A multi-dimensional blow-up problem due to a concentrated nonlinear source in ℝ^{ℕ}
Authors: Chan C. , Tragoonsirisak P. 
Published: April 2015

Let

B
B

be an

N
N

-dimensional ball

{
x
∈

R

N

:

|
x
|

>
R
}

\left \{ x\in \mathbb {R}^{N}:\left \vert x\right \vert >R\right \}

centered at the origin with a radius

R
R

, and

∂
B

\partial B

be its boundary. Also, let

ν

(
x
)

\nu \left ( x\right )

denote the unit inward normal at

x
∈
∂
B

x\in \partial B

, and let

χ

B

(
x
)

\chi _{B}\left ( x\right )

be the characteristic function, which is 1 for

x
∈
B

x\in B

, and

0
0

for

x
∈

R

N

∖
B

x\in \mathbb {R}^{N}\setminus B

. This article studies the following multi-dimensional semilinear parabolic problem with a concentrated nonlinear source on

∂
B

\partial B

:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row u Subscript t Baseline minus white up pointing triangle u equals alpha StartFraction partial-differential chi Subscript upper B Baseline left-parenthesis x right-parenthesis Over partial-differential nu EndFraction f left-parenthesis u right-parenthesis in double-struck upper R Superscript upper N Baseline times left-parenthesis 0 comma upper T right-bracket comma 2nd Row u left-parenthesis x comma 0 right-parenthesis equals psi left-parenthesis x right-parenthesis reverse-solidus for x element-of double-struck upper R Superscript upper N Baseline comma u left-parenthesis x comma t right-parenthesis right-arrow 0 as reverse-solidus StartAbsoluteValue x EndAbsoluteValue right-arrow normal infinity for 0 greater-than t less-than-or-equal-to upper T comma EndLayout">

u

t

−
△
u
=
α

∂

χ

B

(
x
)

∂
ν

f

(
u
)

 in 

R

N

×
(
0
,
T
]
,

u

(
x
,
0
)

=
ψ

(
x
)

\ for 
x
∈

R

N

, 
u

(
x
,
t
)

→
0
 
as\ 

|
x
|

→
∞
 for 
0
>
t
≤
T
,

\begin {array}[c]{c} u_{t}-\triangle u=\alpha {\displaystyle {\frac {\partial \chi _{B}\left ( x\right ) }{\partial \nu }}}f\left ( u\right ) \text { in }\mathbb {R}^{N}\times (0,T],\\ u\left ( x,0\right ) =\psi \left ( x\right ) \text {\ for }x\in \mathbb {R}^{N}\text {, }u\left ( x,t\right ) \rightarrow 0\ \text {as\ }\left \vert x\right \vert \rightarrow \infty \text { for }0>t\leq T, \end {array}

\]
where

α
\alpha

and

T
T

are positive numbers,

f
f

and

ψ
\psi

are given functions such that

f

(
0
)

≥
0

f\left ( 0\right ) \geq 0

,

f

(
u
)

f\left ( u\right )

and

f

′

(
u
)

f^{\prime }\left ( u\right )

are positive for

u
>
0

u>0

,

f

′
′

(
u
)

≥
0

f^{\prime \prime }\left ( u\right ) \geq 0

for

u
>
0

u>0

, and

ψ
\psi

is nontrivial on

∂
B
,

\partial B,

nonnegative, and continuous such that

ψ
→
0

\psi \rightarrow 0

as

|
x
|

→
∞
,

\left \vert x\right \vert \rightarrow \infty ,

∫

R

N

ψ

(
x
)

d
x
>
∞

\int _{\mathbb {R}^{N}}\psi \left ( x\right ) dx>\infty

, and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="white up pointing triangle psi plus alpha left-parenthesis partial-differential chi Subscript upper B Baseline left-parenthesis x right-parenthesis slash partial-differential nu right-parenthesis f left-parenthesis psi left-parenthesis x right-parenthesis right-parenthesis greater-than-or-equal-to 0">

△
ψ
+
α

(
∂

χ

B

(
x
)

/

∂
ν
)

f

(
ψ

(
x
)

)

≥
0

\triangle \psi +\alpha \left ( \partial \chi _{B}\left ( x\right ) /\partial \nu \right ) f\left ( \psi \left ( x\right ) \right ) \geq 0

in

R

N

.

\mathbb {R}^{N}.

It is shown that the problem has a unique nonnegative continuous solution before blowup occurs. We assume that

ψ

(
x
)

=
M

(
0
)

>
ψ

(
y
)

\psi \left ( x\right ) =M\left ( 0\right ) >\psi \left ( y\right )

for

x
∈
∂
B

x\in \partial B

and

y
∉
∂
B

y\notin \partial B

, where

M

(
t
)

=

sup

x
∈

R

N

u

(
x
,
t
)

M\left ( t\right ) =\sup _{x\in \mathbb {R}^{N}}u\left ( x,t\right )

. It is proved that if

u
u

blows up in a finite time, then it blows up everywhere on

∂
B

\partial B

. If, in addition,

ψ
\psi

is radially symmetric about the origin, then we show that if

u
u

blows up, then it blows up on

∂
B

\partial B

only. Furthermore, if

f

(
u
)

≥
κ

u

p

f\left ( u\right ) \geq \kappa u^{p}

, where

κ
\kappa

and

p
p

are positive constants such that

p
>
1

p>1

, then it is proved that for any

α
\alpha

,

u
u

always blows up in a finite time for

N
≤
2

N\leq 2

; for

N
≥
3

N\geq 3

, it is shown that there exists a unique number

α

∗

\alpha ^{\ast }

such that

u
u

exists globally for

α
≤

α

∗

\alpha \leq \alpha ^{\ast }

and blows up in a finite time for

α
>

α

∗

\alpha >\alpha ^{\ast }

. A formula for computing

α

∗

\alpha ^{\ast }

is given.