## Maths Biology - linear instability

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.
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