On Solutions of Evolution Equations with Proportional Time Delay

Transcription

1 On Solutions of Evolution Equations with Proportional Time Delay Weijiu Liu and John C. Clements Department of Mathematics and Statistics Dalhousie University Halifax, Nova Scotia, B3H 3J5, Canada Fax: ( and Abstract This work investigates reaction-diffusion equations of the form y t (x, t = κ y(x, t+ ay(x, λt σ which incorporate a proportional or general linear time delay λt σ where λ, σ, κ > and a is a possibly complex constant. We establish the existence of unique solutions on [, T ] for any T if < λ 1 and for < T < σ/(λ 1 if λ > 1 and σ >. It is shown that if < λ < 1, then there exists a constant µ ( such that these solutions do not grow faster than a polynomial of degree p = 1 ln λ ln κµ a and, moreover, if a < κµ, then p < and the solutions decay to zero at polynomial rate as t. Similar results are obtained for equations involving power time delay terms of the form ay(x, t λ. AMS Subject Classification. 35R1, 35B35, 35K5. Keywords. proportional delay, evolution equations, asymptotic behavior. Supported by the Killam Postdoctoral Fellowship. Supported by NSERC of Canada.

2 1 Introduction Let a be a possibly complex constant and b and λ real constants with λ. Functionaldifferential equations with proportional delay of the form y (t = ay(λt + by(t ( t < (1.1 have found application in the mathematical modeling of electric pantograph dynamics [9, 18] and have even arisen in a partitioning problem in number theory [17]. In [11], Kato and MacLeod showed that this equation leads to a well-posed problem if λ 1 and examined properties of the solutions as t. The dynamics and stability properties of solutions of a generalized form of (1.1 have been studied in [3, 7, 8, 1, 12, 21]. More recently, Brunner [4] has examined the attainable order of m-stage implicit (collocation based Runge-Kutta methods for solving such equations. What is now clear is that properties that hold for constant delay type equations (i.e. which incorporate terms of the form y(t σ do not necessarily carry over to even linear equations with proportional time delay and while constant time delays in reaction-diffusion systems have been studied extensively (see, e.g. [5, 6, 13, 14, 16, 19, 21, 23], there appears to have been little investigation into the corresponding problems involving proportional delay. The objective of this work is simply to employ some of the approaches adopted in [11] together with standard techniques for partial differential equations to determine whether the solution properties established for (1.1 will carry over to reaction-diffusion systems with general linear time delay y t (x, t = κ y(x, t + ay(x, λt σ, x, t >, (1.2 y(x, t =, x Γ, t >, (1.3 y(x, s = y (x, s, x, σ s. (1.4 Here is a nonempty bounded open set in R n with boundary Γ =, is the Laplacian operator, κ > denotes the diffusivity constant, λ >, σ are parameters and y (x, s is an initial state in an appropriate function space. The extension of linear delays θ(t = λt σ to more general expressions θ(t, introduces power delays of the form θ(t = t λ and so we also consider reaction-diffusion problems incorporating such terms y t (x, t = κ y(x, t + ay(x, t λ, x, t >, (1.5 y(x, t =, x Γ, t >, (1.6 y(x, = y (x, x. (1.7 (where the proportional term is in advance of time t for < t < 1. Theorem 2.1 in Section 2 establishes the existence of unique solutions of (1.2-(1.4 on [, T ] for any T if < λ 1 and for < T < σ/(λ 1 if λ > 1 and σ >. In Theorem 2.2 we show that if < λ < 1, then ( the solutions of (1.2-(1.4 do not grow faster than a polynomial of degree p = 1 ln κµ and, moreover, if a < κµ ln λ a, then p < and the solutions decay to zero at a polynomial rate as t. Theorems 3.1 and 3.2 in Section 3 establish the existence, uniqueness and polynomial growth of solutions of (1.5-(1.7 for any 2

