Abstract
AbstractWe 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.
Funder
Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
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