Article,
Entire Solutions with Merging Fronts to Reaction–Diffusion Equations
Journal of Dynamics and Differential Equations, 18 (4): 841--861 (October 2006)
Abstract
We deal with a reaction–diffusion equation u t = u xx + f(u) which has two stable constant equilibria, u = 0, 1 and a monotone increasing traveling front solution u = φ(x + ct) (c > 0) connecting those equilibria. Suppose that u = a (0 < a < 1) is an unstable equilibrium and that the equation allows monotone increasing traveling front solutions u = ψ1(x + c 1 t) (c 1 < 0) and ψ2(x + c 2 t) (c 2 > 0) connecting u = 0 with u = a and u = a with u = 1, respectively. We call by an entire solution a classical solution which is defined for all $$(x, t) R^2$$. We prove that there exists an entire solution such that for t≈ − ∞ it behaves as two fronts ψ1(x + c 1 t) and ψ2(x + c 2 t) on the left and right x-axes, respectively, while it converges to φ(x + ct) as t→∞. In addition, if c > − c 1, we show the existence of an entire solution which behaves as ψ1( − x + c 1 t) in $$xın(-ınfty, (c_1+c)t/2$$ and φ(x + ct) in $$xın(c_1+c)t/2,ınfty)$$ for t≈ − ∞.
