0 (x>0) u(0, t)=(t) fi(t>0), i(t) - 0 for t > to. In this note, we consider some asymptotic behaviours of the solution for the equation of related type with the same boundary data: Our equation is the following: 3 au a = a3u-- f' ()--g(u). u au t ax at at At first, we remark that the existence of global solutions for this problem (3) with boundary data () where fi(t) E C is assumed was completely proved by R. Arima and Y. Hasegawa [] under the conditions: -K1< f'(u)<k0(u+1), g(u) I <-K(u+I u I ), (4) G(u) `- f-g(z)}dz<k3u, g(u), f '(u) E C 1. Throughout this paper, we always assume that f'(u), g(u) satisfy this condition (4). Our results are divided into two parts. The one is the case g(u)=u, the other is the case g(u) = 0. For the first case, we can prove that any solution u(x, t) tends uniformly to zero, when t tends to + oo, under the additional condition (5), which corresponds to the limitation s> 16 3 in (1). For the second case we can show the existence of a threshold value for the boundary data (Prop. 3) and a sort of asymptotic value under another additional conditions (Prop. 4), (9), (11), which is independent of (5). We remark also that the summability in x, of u(x, t) and u~(x, t),

2 No. 10] Asymptotic Behaviour of Solution of Partial Differential Equation 77 which is shown in [], will play an important role in our proofs.. The first case. We assume that g(u)_u and we also assume the following condition (5) (which will be imposed only in this section). There exists a positive constant c such that (5) u f (u) > cu where f (u) = f u f '(z)dz. Then we have PROPOSITION 1. For arbitrary given data J(t) e C appeared in (), the solution of (3) tends uniformly to zero when t tends to + under the condition (5). PROOF. We transform the equation (3) to a system of equations by integration with respect to t and putting f u(x, z)dr=w(x, t), 0 (6) ut uxx-f (u)-w wt=u. We use the following energy form to obtain an energy inequality, here we denote by F(u) the primitive function of f (u) taking F(0) = 0, 7 E t = u ]- u ]-k ux ]-kf u ]-k(u+w)-[l u+w dx, 0 where k > Kl and L is a positive constant so large that it satisfies k < c'< c. Differentiating (7) with respect to t, and by the integration by parts, we obtain (8) E'(t)-f [uxt+(f'(u)+k)u~+(k+l)ux+(k+l)uf (u)--ku ] dx <0 for t>t i where t0 is a constant such that fi(t) 0 in 0 < t < to but p(t) - 0 in t>t. We can conclude from the differential inequality (8), following facts are finite: a) E(t) is non-increasing in t for t > t0. b) 0 < E(t) < E(t ). Consequently lim E(t) exists. c) Integrals: f 00 u _ f ~ dx, f -ut -dx, w dx, uxdx are bounded for t > t d) Integrals: u dxd 0o co z, JJ' ut -- dxdr, J'J' co 00 -ux dxdz, co o~ uxt dxdz. ~o J'J o - If we put (t)=f[-_+---jdx u ux, we can show cp(t)--~0(t---+ o). Be- cause, eo to fa

3 7$ M. YAMAGUTI [vol. 39, f t ~ t to t~ f ~(t) --~(t') I _ ~'(z) dz = [uut + uxuxt] dx dz t o0 1 t o0 1 t eo 1 t o0 1 c udxdz u dxdz + t~ o to ~~ o u dx dr to uxtdx dz By d) above, we can find a constant T for arbitrary given > 0, such that 1 c(t) - co(t') < E for t, t' > T. Then lira co(t) exist and by the t-+oo summability we see lira ~o(t) = 0. Consequently we can prove that max OSx<+o u(x, t) ~-~0 for t-~+ 0, by the Sobolev's lemma. 3. The second case (g(u) = 0). At first we mention some additional conditions for this case: (9 ) f f(u)<0 u >0 (u<0), 0 u a f(u)<0 u >0 (a<u<b), b>u here a and b are two distinct constants. Under this condition (9), we can prove a generalized maximumminimum principle for the solution of (3) for g(u) = 0. That is PROPOSITION. Under the condition (9), if Ba is a constant greater than b, then u(x, t0) < Ba implies always u(x, t) < Ba for t > t0, and u(x, t0)> 0 implies u(x, t) > 0 for t > t0. PROOF. If there is a point (x1, t1) where u(x1, t1) = Bo, then we can consider the set E of (x, t) t > to, 0 < x < + oo such that u(x, t) = B0. We can prove by the contradiction that there is a positive distance o > 0 between the set E and the half straight line t = to 0 < x < +. If not, there should be a sequence of points (fin, zn) which tends to one point of this half line or + 0 point of this half line, it signifies that there exists a point (, to) where u(e, to) = Bo or a sequence of points (fin, to) (n-* + oo) where u(, to) > LP_, by the fact that ut(x, t) is bounded for 0 < t < T. (T is some constant >_ to.) This latter case contradicts to the fact that u(x, to) and ux(x, t0) are summable in 0<x< + oo []. Therefore we find a point (x1, t1) where u(x1, t1) = Bo, u(x, t) < Bo (t0 < t < t1, 0 < x < + oo) and u(x, t1) < B (x < x1). Because u(x, t) is a solution of (), (3), ut = uxx - f (u), u(x1, t1) < 0 and f (Bo) = f (u(x1, t1)). Consequently u(x1, t1) <0, this means that there exists a point (x1, t) (t < t1) such that u(x1, t) > u(x1, t1) = Bo. This is a contradiction. The same argument shows that u(x, to) >_ 0 implies u(x, t) >_ 0 for t > t0. We add still one additional assumption: (10) There exist two positive constants c1 and a1 such that of (u) > c1(u + F(u)) for 0 < u < a1 < a. PROPOSITION 3. Under the assumptions (9) and (10), if 0 < u(x, to) <a1f u(x, t) tends exponentially to zero when t tends to + 0, in the maximum norm. Before entering into the proof, we remark that the

