Global existence and exponential decay rates for the Westervelt equation
Global existence and exponential decay rates for the Westervelt equation
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Westervelt 方程的全局存在性和指数衰减率
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
I. Lasiecka
中科院分区：
文献类型：
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作者：
B. Kaltenbacher;I. Lasiecka
我们考虑用来模拟声音在流体介质中传播的Westvelt方程。
这是一个公认的非线性声学模型，在医学成像和治疗中得到了广泛的应用。
PDE模型由时间演化的二阶模型组成
它既是准线性的，也是退化的。退化程度取决于
声压的波动。
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我们的主要结果是：(1)全局适定性，
(2)能量函数的指数衰减率
既有弱的也有强的解决方案。
证明的基础是：(I)应用适当的不动点定理到偏微分方程解的适当形式，它展示了
基础线性化半群的解析性，
(2)利用与耗散机制有关的衰减率
以及巴尔通向全球福祉的方法。
所获得的结果在任何时候都适用，前提是初始数据
取自一个由方程参数表征的适当的小球。
We consider the Westervelt equation which models propagation of sound in a fluid medium.
This is an accepted in nonlinear acoustics model which finds a multitude of applications in medical imaging and therapy.
The PDE model consists of the second order in time evolution
which is both quasilinear and degenerate. Degeneracy depends on the
fluctuations of the acoustic pressure.
Our main results are : (1) global well-posedness,
(2) exponential decay rates for the energy function corresponding to
both weak and strong solutions.
The proof is based on (i) application of a suitable fixed point theorem applied to an appropriate formulation of the PDE which exhibits
analyticity properties of the underlying linearised semigroup,
(ii) exploitation of decay rates associated with the dissipative mechanism
along with barrier's method leading to global wellposedness.
The obtained result holds for all times, provided that the initial data
are taken from a suitably small ball characterized by the parameters of the equation.