3 T if < λ 1. A simple non-linear proportional time delay equation is considered in Section 4. The analysis of this simple example demonstrates that blowup behavior can be expected for such equations unless strong growth conditions are imposed on the nonlinear terms. 2 Evolution Equations with General Linear Delay Let be a bounded domain (nonempty, open, and connected set in R n with suitably smooth boundary Γ =. We denote by H s ( the usual Sobolev space for fixed s R (see [1, 15]. For s, H s ( denotes the completion of C ( in H s (, where C ( denotes the space of all infinitely differentiable functions on which have compact support in. The norm of L 2 ( is denoted by. Let X be a Banach space and a < b. We denote by C n ([a, b]; X the space of n times continuously differentiable functions defined on [a, b] with values in X and endowed with the usual supremum norm. For brevity, we write C ([a, b]; X simply as C([a, b]; X. Let A be the linear operator Aw = κ w (2.1 with the domain D(A = H 2 ( H 1 (. It is well known that A generates an analytic semigroup e At on L 2 (. Problem (1.2-(1.4 can be transformed into the following equivalent integral equation problem y(t = y (t σ t, (2.2 y(t = e At y + a t e A(t s y(λs σds t >, (2.3 where y(t = y(x, t, y (t = y (x, t. For any given T > and y C([ σ, ]; L 2 (, the solution y of (2.2-(2.3 with y C([, T ]; L 2 ( is called a mild solution of problem (1.2-(1.4. Using the standard bootstrap procedure for delay equations, it is straightforward to show that (1.2-(1.4 is well-posed. Theorem 2.1. Let κ >, σ and a be real numbers. (i If < λ 1, then for any given T > and initial condition y (x, s C([ σ, ]; L 2 (, problem (1.2-(1.4 has unique mild solution with y C([, T ]; L 2 (. (2.4 Moreover, if y (x, s C 1 ([ σ, ]; D(A, problem (1.2-(1.4 has unique classical solution with y C([, T ]; H 2 ( H 1 ( C 1 ([, T ]; L 2 (. (2.5 (ii If λ > 1 and σ >, The conclusions of (i also hold for < T < σ/(λ 1. Proof. Let {y n } (n =, 1, be the sequence defined by y (t if σ t, y (t = e At y ( + a t ea(t s y (λs σds if t σ/λ, e A(t σ/λ y (σ/λ if t > σ/λ; { y (t if σ t, y n (t = e At y ( + a t ea(t s y n 1 (λs σds if t, n = 1, 2,. 3 (2.6 (2.7

4 Then for t σ/λ and t y 1 (t = e At y ( + a = e At y ( + a In the same way, we deduce for t σ λ + σ λ 2 t e A(t s y (λs σds e A(t s y (λs σds = y (t, (2.8 y 1 (t = y (t, σ t σ/λ. (2.9 t y 2 (t = e At y ( + a = e At y ( + a t that e A(t s y 1 (λs σds e A(t s y (λs σds = y 1 (t, (2.1 and y 2 (t = y 1 (t, σ t σ λ + σ λ 2. (2.11 Repeating the above procedure, gives y n (t = y n 1 (t, σ t σ λ + σ λ + + σ, n = 1, 2,. ( λn We now consider the cases: σ > and σ =. Suppose σ >. Then, if λ 1 and T > (or λ > 1 and T < σ/(λ 1, we can take n so large that and σ/λ + σ/λ σ/λ n > T t y n (t = e At y ( + a = e At y ( + a t e A(t s y n 1 (λs σds e A(t s y n (λs σds. (2.13 Thus y n is a mild solution of problem (1.2-(1.4. If σ =, then we have y 1 (t y (t = = a a t t e A(t s y (λsds e A(t s+λs y (ds a K y ( t, (2.14 4