4 No. 10] Asymptotic Behaviour of Solution of Partial Differential Equation 79 same discussion in Proposition shows that if u(x, to) < a1i u(x, t) remains always less than a1 for t > to, PROOF. We consider an energy form E1(t) = u+ u x +F(u) dx, 0 Differentiating with respect to t, we have El (t) _ -- [ux + of (u) + u] dx < -- [ux+c1(u+f(u))] dx< -ce1(t), c>0. 0 Consequently, we obtain E1(t) E1(t0)e-et. By the Sobolev's lemma, we conclude that u(x, t) tends exponentially in the sense of uniform maximum norm. Finally we assume the additional condition: (f'(u) > 0 (u< a), f(u) > cof(u) (u< 0) (11) f'(u)<0 (a<u</9), Lf'(u)>O ($<u), where 0 < a < < b < B. We denote B the point such that F(B) = 0, F(u)>0 for u>b. PROPOSITION 4. Denoting B1 a constant greater than B, and Mt the set of x, (0 < x < + oo) such that u(x, t) > B1, then the measure of the set Mt tends exponentially to zero when t tends to + under conditions (9) and (11). Moreover under same condition, the integral fu(x, t) dx tends also exponentially to 0 when t tends to +00, Mt PROOF. We can construct a times differentiable function ~(u) as follows: (u) = F(u) u < 0 (u)-0 0<u<B (b<b<b) o(u) =X(u) u > B where X(u) satisfies following conditions: X'(u) > 1 X(u), X(u) >0, X"(u) > 0 for u > B (1) X'(B f (B) )=X(B)=0 there exists a positive constant c3 such that X(u) > c3u for u > B1. In fact, taking ~D(u) such that satisfies cp(u), cp'(u), cp''(u) > 0 for u > B cp'(b) = cp(b) = 0, and cp(u) > e4u for u > B1, and setting X(u) = e fcb3 cp(u) we see X(u) satisfies (1). This means that ~(u) satisfies always W(u)f(u) > coo(u). Now we use the following a new energy form

5 730 M. YAMAGUTI [Vol. 39, (13) E(t) = D(u)dx 0 (the summability of D(u) is evident by the fact that u and ux are summable). Differentiating with respect to t, we have E(t)=f ~'(u)utdx = - "(u)uxdx- o'(u) f (u)dx. 0 0 By the condition (1) and remarking that f(u)>f(b) for u > B, we have E'(t)< -coe(t). That is E(t) < E(0)e- t. It follows that c4 udx < fo(u)dx< E(0)e-~~t. Mt 0 Relating Proposition 3, we mention a remark. REMARK. Considering an another energy form which is not positive definite, we can show the existence of a solution of (3) in the case g(u) - 0, which does not tends uniformly to zero when t tends to + 00 In fact, taking E3(t) = f [-ux + F(u) dx, 0 if u(x, to) satisfies ux[(x, to) + F u x t dx 0 u x t 0)>0, 0 then u(x, t) does not tend to zero uniformly; because, if not, for t > T1 (sufficiently large) u(x, t) I <~ for given < a1 < a, then by the same discussion in Proposition, we have 0 < u(x, t) < a1. It signifies that E3(t)=f ux +F(u) dx>0 by the condition (9). - 0 That is a contradiction because we can show always that E3 (t) = - uzdx < 0 (t> to). References [1] J. Nagumo, S. Arimoto, and S. Yoshizawa: An active pulse transmission lines simulating nerve axon, Proceedings of the IRE, 50 (10), (196). [] R. Arima, and Yojiro Hasegawa: On global solutions for mixed problem of a semilinear differential equation, Proc Japan Acad., 39, (1963).