Abstract
We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source
$$\begin{aligned} u_{t} + (-\Delta )^{\frac{\beta }{2}} u =\bigl(1+ \vert x \vert \bigr)^{ \gamma } \int _{0}^{t} (t-s)^{\alpha -1} \vert u \vert ^{p} \bigl\Vert \nu ^{ \frac{1}{q}}(x) u \bigr\Vert _{q}^{r} \,ds \end{aligned}$$
u
t
+
(
−
Δ
)
β
2
u
=
(
1
+
|
x
|
)
γ
∫
0
t
(
t
−
s
)
α
−
1
|
u
|
p
∥
ν
1
q
(
x
)
u
∥
q
r
d
s
for $(x,t) \in \mathbb{R}^{N}\times (0,\infty )$
(
x
,
t
)
∈
R
N
×
(
0
,
∞
)
with initial data $u(x,0)=u_{0}(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})$
u
(
x
,
0
)
=
u
0
(
x
)
∈
L
loc
1
(
R
N
)
, where $p,q,r>1$
p
,
q
,
r
>
1
, $q(p+r)>q+r$
q
(
p
+
r
)
>
q
+
r
, $0<\gamma \leq 2 $
0
<
γ
≤
2
, $0<\alpha <1$
0
<
α
<
1
, $0<\beta \leq 2$
0
<
β
≤
2
, $(-\Delta )^{\frac{\beta }{2}}$
(
−
Δ
)
β
2
stands for the fractional Laplacian operator of order β, the weight function $\nu (x)$
ν
(
x
)
is positive and singular at the origin, and $\Vert \cdot \Vert _{q}$
∥
⋅
∥
q
is the norm of $L^{q}$
L
q
space.
Fundamental solution and discrete random walk model for a time-space fractional diffusion equation of distributed order