1 Introduction
In this paper, we are concerned with the existence and nonexistence of global weak solutions to the following exterior Dirichlet and Neumann problems:
(1.1)
{
∂
k
u
∂
t
k
-
Δ
u
≥
|
u
|
p
in
(
0
,
∞
)
×
D
c
,
u
(
t
,
x
)
≥
f
(
x
)
in
(
0
,
∞
)
×
∂
D
,
∂
i
u
∂
t
i
|
t
=
0
=
u
i
(
x
)
in
D
c
,
i
=
0
,
1
,
…
,
k
-
1
,
(1.2)
{
∂
k
u
∂
t
k
-
Δ
u
≥
|
u
|
p
in
(
0
,
∞
)
×
D
c
,
∂
u
∂
n
+
(
t
,
x
)
≥
f
(
x
)
in
(
0
,
∞
)
×
∂
D
,
∂
i
u
∂
t
i
|
t
=
0
=
u
i
(
x
)
in
D
c
,
i
=
0
,
1
,
…
,
k
-
1
,
where k≥1 is a fixed natural number, D=B(0,1)¯ is the closed unit ball in ℝN, N≥3, and Dc is its complement. Moreover, n+ denotes the outward (relative to Dc) unit normal to ∂D. We obtain the critical exponents for (1.1) and (1.2) in the sense of Fujita. Next, we discuss the behavior of solutions to (1.1) and (1.2) with respect to the initial data. Problems (1.1) and (1.2) are investigated under the assumptions: p>1, ui∈Lloc1(Dc¯), ui≥0, i=0,1,…,k-1, and f∈C(∂D), f≥0, f≢0. Let us mention some motivations for studying problems of the form (1.1) and (1.2).
In the pioneering paper [3], Fujita considered the Cauchy problem
(1.3)
{
∂
u
∂
t
-
Δ
u
=
u
p
in
(
0
,
∞
)
×
ℝ
N
,
u
(
0
,
x
)
=
u
0
(
x
)
in
ℝ
N
,
for which he established the following results:
If 1<p<1+2N and u0>0, then (1.3) admits no global positive solution.
If p>1+2N and u0 is smaller than a small Gaussian, then (1.3) has global positive solutions.
Later, it was shown in [6, 8] that 1+2N belongs to the blow-up case. The exponent p=1+2N is said to be critical in the sense of Fujita. On the other hand, the above results, which hold true for arbitrary non-negative initial data, do not take into consideration the behavior of the solution with respect to u0. In [10], Lee and Ni studied problem (1.3), where u0 is a non-negative bounded continuous function in ℝN and satisfies
u
0
(
x
)
∼
λ
(
1
+
|
x
|
)
-
a
as
|
x
|
→
∞
,
with λ,a>0. They discussed the global existence and nonexistence and the life span of the solution with respect to the parameters λ, a, p and the space dimension N. In [13], Pokhozhaev proved the following result: If u0≥0 satisfies the inequality
T
θ
p
∫
ℝ
N
u
0
(
x
)
e
-
|
x
|
2
4
T
𝑑
x
>
K
ψ
1
p
-
1
for some T>0, where Kψ>0 is a certain constant and θp=2-N(p-1)2(p-1), then (1.3) has no solutions for t∈(0,T).
For the semilinear wave equation
(1.4)
{
∂
2
u
∂
t
2
-
Δ
u
=
u
p
in
(
0
,
∞
)
×
ℝ
N
,
(
u
(
0
,
x
)
,
∂
u
∂
t
(
0
,
x
)
)
=
(
u
0
(
x
)
,
u
1
(
x
)
)
in
ℝ
N
,
we have the following well-known result (see [4, 5, 7, 14, 15, 16, 17] and references therein): Let
p
c
(
N
)
=
(
N
+
1
)
+
[
(
N
+
1
)
2
+
8
(
N
-
1
)
]
1
2
2
(
N
-
1
)
.
Then
If 1<p≤pc(N), then (1.4) has no nontrivial global solutions.
If p>pc(N), then there exist nontrivial global small data solutions.
In [12], Pokhozhaev and Véron studied the nonlinear hyperbolic inequality
(1.5)
{
∂
2
u
∂
t
2
-
Δ
u
≥
|
u
|
p
in
(
0
,
∞
)
×
ℝ
N
,
(
u
(
0
,
x
)
,
∂
u
∂
t
(
0
,
x
)
)
=
(
u
0
(
x
)
,
u
1
(
x
)
)
in
ℝ
N
.
In fact, they considered a more general problem, which includes (1.5) as a special case. For (1.5), they proved the following results:
If
∫
ℝ
N
u
1
(
x
)
𝑑
x
≥
0
and N=1, then there exists no solution to (1.5).
