Title: Interpolating the Stochastic Heat and Wave Equations with Time-independent Noise: Solvability and Exact Asymptotics Show Chinese title

Title: 带有时间独立噪声的随机热方程和波动方程的插值：可解性和精确渐近性

Abstract: In this article, we study a class of stochastic partial differential equations with fractional differential operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which a unique global $L^p(\Omega)$-solution exits for all $p\ge 2$. In this case, we derive exact moment asymptotics following the same strategy in a recent work by Balan et al [1]. In the case when there exits only a local solution, we determine the precise deterministic time, $T_2$, before which a unique $L^2(\Omega)$-solution exits, but after which the series corresponding to the $L^2(\Omega)$ moment of the solution blows up. By properly choosing the parameters, results in this paper interpolate the known results for both stochastic heat and wave equations. Show Chinese abstract

Abstract: 在本文中，我们研究一类受某些时间无关乘法高斯噪声影响的带有分数微分算子的随机偏微分方程。 我们推导出精确的条件，在这些条件下，对于所有$p\ge 2$，唯一全局$L^p(\Omega)$-解存在。 在这种情况下，我们按照 Balan 等人最近的工作 [1] 中的相同策略，推导出精确的矩渐近性。 当仅存在局部解时，我们确定了精确的确定性时间$T_2$，在此时间之前，唯一$L^2(\Omega)$-解存在，但在此时间之后，对应于解的$L^2(\Omega)$矩的级数发散。 通过适当选择参数，本文的结果在已知的随机热方程和波动方程结果之间进行插值。