Consider the following class of conformable time-fractional stochastic equation $$T_{\alpha,t}^a u(x,t)=\lambda\sigma(u(x,t))\dot{W}_t,\,\,\,\,x\in\mathbb{R},\,t\in[a,\infty), \,\,0 0$ is the noise level. Given some precise and suitable conditions on the non-random initial function, we study the asymptotic behaviour of the solution with respect to the time parameter $t$ and the noise level parameter $\lambda$. We also show that when the non-linear term $\sigma$ grows faster than linear, the energy of the solution blows-up at finite time for all $\alpha\in (0,1)$.

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一类适形时间分数随机方程的解的渐近行为和全局解的不存在性

考虑以下一类符合时间分数随机方程 $$T_{\alpha,t}^au(x,t)=\lambda\sigma(u(x,t))\dot{W}_t,\,\ ,\,\,x\in\mathbb{R},\,t\in[a,\infty), \,\,0 0$ 是噪声水平。给定非随机初始函数的一些精确和合适的条件，我们研究解决方案关于时间参数 $t$ 和噪声水平参数 $\lambda$ 的渐近行为。我们还表明，当非线性项 $\sigma$ 的增长速度超过线性时，对于所有 $\alpha\in (0,1)$，解的能量会在有限时间内爆发。