Math 19. Lecture 24Stability Criterion (II)T. JudsonFall 20051 Linear Stability CriterionLet uebe an equilibrium solution to∂u∂t= µ∂2u∂x2+ f(u) (1)∂∂xu(t, 0) =∂∂xu(t, L) = 0. (2)The solution ue(x) is a stable solution toµd2uedx2+ f(ue) = 0 (3)ddxue(0) =ddxue(L) = 0 (4)if 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.• λ ≥ 0• λg = µd2dx2+ z(x)g•dgdxx=0=dgdxx=L= 0A solution is unstable if there is even one such pair (g, λ) that obeys theabove conditions.12 Important Remarks about Stability• For a specific ueand f (u) (hence z(x)), we may or may not b e able tofind such g and λ.• If w is a solution to∂w∂t= µ∂2w∂x2+ z(x)w (5)∂∂xw(t, 0) =∂∂xw(t, L) = 0. (6)and if |w| is very small at all points x, then the function of space andtime, ue(x) + w(t, x), is an approximate solution to (1) and (2).• Conversely, if u(t, x) = ue(x) + w(t, x) is a solution to (1) and (2) forsmall |w|, then w will be an approximate solution for (5) and (6).• If w solution to (5) and (6) such that |w | is very small to begin withfor all x but grows as t → ∞ for some x, then (1) and (2) will have asolution that is close to ue(x) to start with but departs from ue(x) ast → ∞. This solution can be approximated by u(t, x) = ue(x)+w(t, x).Conversely, if (1) and (2) have a solution of the form u( t, x) = ue(x) +w(t, x) that starts at t = 0 for small |w| at all x, then (5) and (6) willhave a solution that starts small and grows with time. This solutioncan be approximated by w when t is small.• If all solutions w to (5) and (6) shrink in absolute value as t → ∞, thenall solutions to (1) and (2) that start near enough to the equilibriumsolution ue(x) at t = 0 will approach ue(x) a t all x as t → ∞.3 Boundary ConditionsThe boundary condtions must match. Let uebe a n equilibrium solution to∂u∂t= µ∂2u∂x2+ f(u) (7)u(t, 0) = u(t, L) = 0. (8)2The solution ue(x) is a stable solution toµd2uedx2+ f(ue) = 0ue(0) = ue(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.• λ ≥ 0• λg = µd2dx2+ z(x)g•dgdxx=0=dgdxx=L= 0A solution is unstable if there is even one such pair (g, λ) that obeys theabove conditions.Readings and References• C. Taubes. Modeling Differential Equations i n Biology. Prentice Hall,Upper Saddle River, NJ, 2001. Chapter 19.• “Direct and Continuous Assessment by Cells of Their Position in aMorphogen Gradient,” pp. 296–300.• “Activin Signalling and Response to a Morphogen Gradient,” pp.
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