Consider the following one-dimensional reaction-diffusion equation: $$u_t= u_{xx}+au$$ on $\Omega=(0,1)$ with Dirichlet boundary conditions with $a>0$ and a nonnegative initial condition $u_0$. If $\frac{\sqrt a}{\pi}\notin\mathbb{N}$, the only steady state is the zero solution. Now if $\frac{\sqrt a}{\pi}\in\mathbb{N}$, then all the functions $u_0(x)=B\sin(\pi x)$ are steady state solutions because they satisfy the equation and boundary condition for every value of $B$.
My question is what is the asymptotic behavior of the solutions? Using Mathematica I conjecture that if $a<\pi^2$ then the solutions decay to $0$. If $a>\pi^2$ then the solutions grow to infinity. If $a=\pi^2$, then the solutions seem to converge to a positive steady state function.