5 where It follows that y 2 (t y 1 (t = K = t a a K max e At. (2.15 t max{t,λt } e A(t s (y 1 (λs y (λsds t a 2 K 2 y ( y 1 (λs y (λs ds t λsds 1 2 a 2 K 2 λ y ( t 2, (2.16 y n (t y n 1 (t 1 n! a n K n λ n 1 y ( t n, n = 1, 2,..., (2.17 Hence, y n converges to y in C([, T ]; L 2 ( if < λ 1. So, again problem (1.2-(1.4 has mild solution y. Uniqueness is easily proved in the usual way and we omit the proof here. To establish the regularity of the solutions, we set z = y t. Then z satisfies (1.2 (1.4 with a replaced by aλ and the initial condition y (x, s replaced by z (x, s = y s (x, s C([ σ, ]; L2 (. Thus, z = y t C([, T ]; L 2 ( and by the elliptic regularity, y C([, T ]; H 2 ( H 1 ( C 1 ([, T ]; L 2 (. We now utilize some of the techniques from [11] to show that the solutions of (1.2-(1.4 do not grow faster than a polynomial. To this end, we let µ be the smallest eigenvalue of with Dirichlet boundary condition φ = µφ, in, (2.18 φ =, on Γ. (2.19 Theorem 2.2. Suppose that κ >, σ, < λ < 1 and a is a real number. Then there exist positive constants C = C(κ, a, σ, λ and µ such that the solution of (1.2-(1.4 satisfies where y(t C y C([ σ,];l 2 ((t + 1 p, t, (2.2 y t (t C y C 1 ([ σ,];l 2 ((t + 1 p 1, t. (2.21 p = 1 ( ln λ ln κµ. (2.22 a Furthermore, the exponent p given by (2.22 is optimal, that is, for any δ >, there exists a solution of (1.2-(1.4 such that (2.2 with p replaced by p δ does not hold. Proof. The basic idea here is to transform the proportional delay λt σ into the usual delay s + τ (τ <. Thus, we set t = e s σ ( 1 λ, τ = ln λ <, u(x, s = e ps y x, e s σ, ( λ 5

6 where p is given by (2.22. Since it follows that u satisfies y t (x, t = [u s (x, s + pu(x, s] exp(ps + s, (2.24 ( y(x, λt σ = y x, λe s λσ 1 λ σ ( = y x, e τ+s σ 1 λ = e p(τ+s u(x, τ + s, (2.25 u s (x, s = κe s u(x, s pu(x, s + ae pτ e s u(x, τ + s, x, < s <, (2.26 u(, s = u(1, s =, < s <. (2.27 Hence, to prove (2.2, it suffices to prove that there exists a positive constant C = C(κ, a, λ such that the solution u of (2.26-(2.27 satisfies Let u(s C y C([ σ,];l 2 (, To prove (2.28, it suffices to establish that and s ln σ 1 λ. (2.28 s = σ ln 1 λ, (2.29 M n = max u(s, s nτ s s nτ τ n =, 1,... (2.3 Integration by parts gives d u 2 (s dx = 2 u(su s (s dx ds = 2κe s u(s u(s dx 2p = 2κe s u(s 2 dx 2p which implies that d ( exp(2ps + 2κµ e s ds M n C(κ, a, λm, n = 1, 2,... (2.31 M C(κ, a, λ y C([ σ,];l 2 (. (2.32 u 2 (s dx 2ae pτ e s u 2 (s dx 2ae pτ e s u(su(s + τ dx u(su(s + τ dx ( note that µ u(x, t 2 u(t 2 for some µ > 2(p + κµ e s u 2 (s dx 2ae pτ e s u(su(s + τ dx, (2.33 u 2 (s dx 2a exp[pτ + (2p + 1s + 2κµ e s ] u(su(s + τ dx. (2.34 6

