Stability of radially symmetric travelling waves in reaction-diffusion equations
Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (3):
341--379 (2004)Abstract
The asymptotic behaviour as t goes to infinity of solutions u(x, t) of the multidimensional parabolic equation ut = u+F (u)
is studied in the "bistable" case. More precisely, we consider the stability of spherically symmetric travelling waves with respect
to small perturbations. First, we show that such waves are stable against spherically symmetric perturbations, and that the
perturbations decay like (log t)/t 2 as t goes to infinity. Next, we observe that this stability result cannot hold for arbitrary
(i.e., non-symmetric) perturbations. Indeed, we prove that there exist small perturbations such that the solution u(x, t) does not
converge to a spherically symmetric profile as t goes to infinity. More precisely, for any direction k S n-1 , the restriction of
u(x, t) to the ray x = kr | r 0 converges to a k-dependent translate of the one-dimensional travelling wave.