MONOSTABLE-TYPE TRAVELING WAVES OF BISTABLE REACTION-DIFFUSION EQUATIONS IN THE MULTI-DIMENSIONAL SPACE
MONOSTABLE-TYPE TRAVELING WAVES OF BISTABLE REACTION-DIFFUSION EQUATIONS IN THE MULTI-DIMENSIONAL SPACE
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发表时间：
2008
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通讯作者：
Y. Morita;H. Ninomiya
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作者：
Y. Morita;H. Ninomiya
我们讨论的是Rn+1中的反应-扩散方程ut=uu+uyy+f(U)，其中(x；y)=(x1；：：；xn；y)2Rn+1，其中(x；y)是Rn中的拉普拉斯算子。假设该方程具有双稳态非线性，即它有两个稳定的常解u=0；1，并且在它们之间有一个不稳定的常解。在不平衡条件R_(10)f(U)Du>；0的情况下，该方程允许平面行波连接两个常解和一个不稳定的驻留解v(X)>；0，其中v+f(V)=0，其中limjxj！然后，我们证明了存在一族行波u=U(x；z)；z=yict，它们在z=i1处连接u=1(或u=0)，在z=1处连接u=v(X)，速度属于半无限区间。利用比较原理，适当地构造了上解和下解，进行了证明。存在定理可以推广到更一般的反应扩散方程。
We are dealing with a reaction-diusion equation ut = ¢u+uyy+f(u) in R n+1 , where (x;y) = (x1;:::;xn;y) 2 R n+1 and ¢ is the Laplacian in R n . 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 R 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 limjxj!1 v(x) = 0. Then we show that there are a family of traveling waves u = U(x;z); z = yict connecting u = 1 (or u = 0) at z = i1 to u = v(x) at z = 1 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-diusion equation.