Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2021-03-26 , DOI: 10.1007/s00033-021-01515-9 Hui Zhang
This paper is concerned with the long-time behavior of Moore-Gibson-Thompson equation with degenerate memory effect, given by
$$\begin{aligned} \tau u_{ttt}+\alpha (x)u_{tt}-b\Delta u_t-c^2\Delta u+\int \limits _0^tg(t-s)div\big (\beta (x)\nabla u(s)\big )ds=0, \quad ~in ~\Omega \times (0,\infty ), \end{aligned}$$where structural damping and molecular relaxation are both locally distributed, satisfying
$$\begin{aligned} \alpha (x)+\beta (x)\ge \delta ,\quad for~a.e. ~x\in \Omega . \end{aligned}$$Under general conditions on memory function g(t), namely, \(g'(t)\le -\xi (t)H(g(t)),\) and some basic restrictions imposed on \(\xi (t)\) and \(H(\cdot )\), we show that the solution energy decays to zero in a general rate. Moreover, through specific examples, our result allows to deal with a much larger class of g(t), including but are not limited to polynomial or exponential decay. This will improve some related results in the literature.
中文翻译：
具有局部和简并记忆效应的Moore-Gibson-Thompson方程的长时间行为
本文关注具有退化记忆效应的Moore-Gibson-Thompson方程的长时间行为，由下式给出：
$$ \ begin {aligned} \ tau u_ {ttt} + \ alpha（x）u_ {tt} -b \ Delta u_t-c ^ 2 \ Delta u + \ int \ limits _0 ^ tg（ts）div \ big（\ beta（x）\ nabla u（s）\ big）ds = 0，\ quad〜in〜\ Omega \ times（0，\ infty），\ end {aligned} $$结构阻尼和分子弛豫都局部分布
$$ \ begin {aligned} \ alpha（x）+ \ beta（x）\ ge \ delta，\ quad for〜ae〜x \ in \ Omega。\ end {aligned} $$在一般情况下，关于记忆函数g（t），即\（g'（t）\ le-\ xi（t）H（g（t）），\）和对\（\ xi（t ）\）和\（H（\ cdot）\），我们证明了溶液能量以一般速率衰减到零。此外，通过特定的例子，我们的结果允许处理更大的g（t）类，包括但不限于多项式或指数衰减。这将改善文献中的一些相关结果。
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