CAJ | 학술논문

在R3的有界区域 D上考虑了如下具有临界增长率的非自治随机波动方程的长时间渐近行为：ut ＋αut －Δu＋ f （u ，t）＝ g（x ，t）＋ uodWd t 。其中，W 是一维双边标准Wiener过程，f是具有临界增长率、时间依赖的非线性项，g是时间依赖的外力项。此方程的解导出一个具有2个参数的随机无穷维动力过程。证明了此随机无穷维动力过程的拉回吸引子的存在性。非线性项 f 的非紧性是研究此无穷维动力过程的渐近行为的难点所在。利用构造压缩函数的技术性方法来解决了这一难点。
재R3적유계구역 D상고필료여하구유림계증장솔적비자치수궤파동방정적장시간점근행위：ut ＋αut －Δu＋ f （u ，t）＝ g（x ，t）＋ uodWd t 。기중，W 시일유쌍변표준Wiener과정，f시구유림계증장솔、시간의뢰적비선성항，g시시간의뢰적외력항。차방정적해도출일개구유2개삼수적수궤무궁유동력과정。증명료차수궤무궁유동력과정적랍회흡인자적존재성。비선성항 f 적비긴성시연구차무궁유동력과정적점근행위적난점소재。이용구조압축함수적기술성방법래해결료저일난점。
We consider the longtime behavior of the following non‐autonomous stochastic wave equa‐tion with the critical exponent on a bounded domain D∈ R 3 :ut +αut -Δu+ f (u, t)= g(x, t)+ uodWd t. Where W is a one‐dimensional two‐sided standard Wiener process, f is the time‐dependent nonlineari‐ty with critical exponent, g is a given external time‐dependent forcing. The solution map of this equation defines a two‐parameter stochastic infinite dimensional dynam‐ical process. We prove the existence of pullback attractors of the dynamical process. The nonlinearity f is not precompact, this is the essential difficulty in studying the asymptotic behavior. We solve this difficulty by using a technical method of constructing contractive functions.