In this paper we study the asymptotic profile (as t→∞) of the solution to the Cauchy problem for the linear plate equation utt+Δ2u−λ(t)Δu+ut=0 when λ=λ(t) is a decreasing function, assuming initial data in the energy space and verifying a moment condition. For sufficiently small data, we find the critical exponent for global solutions to the corresponding problem with power nonlinearity utt+Δ2u−λ(t)Δu+ut=|u|p. In order to do that, we assume small data in the energy space and, possibly, in L1. In this latter case, we also determinate the asymptotic profile of the solution to the semilinear problem for supercritical power nonlinearities.

中文翻译：

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具有时间相关系数的半线性阻尼板方程的渐近曲线和临界指数

本文研究了当线性方程utt +Δ2u-λ（t）Δu+ ut = 0时线性方程Cauty问题解的渐近分布（t→∞） ，假设能量空间中的初始数据并验证力矩条件。对于足够小的数据，我们找到了幂非线性为utt +Δ2u-λ（t）Δu+ ut = | u | p的相应问题的整体解的临界指数。为了做到这一点，我们假设在能量空间中以及可能在L1中有少量数据。在后一种情况下，我们还为超临界功率非线性确定半线性问题解的渐近曲线。