Travelling waves for a nonlocal double-obstacle problem

Abstract

Existence, uniqueness and regularity properties are established for monotone travelling waves of a convolution double-obstacle problemut =J*u−u−f (u),the solution u(x, t) being restricted to taking values in the interval [−1, 1]. When u=±1, the equation becomes an inequality. Here the kernel J of the convolution is nonnegative with unit integral and f satisfies f(−1)>0>f(1). This is an extension of the theory in Bates et al. (1997), which deals with this same equation, without the constraint, when f is bistable. Among many other things, it is found that the travelling wave profile u(x−ct) is always ±1 for sufficiently large positive or negative values of its argument, and a necessary and sufficient condition is given for it to be piecewise constant, jumping from −1 to 1 at a single point.