If
∫
ℝ
N
u
1
(
x
)
𝑑
x
≥
0
,
N
≥
2
and 1<p≤N+1N-1, then there exists no solution to (1.5).
If p>N+1N-1 and N≥2, then (1.5) admits a positive solution such that
∫
ℝ
N
u
1
(
x
)
𝑑
x
≥
0
.
In [1], Bandle and Levine studied the exterior problem in the N-dimensional case, N≥3:
(1.6)
{
∂
u
∂
t
-
Δ
u
=
u
p
in
(
0
,
∞
)
×
D
c
,
u
(
t
,
x
)
=
0
in
(
0
,
∞
)
×
∂
D
,
u
(
0
,
x
)
=
u
0
(
x
)
in
D
c
.
They proved that the critical exponent for (1.6) is still equal to Fujita’s exponent 1+2N.
In [9] (see also [11]), Laptev studied problem (1.1) in the case f≡0. He proved that in such case, under the condition
∂
k
-
1
u
∂
t
k
-
1
|
t
=
0
≥
0
,
if
1
<
p
<
N
+
2
k
N
-
2
+
2
k
,
then there is no global nontrivial solution.
In [18], Zhang studied several exterior boundary value problems with nontrivial boundary conditions. In particular, he considered the semilinear exterior problem in the N-dimensional case, N≥3:
(1.7)
{
∂
u
∂
t
-
Δ
u
=
u
p
in
(
0
,
∞
)
×
D
c
,
u
(
t
,
x
)
=
f
(
x
)
in
(
0
,
∞
)
×
∂
D
,
u
(
0
,
x
)
=
u
0
(
x
)
in
D
c
,
where f≥0 and f≢0. He established that the critical exponent for (1.7) passes from 1+2N to a bigger number given by
(1.8)
p
=
N
N
-
2
.
He proved also that the same result holds true if instead of u(t,x)=f(x) in (0,∞)×∂D, a Neumann boundary condition ∂u∂n+(t,x)=f(x) in (0,∞)×∂D is considered. In the same paper, Zhang considered the semilinear wave equation in the N-dimensional case, N≥3,
(1.9)
{
∂
2
u
∂
t
2
-
Δ
u
=
|
u
|
p
in
(
0
,
∞
)
×
D
c
,
∂
u
∂
n
+
(
t
,
x
)
=
f
(
x
)
in
(
0
,
∞
)
×
∂
D
,
(
u
(
0
,
x
)
,
∂
u
∂
t
(
0
,
x
)
)
=
(
u
0
(
x
)
,
u
1
(
x
)
)
in
D
c
.
He established that the critical exponent for (1.9) is still given by (1.8). He also mentioned that it would be interesting to know whether the Dirichlet boundary-value problem corresponding to (1.9) also has a similar critical behavior.
Let us mention some remarks concerning the used approach in [18] for studying (1.7). In fact, the main idea consists in using a change of variable in order to reduce (1.7) to another problem with a trivial boundary condition. However, due to this change of variable, a new source term appears in the newly obtained problem. In order to deal with this source term, the maximum principle for the heat equation is used. Next, the method consists in showing that the Lp-norm of the solution blows up in a certain selected compact region. We note that this region depends on the considered problem. Clearly, this approach cannot be applied for studying the Dirichlet boundary-value problem corresponding to (1.9) (or more generally for studying (1.1)) since the maximum principle cannot be used in this case.
Motivated by the above facts, we are concerned in this paper with the critical exponents for problems (1.1) and (1.2). An extension of the approach in [18] is proposed for solving these questions. Note that in our method, no maximum principle is needed and the selected region is the same for any integer k≥1 and any boundary condition (Dirichlet or Neumann). Moreover, inspired by [10, 13], we study the behavior of solutions to problems (1.1) and (1.2) with respect to the initial data.
To better state our main results, let us provide the definitions of solutions to (1.1) and (1.2).
Definition 1.1.
