Author:
Asogwa Sunday A.,Foondun Mohammud,Mijena Jebessa B.,Nane Erkan
Abstract
AbstractWe will look at reaction–diffusion type equations of the following type, $$\begin{aligned} \partial ^\beta _tV(t,x)=-(-\Delta )^{\alpha /2} V(t,x)+I^{1-\beta }_t[V(t,x)^{1+\eta }]. \end{aligned}$$
∂
t
β
V
(
t
,
x
)
=
-
(
-
Δ
)
α
/
2
V
(
t
,
x
)
+
I
t
1
-
β
[
V
(
t
,
x
)
1
+
η
]
.
We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when $$0<\eta \leqslant \eta _c$$
0
<
η
⩽
η
c
, there is no global solution other than the trivial one while for $$\eta >\eta _c$$
η
>
η
c
, non-trivial global solutions do exist. The critical parameter $$\eta _c$$
η
c
is shown to be $$\frac{1}{\eta ^*}$$
1
η
∗
where $$\begin{aligned} \eta ^*:=\sup _{a>0}\left\{ \sup _{t\in (0,\,\infty ),x\in \mathbb {R}^d}t^a\int _{\mathbb {R}^d}G(t,\,x-y)V_0(y)\,\mathrm{d}y<\infty \right\} \end{aligned}$$
η
∗
:
=
sup
a
>
0
sup
t
∈
(
0
,
∞
)
,
x
∈
R
d
t
a
∫
R
d
G
(
t
,
x
-
y
)
V
0
(
y
)
d
y
<
∞
and $$G(t,\,x)$$
G
(
t
,
x
)
is the heat kernel of the corresponding unforced operator. $$V_0$$
V
0
is a non-negative initial function. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the fractional time derivative induces a significant change in the behavior of the solution.
Funder
University of Strathclyde
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis
Cited by
6 articles.
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