Boundedness and stabilization in a quasilinear forager–exploiter model with volume-filling effects
This paper deals with a forager–exploiter model involving nonlinear diffusions and volume-filling effects u t = ∇ · ( ( u + 1 ) m ∇ u ) - ∇ · ( S 1 ( u ) ∇ w ) , x ∈ Ω , t > 0 , v t = ∇ · ( ( v + 1 ) l ∇ v ) - ∇ · ( S 2 ( v ) ∇ u ) , x ∈ Ω , t > 0 , w t = Δ w - ( u + v ) w - μ w + r ( x , t )...
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| Vydané v: | Zeitschrift für angewandte Mathematik und Physik Ročník 73; číslo 5 |
|---|---|
| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Cham
Springer International Publishing
01.10.2022
Springer Nature B.V |
| Predmet: | |
| ISSN: | 0044-2275, 1420-9039 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | This paper deals with a forager–exploiter model involving nonlinear diffusions and volume-filling effects
u
t
=
∇
·
(
(
u
+
1
)
m
∇
u
)
-
∇
·
(
S
1
(
u
)
∇
w
)
,
x
∈
Ω
,
t
>
0
,
v
t
=
∇
·
(
(
v
+
1
)
l
∇
v
)
-
∇
·
(
S
2
(
v
)
∇
u
)
,
x
∈
Ω
,
t
>
0
,
w
t
=
Δ
w
-
(
u
+
v
)
w
-
μ
w
+
r
(
x
,
t
)
,
x
∈
Ω
,
t
>
0
under homogeneous Neumann boundary conditions in a smooth bounded domain
Ω
⊂
R
n
with
n
≥
1
, where
μ
>
0
,
m
,
l
∈
R
and
r
∈
C
1
(
Ω
¯
×
[
0
,
∞
)
)
∩
L
∞
(
Ω
×
(
0
,
∞
)
)
is a given nonnegative function, the initial data
u
0
,
v
0
,
w
0
satisfy
0
≤
u
0
,
v
0
≤
1
and
w
0
≥
0
. Volume-filling effects account for an ordinary form by taking
S
1
(
u
)
=
u
(
1
-
u
)
,
S
2
(
v
)
=
v
(
1
-
v
)
.
It is proved that the corresponding initial-boundary value problem admits a unique global bounded classical solution. Furthermore, if
r
satisfies
∫
t
t
+
1
∫
Ω
r
→
0
as
t
→
∞
, then the global bounded classical solution (
u
,
v
,
w
) that converges to
1
|
Ω
|
∫
Ω
u
0
,
1
|
Ω
|
∫
Ω
v
0
,
0
as
t
→
∞
. |
|---|---|
| Bibliografia: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0044-2275 1420-9039 |
| DOI: | 10.1007/s00033-022-01821-w |