Klein-Gordon type decay rates for wave equations with time-dependent coefficients
Klein-Gordon type decay rates for wave equations with time-dependent coefficients
复制标题
具有瞬态系数的波动方程的 Klein-Gordon 型衰减率
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
K. Yagdjian
中科院分区：
文献类型：
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作者：
M. Reissig;K. Yagdjian
本文证明了Klein-Gordon型方程utt 2（t）B 2（t）（4 um 2 u）= 0的柯西问题解的Lp Lq衰减估计.该系数由一个增光滑函数和一个振荡光滑有界函数B组成，它们是一致分离的.而且，m2是一个正的常数.我们研究的假设和B的一个可以预期的经典克莱因-戈登衰变率n（1 - 1）的衰变率的一个重要组成部分。1.导论.为了证明非线性波动方程柯西问题解的整体存在性结果，线性波动方程解的所谓Lp Lq衰减估计起着至关重要的作用（7），（8），（11）。这就是柯西问题utt 4的解u = u（t;x）的如下估计：u = 0; u（0;x）= 0; ut（0;x）= u 1（x）;
This work is concerned with the proof ofLp Lq decay estimates for solutions of the Cauchy problem for the Klein-Gordon type equation utt 2 (t)b 2 (t)(4u m 2 u) = 0. The coecient consists of an increasing smooth function and an oscillating smooth and bounded function b which are uniformly separated from zero. Moreover, m 2 is a positive constant. We study under which assumptions for and b one can expect as an essential part of the decay rate the classical Klein-Gordon decay rate n ( 1 1 ). 1. Introduction. To prove global existence results for the solutions of the Cauchy problem for nonlinear wave equations so-called Lp Lq decay estimates for the solutions of the linear wave equation play an essential role (7),(8),(11). That is the following estimate for the solution u = u(t;x) of the Cauchy problem utt4 u = 0; u(0;x) = 0; ut(0;x) = u1(x);