On the Cauchy problem for reaction-diffusion equations
Authors: Wang Xuefeng 
Published: June 2012

The simplest model of the Cauchy problem considered in this paper is the following

(
∗
)

(\ast )

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column u Subscript t Baseline equals normal upper Delta u plus u Superscript p Baseline comma 2nd Column a m p semicolon x element-of upper R Superscript n Baseline comma t greater-than 0 comma u greater-than-or-equal-to 0 comma p greater-than 1 comma 2nd Row 1st Column u vertical-bar Subscript t equals 0 Baseline equals phi element-of upper C Subscript upper B Baseline left-parenthesis upper R Superscript n Baseline right-parenthesis comma 2nd Column a m p semicolon phi greater-than-or-equal-to 0 comma phi not-identical-to 0 period EndLayout">

u
t

=
Δ
u
+

u
p

,

a
m
p
;

x
∈

R
n

,
t
>
0
,
u
≥
0
,
p
>
1
,

u

|

t
=
0

=
ϕ
∈

C
B

(

R
n

)
,

a
m
p
;

ϕ
≥
0
,
ϕ

≢

0.

\begin {array}{*{20}{c}} {{u_t} = \Delta u + {u^p},} \hfill & {x \in {R^n},t > 0,u \geq 0,p > 1,} \hfill \\ {u{|_{t = 0}} = \phi \in {C_B}({R^n}),} \hfill & {\phi \geq 0,\phi \,\not \equiv \,0.} \hfill \\ \end {array} \;

\]
It is well known that when

1
>
p
≤
(
n
+
2
)

/

n

1 > p \leq (n + 2)/n

, the local solution of

(
∗
)

(\ast )

blows up in finite time as long as the initial value

ϕ
\phi

is nontrivial; and when

p
>
(
n
+
2
)

/

n

p > (n + 2)/n

, if

ϕ
\phi

is "small",

(
∗
)

(\ast )

has a global classical solution decaying to zero as

t
→
+
∞

t \to + \infty

, while if

ϕ
\phi

is "large", the local solution blows up in finite time. The main aim of this paper is to obtain optimal conditions on

ϕ
\phi

for global existence and to study the asymptotic behavior of those global solutions. In particular, we prove that if

n
≥
3

n \geq 3

,

p
>
n

/

(
n
−
2
)

p > n/(n - 2)

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to phi left-parenthesis x right-parenthesis less-than-or-equal-to lamda u Subscript s Baseline left-parenthesis x right-parenthesis equals lamda left-parenthesis StartFraction 2 left-parenthesis n minus 2 right-parenthesis Over left-parenthesis p minus 1 right-parenthesis squared EndFraction left-parenthesis p minus StartFraction n Over n minus 2 EndFraction right-parenthesis right-parenthesis Superscript 1 slash left-parenthesis p minus 1 right-parenthesis Baseline StartAbsoluteValue x EndAbsoluteValue Superscript negative 2 slash left-parenthesis p minus 1 right-parenthesis">

0
≤
ϕ
(
x
)
≤
λ

u
s

(
x
)
=
λ

(

2
(
n
−
2
)

(
p
−
1
)

2

(

p
−

n

n
−
2

)

)

1

/

(
p
−
1
)

|

x

|

−
2

/

(
p
−
1
)

0 \leq \phi (x) \leq \lambda {u_s}(x) = \lambda {\left ( {\frac {{2(n - 2)}} {{{{(p - 1)}^2}}}\left ( {p - \frac {n} {{n - 2}}} \right )} \right )^{1/(p - 1)}}|x{|^{ - 2/(p - 1)}}

\]
(

u
s

{u_s}

is a singular equilibrium of

(
∗
)

(\ast )

) where

0
>
λ
>
1

0 > \lambda > 1

, then

(
∗
)

(\ast )

has a (unique) global classical solution

u
u

with

0
≤
u
≤
λ

u
s

0 \leq u \leq \lambda {u_s}

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x comma t right-parenthesis less-than-or-equal-to left-parenthesis left-parenthesis lamda Superscript 1 minus p Baseline minus 1 right-parenthesis left-parenthesis p minus 1 right-parenthesis t right-parenthesis Superscript negative 1 slash left-parenthesis p minus 1 right-parenthesis Baseline period">

u
(
x
,
t
)
≤

(
(

λ

1
−
p

−
1
)
(
p
−
1
)
t

)

−
1

/

(
p
−
1
)

.

u(x,t) \leq {(({\lambda ^{1 - p}} - 1)(p - 1)t)^{ - 1/(p - 1)}}.

\]
(This result implies that

u
0

≡
0

{u_0} \equiv 0

is stable w.r.t. to a weighted

L
∞

{L^\infty }

topology when

n
≥
3

n \geq 3

and

p
>
n

/

(
n
−
2
)

p > n/(n - 2)

.) We also obtain some sufficient conditions on

ϕ
\phi

for global nonexistence and those conditions, when combined with our global existence result, indicate that for

ϕ
\phi

around

u
s

{u_s}

, we are in a delicate situation, and when

p
p

is fixed,

u
0

≡
0

{u_0} \equiv 0

is "increasingly stable" as the dimension

n
↑
+
∞

n \uparrow + \infty

. A slightly more general version of

(
∗
)

(\ast )

is also considered and similar results are obtained.