7 Integrating from s (n + 1τ to s, we obtain exp(2ps + 2κµ e s u 2 (s dx exp(2p(s (n + 1τ + 2κµ exp(s (n + 1τ u 2 (s (n + 1τ dx s 2a exp[pτ + (2p + 1t + 2κµ e t ] u(tu(t + τ dxdt s (n+1τ Mn 2 exp[2p(s (n + 1τ + 2κµ exp(s (n + 1τ] s + 2 a M n M n+1 exp[pτ + (2p + 1t + 2κµ e t ]dt. (2.35 On the other hand, s s (n+1τ e t s (n+1τ exp[(2p + 1t + 2κµ e t ]dt s = 2p + 2κµ e exp(2pt + 2κµ e t s exp(2pt + 2κµ t e t d ( e t dt s (n+1τ s (n+1τ dt 2p + 2κµ e t e t = 2p + 2κµ e exp(2pt + 2κµ e t s s exp(2pt + 2κµ t e t 2pe t s (n+1τ s (n+1τ (2p + 2κµ e t dt 2 e t 2p + 2κµ e exp(2pt + 2κµ e t s s + 2 p exp(2ps + 2κµ t e s e t s (n+1τ s (n+1τ (2p + 2κµ e t dt 2 e t = 2p + 2κµ e exp(2pt + 2κµ e t s p exp(2ps + 2κµ t e s 1 s s (n+1τ κµ 2p + 2κµ e t s (n+1τ ( = e s p exp(2ps + 2κµ e s p exp(2ps + 2κµ e s + κµ 2p + 2κµ e s κµ (2p + 2κµ exp(s (n + 1τ exp(s (n + 1τ 2p + 2κµ exp(s (n + 1τ exp[2p(s (n + 1τ + 2κµ exp(s (n + 1τ] es exp(2ps + 2κµ e s p exp(2ps + 2κµ e s + 2p + 2κµ e s κµ (2p + 2κµ exp(s (n + 1τ exp(s (n + 1τ 2p + 2κµ exp(s (n + 1τ exp[2p(s (n + 1τ + 2κµ exp(s (n + 1τ] (2.36 7

8 and it follows from (2.35 that exp(2ps + 2κµ e s u 2 (s dx M 2 n exp[2p(s (n + 1τ + 2κµ exp(s (n + 1τ] 2 a M nm n+1 exp(pτ + s (n + 1τ 2p + 2κµ exp(s (n + 1τ + 2 a e pτ e s exp(2ps + 2κµ e s M n M n+1 2p + 2κµ e s exp[2p(s (n + 1τ + 2κµ exp(s (n + 1τ] + 2 a e pτ M n M n+1 p exp(2ps + 2κµ e s κµ (2p + 2κµ exp(s (n + 1τ. (2.37 Let f(s = We now establish that 2 a e pτ e s 2p + 2κµ e s, (2.38 σ n = 1 f(s (n + 1τ + 2 ap e pτ κµ (2p + 2κµ exp(s (n + 1τ. (2.39 M n+1 (1 + σ n M n. (2.4 If M n+1 M n, then (2.4 holds. Therefore, we may as well assume that M n+1 > M n. Since the function f(s = 2 a epτ e s 2p + 2κµ e s is increasing on [,, it follows from (2.22 that From (2.37 that f(s a epτ κµ = 1. (2.41 u 2 (s dx M n M n+1 [1 f(s (n + 1τ] exp[2p(s (n + 1τ + 2κµ exp(s (n + 1τ] exp( 2ps 2κµ e s + 2 a epτ M n M n+1 e s 2 ap e pτ M n M n+1 + 2p + 2κµ e s κµ (2p + 2κµ exp(s (n + 1τ M n M n+1 [1 f(s (n + 1τ] + a epτ M n M n+1 ap e pτ M n M n+1 + κµ κµ (p + κµ exp(s (n + 1τ = M n M n+1 [1 f(s (n + 1τ] (by ( M n M n+1 + ap e pτ M n M n+1 κµ (p + κµ exp(s (n + 1τ, (2.42 8