Let ui∈Lloc1(Dc¯), i=0,1,…,k-1, and f∈C(∂D), f≥0. We say that u is a local weak solution to (1.1) if there exists 0<T<∞ such that u∈Llocp(QT¯), where QT=[0,T]×Dc, and such that it satisfies
∫
Q
T
|
u
|
p
φ
𝑑
x
𝑑
t
-
∫
[
0
,
T
]
×
∂
D
∂
φ
∂
n
+
f
𝑑
ν
(
x
)
𝑑
t
+
∑
i
=
1
k
(
-
1
)
i
+
1
∫
D
c
u
k
-
i
(
x
)
∂
i
-
1
φ
∂
t
i
-
1
(
0
,
x
)
𝑑
x
(1.10)
≤
(
-
1
)
k
∫
Q
T
u
∂
k
φ
∂
t
k
𝑑
x
𝑑
t
-
∫
Q
T
u
Δ
φ
𝑑
x
𝑑
t
for every non-negative function φ∈Ct,xk,2(QT)∩C(QT¯) with
∂
i
-
1
φ
∂
t
i
-
1
(
0
,
⋅
)
∈
C
(
D
c
¯
)
, i=1,2,…,k, ∂kφ∂tk,Δφ∈C(QT¯),
φ
(
T
,
⋅
)
=
⋯
=
∂
k
-
1
φ
∂
t
k
-
1
(
T
,
⋅
)
≡
0
,
φ
|
∂
D
=
0
and there exists R>1 such that φ(⋅,x)≡0 for |x|≥R,
∂
φ
∂
n
+
∈
L
∞
(
[
0
,
T
]
×
∂
D
)
, ∂φ∂n+(t,x)≤0, (t,x)∈[0,T]×∂D,
where dν(x) stands for the surface measure on ∂D. Moreover, if T>0 can be arbitrarily chosen, then u is said to be a global weak solution to (1.1).
Definition 1.2.
Let ui∈Lloc1(Dc¯), i=0,1,…,k-1, and f∈C(∂D), f≥0. We say that u is a local weak solution to (1.2) if there exists 0<T<∞ such that u∈Llocp(QT¯) and it satisfies
∫
Q
T
|
u
|
p
φ
𝑑
x
𝑑
t
+
∫
[
0
,
T
]
×
∂
D
f
φ
𝑑
ν
(
x
)
𝑑
t
+
∑
i
=
1
k
(
-
1
)
i
+
1
∫
D
c
u
k
-
i
(
x
)
∂
i
-
1
φ
∂
t
i
-
1
(
0
,
x
)
𝑑
x
(1.11)
≤
(
-
1
)
k
∫
Q
T
u
∂
k
φ
∂
t
k
𝑑
x
𝑑
t
-
∫
Q
T
u
Δ
φ
𝑑
x
𝑑
t
for every non-negative function φ∈Ct,xk,2(QT)∩C(QT¯) with
∂
i
-
1
φ
∂
t
i
-
1
(
0
,
⋅
)
∈
C
(
D
c
¯
)
, i=1,2,…,k, ∂kφ∂tk,Δφ∈C(QT¯),
φ
(
T
,
⋅
)
=
⋯
=
∂
k
-
1
φ
∂
t
k
-
1
(
T
,
⋅
)
≡
0
,
there exists R>1 such that φ(⋅,x)≡0, for |x|≥R,
∂
φ
∂
n
+
(
t
,
x
)
=
0
, (t,x)∈[0,T]×∂D.
Moreover, if T>0 can be arbitrarily chosen, then u is said to be a global weak solution to (1.2).
Now, we are ready to state the main results obtained in this paper. Our first result concerns the critical exponent for (1.1).
Theorem 1.3.
Let N≥3. Suppose that ui∈Lloc1(Dc¯), ui≥0, i=0,1,…,k-1. Then NN-2 is the critical exponent for (1.1), i.e., we have the following:
Let
f
∈
C
(
∂
D
)
, f≥0 and f≢0. If
1
<
p
<
N
N
-
2
,
then (
1.1
) admits no global weak solution.
If
p
>
N
N
-
2
,
then (
1.1
) admits solutions for some
u
i
≥
0
, i=0,1,…,k-1, and f∈L∞(∂D), f≥0, such that ∥f∥L∞(∂D) is sufficiently small.
As is shown by the following theorem, problem (1.2) has the same critical behavior as problem (1.1).
Theorem 1.4.
Let N≥3. Suppose that ui∈Lloc1(Dc¯), ui≥0, i=0,1,…,k-1. Then NN-2 is the critical exponent for (1.2), i.e., we have the following:
Let
f
∈
C
(
∂
D
)
, f≥0 and f≢0. If
1
<
p
<
N
N
-
2
,
then (
1.2
) admits no global weak solution.
If
p
>
N
N
-
2
,
then (
1.2
) admits solutions for some
u
i
≥
0
, i=0,1,…,k-1, and f∈L∞(∂D), f≥0, such that ∥f∥L∞(∂D) is sufficiently small.
Remark 1.5.
Observe that the critical exponent for (1.1) and (1.2) is independent of k. Moreover, the result given by Theorem 1.3 contains an answer to an open question given by Zhang (see [18, Remark 1.5, p. 454]) concerning the critical behavior of the Dirichlet boundary-value problem corresponding to (1.9).
Remark 1.6.
Comparing our result given by Theorem 1.3 with that obtained by Laptev [9], we observe that the critical exponent for problem (1.1) passes from
N
+
2
k
N
-
2
+
2
k
(which is the critical exponent for (1.1) with f≡0) to NN-2.
