MONOSTABLE-TYPE TRAVELING WAVES OF BISTABLE REACTION-DIFFUSION EQUATIONS IN THE MULTI-DIMENSIONAL SPACE BY YOSHIHISA MORITA AND HIROKAZU NINOMIYA Dedicated to Professor Mimura on the occasion of his 65th birthday
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چکیده
We are dealing with a reaction-diffusion equation ut = ∆u+uyy+f(u) in R , where (x, y) = (x1, . . . , xn, y) ∈ R n+1 and ∆ is the Laplacian in R. Suppose that the equation has a bistable nonlinearity, namely it has two stable constant solutions u = 0, 1 and an unstable one between those. With the unbalanced condition ∫ 1 0 f(u)du > 0 the equation allows planar traveling waves connecting two constant solutions and an unstable standing solution v(x) > 0 of ∆v+ f(v) = 0 with lim|x|→∞ v(x) = 0. Then we show that there are a family of traveling waves u = U(x, z), z = y − ct connecting u = 1 (or u = 0) at z = −∞ to u = v(x) at z = ∞ with speeds belonging to a half infinite interval. The proof is carried out by using the comparison principle and constructing a subsolution and a supersolution appropriately. The existence theorem can be extended to a more general reaction-diffusion equation.
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تاریخ انتشار 2008