Abstract
We consider the following fractional reaction-diffusion equation
u
t
(
t
)
+
∂
t
∫
0
t
g
α
(
s
)
A
u
(
t
−
s
)
d
s
=
t
γ
f
(
u
)
,
$$ u_t(t) + \partial_t \int\nolimits_{0}^{t} g_{\alpha}(s) \mathcal{A} u(t-s) ds = t^{\gamma} f(u),$$
where g
α
(t) = t
α−1/Γ(α) (0 < α < 1), f ∈ C([0, ∞)) is a non-decreasing function, γ > −1, and
A
$\mathcal{A}$
is an elliptic operator whose fundamental solution of its associated parabolic equation has Gaussian lower and upper bounds. We characterize the behavior of the functions f so that the above fractional reaction-diffusion equation has a bounded local solution in L
r
(Ω), for non-negative initial data u
0 ∈ L
r
(Ω), when r > 1 and Ω ⊂ ℝ
N
is either a smooth bounded domain or the whole space ℝ
N
. The case r = 1 is also studied.