ISSN:
1432-0673
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation ut—uxx—∞;(u)=O, x∈(—∞, ∞) , in the case ∞(0)=∞(1)=0, ∞′(0)〈0, ∞′(1)〈0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given. 1. Let u(x, 0)=u 0(x) satisfy 0≦u 0≦1. Let $$a\_ = \mathop {\lim \sup u0}\limits_{x \to - \infty } {\text{(}}x{\text{), }}\mathop {\lim \inf u0}\limits_{x \to \infty } {\text{(}}x{\text{)}}$$ . Then u approaches a translate of U uniformly in x and exponentially in time, if a− is not too far from 0, and a+ not too far from 1. 2. Suppose $$\int\limits_{\text{0}}^{\text{1}} {f{\text{(}}u{\text{)}}du} 〉 {\text{0}}$$ . If a − and a + are not too far from 0, but u0 exceeds a certain threshold level for a sufficiently large x-interval, then u approaches a pair of diverging travelling fronts. 3. Under certain circumstances, u approaches a “stacked” combination of wave fronts, with differing ranges.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00250432
Permalink
