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Travelling waves for a nonlocal double-obstacle problem

Published online by Cambridge University Press:  01 December 1997

PAUL C. FIFE
Affiliation:
Mathematics Department, University of Utah, Salt Lake City, Utah 84112, USA

Abstract

Existence, uniqueness and regularity properties are established for monotone travelling waves of a convolution double-obstacle problem

ut =J*uuf (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(xct) 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.

Type
Research Article
Copyright
1997 Cambridge University Press

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