Convergence and sharp thresholds for propagation in nonlinear diffusion problems

Yihong Du, Hiroshi Matano
2010 Journal of the European Mathematical Society (Print)  
We study the Cauchy problem where f (u) is a locally Lipschitz continuous function satisfying f (0) = 0. We show that any nonnegative bounded solution with compactly supported initial data converges to a stationary solution as t → ∞. Moreover, the limit is either a constant or a symmetrically decreasing stationary solution. We also consider the special case where f is a bistable nonlinearity and the case where f is a combustion type nonlinearity. Examining the behavior of a parameter-dependent
more » ... arameter-dependent solution u λ , we show the existence of a sharp threshold between extinction (i.e., convergence to 0) and propagation (i.e., convergence to 1). The result holds even if f has a jumping discontinuity at u = 1.
doi:10.4171/jems/198 fatcat:6yy7m7uqjrclvn5rgvebny7xna