Asymptotic behavior of solutions for the damped wave equation with slowly decaying data
Asymptotic behavior of solutions for the damped wave equation with slowly decaying data
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DOI：
10.1016/j.jmaa.2007.05.068
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发表时间：
2008-02
影响因子：
1.3
通讯作者：
T. Narazaki;K. Nishihara
中科院分区：
文献类型：
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作者：
T. Narazaki;K. Nishihara
如果数据量很小且衰减缓慢，可能为c1(1+|x|)−kN, 0<k≤1，则半线性热方程的临界指数为ρc(k)=1+2kN。本文证明了在超临界情况下，任意维空间RN上的半线性热方程的Cauchy问题存在一个唯一的时间全局解，其渐近曲线由以下条件给出：假设数据ϕ0满足lim|x|→∞< x > kNϕ0(x)=c1（≠0）。即使在超临界情况下的半线性阻尼波动方程中，在低维空间RN， N=1,2,3中，当lim|x|→∞< x > kN(u0+u1)(x)=c1（≠0）时，具有数据（u,ut）(0,x)=(u0, x)的时间全局解u具有相同的渐近剖面Φ0（t,x）。这些证明是通过对显式解公式的初等估计给出的。
We consider the Cauchy problem for the damped wave equation and the heat equation If the data is small and slowly decays likely c1(1+|x|)−kN, 0<k⩽1, then the critical exponent is ρc(k)=1+2kN for the semilinear heat equation. In this paper it is shown that in the supercritical case there exists a unique time global solution to the Cauchy problem for the semilinear heat equation in any dimensional space RN, whose asymptotic profile is given by provided that the data ϕ0satisfies lim|x|→∞〈x〉kNϕ0(x)=c1(≠0). Even in the semilinear damped wave equation in the supercritical case a time global solution u with the data (u,ut)(0,x)=(u0,u1)(x) is shown in low dimensional spaces RN, N=1,2,3, to have the same asymptotic profile Φ0(t,x) provided that lim|x|→∞〈x〉kN(u0+u1)(x)=c1(≠0). Those proofs are given by elementary estimates on the explicit formulas of solutions.