summary3 - Summary for Stability for Reaction-Diusion...

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Summary for Stability for Reaction-Diffusion Problems One Equation ∂c ∂t = D 2 c ∂x 2 + f ( c ) Uniform Steady-state : f c ) = 0 Stability : A steady-state is asymptotically stable if f 0 c ) < 0 and it is unstable if f 0 c ) > 0 Two Equations ∂u ∂t = D 1 2 u ∂x 2 + f ( u, v ) ∂v ∂t = D 2 2 v ∂x 2 + g ( u, v ) Uniform Steady-state : f u, ¯ v ) = 0, g u, ¯ v ) = 0 The Jacobian J is J = ∂f ∂u ∂f ∂v ∂g ∂u ∂g ∂v Also, let Q = D 2 ∂f ∂u + D 1 ∂g ∂v In the theorem below, J and Q are evaluated at (¯ u, ¯ v ). Theorem : Assuming that D 1 and D 2 are positive, then a uniform steady-state is asymp- totically stable if: 1. tr( J ) < 0 and 2. One of the following hold: a) Q 0 and det( J ) > 0 or b) Q > 0 and 4 D
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