9 which implies (2.4. Therefore, to prove (2.31, it suffices to prove that the product Π n=1 (1+ σ n is convergent or, equivalently, that the series n=1 ln(1 + σ n is convergent. This in turn is equivalent to showing that that the improper integral 1 ln (2 a epτ e τs p + κµ e + ap e pτ ds τs κµ (p + κµ e τs is convergent which follows from ln lim s = lim s ( 2 a epτ e τs p+κµ e τs + (1 + s 2 ( 1 d ln 2 a epτ e τs ds p+κµ + e τs ap e pτ κµ (p+κµ e τs d dx (1 + s2 1 ap e pτ κµ (p+κµ e τs ( pτ a e pτ e τs = lim s (p + κµ e τs + κµ τ ap e pτ e τs (1 + s κµ (p + κµ e τs 2 2s =. (2.43 It remains to prove (2.32. Since is equivalent to it suffices to prove that Set ln σ 1 λ s ln σ 1 λ ln λ t = e s σ 1 λ σ λ, max y(t C(κ, a, t σ/λ λ y C([ σ,];l 2 (. (2.44 ϕ(t = max s t y(s 2. Let C = C(κ, a, λ denote a generic positive constant that may vary from line to line. Multiplying (1.2 by y and integrating over by parts, we obtain for t 1 t d ds ( y(s 2 2 a y(sy(λs σ dx Then C y C([ σ,];l 2 ( + Cϕ(s. (2.45 y(t 1 2 (Ct y C([ σ,];l 2 ( + C (Ct y C([ σ,];l 2 ( + C 9 t1 t ϕ(sds ϕ(sds. (2.46

10 and it follows that ϕ(t (Ct + 1 y C([ σ,];l 2 ( + C t ϕ(sds. (2.47 By Gronwall s inequality (see, e.g. [22, p.9], we deduce that for t σ/λ ϕ(t C(κ, a, λ y C([ σ,];l 2 (. (2.48 To prove (2.21, we set z = y t. Then z satisfies (1.2 (1.4 with a replaced by aλ and the initial condition y (x, s replaced by z (x, s = y s (x, s C([ σ, ]; L2 (. Hence, we can apply (2.2 to z with p replaced by p 1 = 1 ( ln λ ln κµ = p 1 (2.49 aλ and obtain (2.21. To prove that p is optimal, we argue by contradiction. Suppose that there exists a δ > such that y(t C y C([ σ,];l 2 ((t + 1 p δ, t (2.5 for all solutions of (1.2-(1.4. Let φ be the eigenfunction of (2.18-(2.19 corresponding to the eigenvalue µ and ψ(t the solution of ψ t (t = κµ ψ(t + aψ(λt, (2.51 ψ( = 1. (2.52 One can easily verify that y = φ ψ is a solution (1.2-(1.4 with σ = and y = φ. It then follows from (2.5 that ψ(t C(t + 1 p δ, t (2.53 which, by Theorem 3 of [11], implies ψ. But this is impossible since ψ( = 1. Remark 2.1. If a < κµ, then p < (note that ln λ <. (1.2-(1.4 decays to zero at a polynomial rate as t. Therefore, the solution of Remark 2.2. Let us compare the case λ = 1 with the case < λ < 1. For simplicity, we assume that σ =. If λ = 1 and a > κµ, problem (1.2-(1.4 has an exponentially growing solution y = φ exp((a κµ t, where φ is the eigenfunction of (2.18-(2.19 corresponding to µ. On the other hand, if < λ < 1, Theorem 2.2 shows that all solutions grow at most polynomially. This means that the proportionally delayed term y(x, λt has a dissipative effect in the case a > κµ. For a < κµ, if λ = 1, all solutions of (1.2-(1.4 satisfies y(t y exp((a κµ t. while the solutions decay probably only polynomially if < λ < 1. This shows that the proportionally delayed term y(x, λt has an anti-dissipative effect in the case a < κµ. 1

