We consider a reaction-diffusion equation ut = uxx + f (u), where f has exactly three zeros 0, α and 1 (0 < α < 1), fu(0) < 0, fu(1) < 0 andThen, the equation has a travelling wave solution u(x, t) = φ(x − ct) with φ(−∞) = 0 and φ(+∞) = 1. Known results suggest that for an initial state u0(x) with lim x→±∞ u0(x) > α having two interfaces at a large distance, u(x, t) approaches a pair of travelling wave solutions φ(x − p1(t)) + φ(−x + p2(t)) for a long time, and then the travelling fronts eventually disappear by colliding with each other. While our results establish this process, they show that there is a (backward) global solution ψ(x, t) and that the annihilation process is approximated by a solution ψ(x−x0, t−t0). §1. IntroductionIn this paper, we consider the scalar bistable reaction-diffusion equationwhere BU (R) is the space of bounded uniformly continuous functions from R to R with the supremum norm, and the reaction term f satisfies the following conditions:

We consider the nonlocal analogue of the Fisher-KPP equationwhere μ is a Borel-measure on R with μ(R) = 1 and f satisfies f (0) = f (1) = 0 and f > 0 in (0, 1). We do not assume that μ is absolutely continuous with respect to the Lebesgue measure. The equation may have a standing wave solution whose profile is a monotone but discontinuous function. We show that there is a constant c * such that it has a traveling wave solution with speed c when c ≥ c * while no traveling wave solution with speed c when c < c * , provided R y∈R e −λy dμ(y) < +∞ for some positive constant λ. In order to prove it, we modify a recursive method for abstract monotone discrete dynamical systems by Weinberger. We note that the monotone semiflow generated by the equation is not compact with respect to the compact-open topology. We also show that it has no traveling wave solution, provided f (0) > 0 and R y∈R e −λy dμ(y) = +∞ for all positive constants λ. §1. IntroductionIn 1930, Fisher [8] introduced the reaction-diffusion equation u t = u xx + u(1 − u) as a model for the spread of an advantageous form (allele) of a single gene in a population of diploid individuals. He [9] found that there is a constant c * such that the equation has a traveling wave solution with speed c when c ≥ c *

We consider traveling fronts to the nonlocal bistable equationwhere μ is a Borel-measure on R with μ(R) = 1 and f satisfies f (0) = f (1) = 0, f < 0 in (0, α) and f > 0 in (α, 1) for some constant α ∈ (0, 1). We do not assume that μ is absolutely continuous with respect to the Lebesgue measure. We show that there are a constant c and a monotone function φ with φ(−∞) = 0 and φ(+∞) = 1 such that u(t, x) := φ(x+ct) is a solution to the equation, provided f (α) > 0. In order to prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation.

We consider the blow-up problem of a semilinear heat equation,where is a bounded smooth domain in R N , T D > 0, D > 0, and p > 1. We study the blowup time, the location of the blow-up set, and the blow-up profile of the blow-up solution for sufficiently large D. In particular, we prove that, for almost all initial data , if D is sufficiently large, then the solution blows-up only near the maximum points of the orthogonal projection of the initial data from L 2 ( ) onto the second Neumann eigenspace.

No abstract