We consider the Cauchy-type problem associated to the time fractional partial differential equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+\partial _t^{\beta }u-\varDelta u=g(t,x), &{} t>0, \ x\in {\mathbb {R}}^n \\ u(0,x)=u_0(x), \end{array}\right. } \end{aligned}$$

with \(\beta \in (0,1)\), where the fractional derivative \(\partial _t^{\beta }\) is in Caputo sense. We provide a sufficient condition on the right-hand term g(t, x) to obtain a solution in \({\mathcal {C}}_b([0,\infty ),H^s)\). We exploit a dissipative-smoothing effect which allows to describe the asymptotic profile of the solution in low space dimension.

中文翻译：

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两项时间分数扩散问题的渐近分布

我们考虑与时间分数偏微分方程相关的柯西型问题：

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+\partial _t^{\beta }u-\varDelta u=g(t,x), &{} t> 0, \x\in {\mathbb {R}}^n \\u(0,x)=u_0(x), \end{array}\right. } \end{对齐}$$

与\(\beta \in (0,1)\)，其中分数导数\(\partial _t^{\beta }\)在卡普托意义上。我们提供了关于右手项g ( t ,  x ) 的充分条件，以获得\({\mathcal {C}}_b([0,\infty ),H^s)\)中的解。我们利用耗散平滑效应来描述解决方案在低空间维度中的渐近分布。