11 3 Evolution Equations with Power Time Delays We now consider the linear equation with power time delay y t (x, t = κ y(x, t + ay(x, t λ, x, t >, (3.1 y(x, t =, x Γ, t >, (3.2 y(x, = y (x, x. (3.3 Note that the proportional term is in advance of time t for < t < 1. This problem could be transformed into a standard singular problem with the change of variable τ = t λ, but to conform with our approach to the general linear delay case, we transform it here into the following integral equation t y(t = e At y + a e A(t s y(s λ ds, t >, (3.4 where A is the operator defined by (2.1. For any given T > and y L 2 (, the solution y of this integral equation with y C([, T ]; L 2 ( is called a mild solution of problem (3.1-(3.3. Theorem 3.1. Suppose that κ >, < λ 1 and a is a real number. Then for any given T > and initial condition y (x L 2 (, problem (3.1-(3.3 has unique mild solution with Moreover, if y (x, s D(A, then y is a classical solution with y C([, T ]; L 2 (. (3.5 y C([, T ]; H 2 ( H 1 ( C 1 ([, T ]; L 2 (. (3.6 Proof. We define a successive sequence {y n } (n =, 1, by We then have where y (t = e At y, (3.7 y n (t = e At y + a t y 1 (t y (t = K = e A(t s y n 1 (s λ ds, n = 1, 2,. (3.8 = a a t t e A(t s y (s λ ds e A(t s+sλ y ds a K y t, (3.9 max e At. (3.1 t max{t,t λ } 11

12 It therefore follows that y 2 (t y 1 (t = = t a a K e A(t s (y 1 (s λ y (s λ ds t a 2 K 2 y y 1 (s λ y (s λ ds t s λ ds 1 λ + 1 a 2 K 2 y t λ+1, (3.11 y n (t y n 1 (t a n K n y t 1+λ+ +λn 1 n 1 i=1 (1 + λ + +, n = 3, 4,..., (3.12 λi Hence, y n converges to y in C([, T ]; L 2 ( since 1 + λ + + λ n 1 n for < λ 1. So, problem (3.1-(3.3 has unique mild solution y. The uniqueness can be easily proved in the usual way and we omit it. To prove the regularity of solutions, we set z = y t. Then z satisfies (3.1-(3.3 with a replaced by aλt λ 1 and the initial condition y (x replaced by z (x = κ y (x + ay (x L 2 (. We can readily check that the above arguments are also true if a is replaced by aλt λ 1. Therefore, we have z = y t C([, T ]; L 2 ( and then, by the elliptic regularity, we deduce that y C([, T ]; H 2 ( H 1 ( C 1 ([, T ]; L 2 (. Using an approach similar to that in Theorem 2.2, we derive the following result regarding the asymptotic behavior of solutions of (3.1-(3.3. Theorem 3.2. Suppose that κ >, < λ < 1 and a is a real number. Let p > 1 λ(1 λ. (3.13 Then there exists a positive constant C = C(κ, a, λ such that the solution of (3.1-(3.3 satisfies y(t C y (t + 1 p, t (3.14 Proof. The proof is similar to that of Theorem 2.2. Set ( t = e es, τ = ln λ <, u(x, s = e pes y x, e es. (3.15 Since y t (x, t = (u s (x, s + pe s u(x, s exp((p 1e s s, (3.16 y(x, t λ = y (x, e λes = e pλes u(x, τ + s, (

13 it follows that u satisfies u s (x, s = κ exp(s + e s u(x, s pe s u(x, s + a exp[s + e s + (λ 1pe s ]u(x, τ + s (3.18 x, < s <, u(, s = u(1, s = < s <. (3.19 As in the proof of Theorem 2.2, to prove (3.14, it suffices to prove that the solution u of (3.18-(3.19 is bounded. Set M n = max u(s, n =, 1,... (3.2 nτ s nτ τ It suffices to prove M n is bounded. Integrating by parts, we obtain d u 2 (s dx = 2 u(su s (s dx ds = 2κ exp(s + e s u(s u(s dx 2pe s u 2 (s dx 2a exp[s + e s + (λ 1pe s ] u(su(s + τ dx = 2κ exp(s + e s u(s 2 dx 2pe s u 2 (s dx 2a exp[s + e s + (λ 1pe s ] u(su(s + τ dx ( again with µ u(x, t 2 u(t 2 2[pe s + κµ exp(s + e s ] u(s 2 dx 2a exp[s + e s + (λ 1pe s ] u(su(s + τ dx, (3.21 which implies that d ( exp(2pe s + 2κµ exp(e s u 2 (s dx ds 2a exp[2pe s + 2κµ exp(e s + s + e s + (λ 1pe s ] Integrating (3.22 from (n + 1τ to s, we obtain exp(2pe t + 2κµ exp(e t u 2 (t dx s (n+1τ s 2a exp[2pe t + 2κµ exp(e t + t + e t + (λ 1pe t ] (n+1τ u(su(s + τ dx. (3.22 u(tu(t + τ dxdt 2τ a M n M n+1 exp[2pe s + 2κµ exp(e s + s + e s + (λ 1pe (n+1τ ], (3.23 which implies that M 2 n+1 M 2 n 2τ a M nm n+1 exp[ (n + 2τ + e (n+2τ + (λ 1pe (n+1τ ]. (

