In this work, we interest in the study of the wellposedness and the stability of fractional stochastic nonlinear heat equation in the Hilbert space \( L^{2}(0,1) \); perturbed by a trace-class noise and driven by the fractional Laplacian. Precisely, we use the fixed point theorem to prove the wellposedness of the problem. Moreover, we prove the \( p^{th} \)-moment exponential stability and the almost surely exponential stability by imposing an additional assumption. Some examples are considered in order to confirm and support the validity of our theoretical results.

中文翻译：

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希尔伯特空间分数随机非线性热方程的适定性和稳定性

在这项工作中，我们感兴趣的是研究希尔伯特空间\(L^{2}(0,1)\)中分数随机非线性热方程的适定性和稳定性；由微量级噪声扰动并由分数拉普拉斯算子驱动。确切地说，我们使用不动点定理来证明问题的适定性。此外，我们通过附加假设证明了\( p^{th} \) -矩指数稳定性和几乎可以肯定的指数稳定性。为了证实和支持我们的理论结果的有效性，我们考虑了一些例子。