Optimal l∞ decay for solutions to the wave equation with a potential
Optimal l∞ decay for solutions to the wave equation with a potential
复制标题
具有势能的波动方程解的最优 l∞ 衰减
DOI：
10.1080/03605309408821056
复制
发表时间：
1994
影响因子：
1.9
通讯作者：
M. Beals
中科院分区：
文献类型：
--
作者：
M. Beals
我们继续研究全球时间估计，以解决[2]、[3]中开始的问题。我们感兴趣的是测量 u (t) 的适当范数的衰减率，即 t+ oo。这种估计在线性和非线性理论中都很有用，如 Ginibre-Velo [6]、Strauss [13]。例如，如果 V r 0，则估计值是众所周知的，并且可以应用于分析某些非线性方程组解的存在性和散射特性，其中线性化方程为 ou= 0。只要 (0.1)、(0.2) 解的类似估计成立，Marshall-Strauss-Wainger [9]、Strauss [l21] 的结论就可以应用于线性化为 (0.1) 的非线性方程组。如果没有f 的额外平滑度假设为 (0.21，当 f E LD (IRn), llp= 112+ ll (n+ 1) 时，V s 0 情况下的最佳衰减发生。对于 Up'= 112-ll (n+ 11，u (t) 的 LP'(IRn) 范数则像 td 一样衰减，d=(n-lY (n+ 1)。参见 Strichartz
We continue the study of global time estimates for solutions of the problem begun in [2],[3]. We are interested in measuring the decay rate of appropriate norms of u (t) as t+ oo. Such estimates are useful in both linear and nonlinear theory, as in Ginibre-Velo [6], Strauss [13]. For example, if V r 0, the estimates are well-known and can be applied to the analysis of the existence and scattering properties of solutions to certain nonlinear equations for which the linearized equation is ou= 0. Whenever the analogous estimates for solutions to (0.1),(0.2) hold, the conclusions of Marshall-Strauss-Wainger [9], Strauss [l21, can be applied to nonlinear equations for which (0.1) is the linearization.If no extra smoothness for f is assumed in (0.21, the optimal decay in the case V s 0 occurs when f E LD (IRn), llp= 112+ ll (n+ 1). For Up'= 112-ll (n+ 11, the LP'(IRn) norm of u (t) then decays like td, d=(n-lY (n+ 1). See Strichartz