14 Set We then have δ = e τ + (λ 1p < (by (3.13. (3.25 M n+1 [1 + 2 τ a e τ exp[ (n + 1τ + δe (n+1τ ]]M n. (3.26 If M n+1 M n, then (3.26 clearly holds. If M n+1 > M n, then (3.26 follows from (3.24. Therefore, to prove that M n is bounded, it suffices to prove that the product Π n=1[1 + 2 τ a e τ exp[ (n + 1τ + δe (n+1τ ]] is convergent, or equivalently, to show that the improper integral ( ln τ a e τ exp[ τs + δe τs ] ds 1 is convergent. But this follows as in the proof of Theorem Nonlinear Equations We now consider the nonlinear proportional delay boundary value problem y t (x, t = κy xx (x, t + ay(x, t[1 y(x, λt], < x < 1, t >, (4.1 y(, t = y(1, t =, t >, (4.2 y(x, = y (x, < x < 1. (4.3 Again, the problem can be transformed into the following integral equation y(t = e At y + a t e A(t s y(s[1 y(λs]ds, t >, (4.4 where A is the operator defined by (2.1. The existence and uniqueness of a local solution of (4.1-(4.3 can be readily established using the Banach contraction fixed point theorem, but a determination of the of asymptotic behavior of such solutions presents some serious difficulties. Indeed, ( it may not have a global solution as the following simple example demonstrates. Let y (x = 1 sin(5πx. For a = 5, it can be seen from Figure 4.1 that the energy E(t = 1 y 2 (x, tdx, of the solution without delay (i.e. λ = 1 decays rapidly to zero while the energy of the solution with delay (i.e. λ =.5 decreases initially and then increases rapidly after t = 2.5 and may blow up within finite time. The proofs of Theorems 2.1, 2.2, 3.1 and 3.2 provide the foundation for applying standard bootstrap methods (cf [2] to questions of existence, uniqueness and properties of solutions of semilinear (and more general proportional delay equations of the form y t (x, t = κ y(x, t + f(t, y(x, t, y(x, θ(t, x, t >. and this will be the focus of a future work. 14

15 E.5 a=5, lambda=1 E 3 a=5, lambda= t-axis t-axis Figure 4.1: Energy curves of solutions of the equation y t (x, t = y xx (x, t+ay(x, t[1 y(x, λt] with the initial condition y (x = 1 sin(5πx. References [1] R. Adams, Sobolev Spaces, Academic Press, New York, [2] W.F. Ames, Numerical Methods for Partial Differential Equations, Second edition, Academic Press, New York, [3] H. Brunner, The numerical solution of neutral Volterra integro-differential equations with delay arguments, Ann. Numer. Math., 1 ( [4] H. Brunner, On the discretization of differential and Volterra integral equations with variable delay, BIT, 37;1 ( [5] G. Friesecke, Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynamics and Differential Equations 5 (1993, [6] G. Friesecke, Exponentially growing solutions for a delay-diffusion equation with negative feedback, J. Differential Equations 98 (1992, no. 1,

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