We consider the Cauchy-type problem associated to the time fractional partial differential equation: $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u+\partial _t^{\beta }u-\varDelta u=g(t,x), &{} t>0, \ x\in {\mathbb {R}}^n \\ u(0,x)=u_0(x), \end{array}\right. } \end{aligned}$$ ∂ t u + ∂ t β u - Δ u = g ( t , x ) , t > 0 , x ∈ R n u ( 0 , x ) = u 0 ( x ) , with $$\beta \in (0,1)$$ β ∈ ( 0 , 1 ) , where the fractional derivative $$\partial _t^{\beta }$$ ∂ t β is in Caputo sense. We provide a sufficient condition on the right-hand term g(t, x) to obtain a solution in $${\mathcal {C}}_b([0,\infty ),H^s)$$ C b ( [ 0 , ∞ ) , H s ) . We exploit a dissipative-smoothing effect which allows to describe the asymptotic profile of the solution in low space dimension.