In this paper we focus on the initial value problem for quasi-lineardissipative plate equation in multi-dimensional space $(n\geq2)$. This equationverifies the decay property of the regularity-loss type, which causes thedifficulty in deriving the global a priori estimates of solutions. We overcomethis difficulty by employing a time-weighted $L^2$ energy method which makesuse of the integrability of $\|(\p^2_xu_t,\p^3_xu)(t)\|_{L^{\infty}}$. This$L^\infty$ norm can be controlled by showing the optimal $L^2$ decay estimatesfor lower-order derivatives of solutions. Thus we obtain the desired a prioriestimate which enables us to prove the global existence and asymptotic decay ofsolutions under smallness and enough regularity assumptions on the initialdata. Moreover, we show that the solution can be approximated by asimple-looking function, which is given explicitly in terms of the fundamentalsolution of a fourth-order linear parabolic equation.
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机译：在本文中，我们关注多维空间$（n \ geq2）$中的拟线性耗散板方程的初值问题。该方程式验证了正则损失类型的衰减特性，这导致难以求出解的全局先验估计。我们通过采用时间加权的$ L ^ 2 $能量方法克服了这一难题，该方法利用了$ \ |（\ p ^ 2_xu_t，\ p ^ 3_xu）（t）\ | _ {L ^ {\ infty}}的可积性$。这个L ^ \ infty $范数可以通过显示解决方案低阶导数的最佳$ L ^ 2 $衰减估计来控制。因此，我们获得了所需的先验估计，使我们能够证明在初始数据的较小性和足够规律性假设下，解的整体存在性和渐近衰减。此外，我们表明，该解决方案可以通过看似简单的函数来近似，该函数由四阶线性抛物方程的基本解明确给出。
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