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

• Notes
• 2

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.
• Fall '13
• MarkHolmes
• Math, Equations, Instability, Stability theory, uniform steady-state

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern