# Maths Biology - linear instability

by Auron87
Tags: biology, instability, linear, maths
 P: 12 1. The problem statement, all variables and given/known data Given the linear diffusion equation with a linear source term, $$\frac{\partial u}{\partial t} =\frac{\partial^2u}{\partial x^2} + au$$ where a is a positive constant and inital data u(x,0) = u0(x) use a linear instability analysis to show u identically equal to 0 is always unstable no matter how small a > 0. 2. Relevant equations 3. The attempt at a solution To be honest I really don't know where to start. We've seen a similar example without the 'au' term and I'm struggling to follow through that. We've been told that the differential equation should come to {sigma}*u = -k^2*u + au but I'm not sure how to get to there. From this I can tell that sigma could be positive or negative whereas in his example, sigma was always negative which meant that u identically equal to 0 was always stable. I'm genuinely unsure of where to go with this problem. Thanks for any help.