Math 19. Lecture 22Pattern Formation (II)T. JudsonFall 20051 StabilitySuppose that ue(x) is a solution toµd2uedx2+ f(ue) = 0subject to the boundary conditions. Let w(x) be a small perturbation ofue(x) at t = 0, and setu(0, x) = ue(x) + w(x)and move forward in time to obtain a solution to∂u∂t= µ∂2u∂x2+ f(u) (1)∂∂xu(t, 0) =∂∂xu(t, L) = 0. (2)that is equal to ue(x) + w(x) at t = 0If w(x) is small enough, then the resulting solution u(t, x) to ( 1) and (2)that has the property u(0, x) = ue(x) + w(x) has the property that at everyx, the values of u(t, x) → ue(x) as t → ∞.A solution is unstable if there is an arbitra r ily small (but not identicallyzero) perturbation w(x) such that u(t, x) does not approach ue(x) for at leastone x as t → ∞.12 Linear StabilityThis definition satisfies our intuition, but stability may be impossible toverify for a given f . We give a stronger condition for stability below, li nearstability.• Linear stability ⇒ Stability• Stability 6⇒ Linear stability• Linear stability guarantees stability against slight changes in the equa-tion not just slight changes in the starting function u(0, x).The definition of linear stability is somewhat technical, but it is more relevantin the r eal world.We first construct a new function z(x) from the function f and from theequilibrium solution ue(x) toµd2uedx2+ f(ue) = 0ddxue(0) =ddxue(L) = 0.Define z(x) byz(x) =dfduu=ue.For example, If f (u) = r1u − r2u2, where r1, r2> 0, thenz(x) = r1− 2r2ue(x).3 Linear Stability CriterionThe solution ue(x) is a stable solution toµd2uedx2+ f(ue) = 0ddxue(0) =ddxue(L) = 0if and only if there is no pair (g, λ) , where g( x) is some function that isnot identically zero for 0 ≤ x ≤ L, where λ ∈ R, and where the followingconstraints are satisfied.2• λ ≥ 0• λg = µd2gdx2+ z(x)g•dgdxx=0=dgdxx=L= 0A solution is unstable if there is even one such pair (g, λ) that obeys theabove conditions.4 An ExampleLet∂u∂t=∂2u∂x2+ 5u(2 − u)for all t and for 0 ≤ x ≤ 1. Assume that we have boundary conditions∂∂xu(t, 0) =∂∂xu(t, 1) = 0.The solutions that are independent of t and x are u = 0 and u = 2.To check stability, we ask whether there is a pair (g, λ), where λ ≥ 0 andg(x) 6≡ 0 and g is a solution toλg =d2gdx2+ f′(ue)g.Since f(u) = 5u(2 − u) and ue= 0 or ue= 2,f′(0) = 10,f′(2) = −10.Also, g must satisfydgdxx=0=dgdxx=1= 0.If such a (g, λ) exists, then the equilibrium solution ueis unstable.3Readings and References• C. Taub es. Modeling Differential Equations i n Biology. Prentice Hall,Upper Saddle River, NJ, 200 1. Chapter 18.• “D ynanamics of Stripe Formation,” pp. 280–282.• “A Reaction-Diffusion Wave on the Skin of the Marine Angelfish,” pp.282–286.• “Letters to Nature,” pp.
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