Propagation of Reactions in Inhomogeneous Media

Abstract

Abstract. Consider reaction-diffusion equation u t = ∆u + f (x, u) with x ∈ R d and general inhomogeneous ignition reaction f ≥ 0 vanishing at u = 0, 1. Typical solutions 0 ≤ u ≤ 1 transition from 0 to 1 as time progresses, and we study them in the region where this transition occurs. Under fairly general qualitative hypotheses on f we show that in dimensions d ≤ 3, the Hausdorff distance of the super-level sets {u ≥ ε} and {u ≥ 1 − ε} remains uniformly bounded in time for each ε ∈ (0, 1). Thus, u remains uniformly in time close to the characteristic function of {u ≥ 1 2 } in the sense of Hausdorff distance of super-level sets. We also show that {u ≥ 1 2 } expands with average speed (over any long enough time interval) between the two spreading speeds corresponding to any x-independent lower and upper bounds on f . On the other hand, these results turn out to be false in dimensions d ≥ 4, at least without further quantitative hypotheses on f . The proof for d ≤ 3 is based on showing that as the solution propagates, small values of u cannot escape far ahead of values close to 1. The proof for d ≥ 4 involves construction of a counter-example for which this fails.Such results were before known for d = 1 but are new for general non-periodic media in dimensions d ≥ 2 (some are also new for homogeneous and periodic media). They extend in a somewhat weaker sense to monostable, bistable, and mixed reaction types, as well as to transitions between general equilibria u − < u + of the PDE, and to solutions not necessarily satisfying u − ≤ u ≤ u + .

Large time monotonicity of solutions of reaction–diffusion equations in RN

“…Other time-monotonicity results have been obtained in [3] for time-global transition fronts of space-heterogeneous reaction-diffusion equations of the type (1.1) connecting two stable limiting points. In [19], the time-monotonicity of the solutions u of equations u t = ∆u + f (x, u) with reactions f of the ignition type or involving a weak Allee effect has been established for large times in the set where 0 < ε ≤ u(t, x) ≤ 1 − ε < 1, for any ε > 0 small enough. Lastly, we refer to [7] for some results on time-monotonicity for small t and large x for the solutions of the homogeneous equation u t = ∆u + g(u) which are initially compactly supported.…”

On the mean speed of bistable transition fronts in unbounded domains

“…Lastly, even in the homogeneous space R N , non-standard transition fronts which are not invariant in any moving frame were also constructed in [24] under assumptions (1.2)-(1.5). More generally speaking, there is now a large literature devoted to transition fronts for bistable reactions in homogeneous or heterogeneous settings [6,13,18,22,48,53,60], as well as for other types of homogeneous or space/time dependent reactions in dimension 1 [16,29,30,34,35,37,40,42,43,51,52,57,58] and in higher dimensions [1,9,38,39,49,50,59,61].…”

Symmetrization and anti-symmetrization in parabolic equations

“…This is left as an open question. A positive answer is given in [11] when f is of combustion-type and is allowed to have some x-dependence, but not t-dependence, in dimension N ≤ 3, together with a counter-example in dimension N > 3. Let us mention that the interface {θ ′ < u < θ} may not have bounded width if the initial datum does not decay sufficiently fast at infinity, see [9,Theorem 8.4].…”