Exponentially growing solutions for a delay-diffusion equation with negative feedback

The scalar partial delay differential equation u_t - μΔu = u(t)(1 - u(t - τ)) is investigated. It was known that in one space dimension, all nonnegative solutions of the initial-boundary-value problem stay bounded as t → ∞. Surprisingly, this ceases to remain true in two or more dimensions: if the delay is large and the diffusion coefficient small, there exists a large set of trajectories whose total mass tends exponentially to infinity as t → ∞. These trajectories "escape" the negative feedback of the equation by transporting their mass around along periodic paths in space, where the paths can be prescribed by the initial data.