Remark 1.7.
Given a function g:(0,∞)×ℝN∖{0}→ℝ, let 𝒦g:(0,∞)×ℝN∖{0}→ℝ be its Kelvin transform, which is defined by
(
𝒦
g
)
(
t
,
x
)
=
|
x
|
2
-
N
g
(
t
,
x
|
x
|
2
)
,
(
t
,
x
)
∈
(
0
,
∞
)
×
ℝ
N
∖
{
0
}
.
It can be easily seen that u is a solution to (1.1) if and only if v=𝒦u is a solution to the interior problem
(1.12)
{
∂
k
v
∂
t
k
-
|
y
|
4
Δ
v
≥
|
y
|
(
p
-
1
)
(
N
-
2
)
|
v
|
p
in
(
0
,
∞
)
×
D
̊
∖
{
0
}
,
v
(
t
,
y
)
≥
f
(
y
)
in
(
0
,
∞
)
×
∂
D
,
∂
i
v
∂
t
i
|
t
=
0
=
(
𝒦
u
i
)
(
y
)
in
D
̊
∖
{
0
}
,
i
=
0
,
1
,
…
,
k
-
1
,
where D̊ denotes the interior of D. Therefore, by Theorem 1.3, the critical exponent for (1.12) is NN-2. The same remark holds true for the corresponding interior problem to (1.2) obtained via the Kelvin transform.
Our next result concerns the behavior of solutions to problems (1.1) and (1.2) with respect to the initial data.
Theorem 1.8.
Let p>1, f∈C(∂D), f≥0 and ui∈Lloc1(Dc¯), ui≥0, i=0,1,…,k-1. Suppose that for some j∈{1,2,…,k} we have
(1.13)
u
k
-
j
(
x
)
≥
A
|
x
|
γ
for |x| large enough, where A>0 is a certain constant (independent of x) and γ>-N. If
(1.14)
k
p
p
-
1
>
j
-
k
γ
2
,
then
problem (
1.1
) admits no global weak solution,
problem (
1.2
) admits no global weak solution.
Let us give some examples to illustrate the results given by Theorem 1.8.
Example 1.9.
Suppose that (1.13) is satisfied with
γ
≥
2
(
j
-
k
)
k
.
Then (1.14) is satisfied for every p>1. Consequently, for every p>1, problems (1.1) and (1.2) admit no global weak solutions.
Example 1.10.
Suppose that (1.13) is satisfied with
-
N
<
γ
<
2
(
j
-
k
)
k
.
Then (1.14) is satisfied for every
(1.15)
1
<
p
<
j
-
k
γ
2
j
-
k
γ
2
-
k
.
Consequently, for every p satisfying (1.15), problems (1.1) and (1.2) admit no global weak solutions.
The rest of the paper is organized as follows: In Section 2, we provide some lemmas and preliminary estimates. In Section 3, we give the proofs of Theorems 1.3 and 1.4. The proof of Theorem 1.8 is given in Section 4.
2 Preliminaries
Let us denote by F the solution to the N-dimensional (N≥3) exterior problem
(2.1)
{
-
Δ
F
=
0
in
D
c
,
F
=
0
on
∂
D
,
F
(
x
)
→
1
as
|
x
|
→
∞
.
Then, by Poisson’s integral formula (see Dautray and Lions [2]), we have
F
(
x
)
=
1
-
1
|
x
|
N
-
2
+
1
σ
N
∫
∂
D
|
x
|
2
-
1
|
y
-
x
|
N
F
(
y
)
𝑑
ν
(
y
)
,
x
∈
D
c
,
where σN denotes the total surface area of the unit sphere in ℝN. Taking into consideration the boundary condition in (2.1), we get
(2.2)
F
(
x
)
=
1
-
1
|
x
|
N
-
2
,
x
∈
D
c
.
Moreover, we have
(2.3)
∇
F
(
x
)
=
(
N
-
2
)
|
x
|
N
x
,
x
∈
D
c
.
Further, let Φ∈C0∞([0,∞)) be a function satisfying
0
≤
Φ
≤
1
,
Φ
(
σ
)
=
{
1
if
0
≤
σ
≤
1
,
0
if
σ
≥
2
.
Given 0<T<∞, we introduce the functions
(2.4)
φ
(
t
,
x
)
=
ζ
T
(
t
)
ψ
T
(
x
)
,
(
t
,
x
)
∈
Q
T
,
and
(2.5)
𝒩
(
t
,
x
)
=
ζ
T
(
t
)
Ψ
T
(
x
)
,
(
t
,
x
)
∈
Q
T
,
where
(2.6)
ζ
T
(
t
)
=
T
-
λ
(
T
-
t
)
λ
,
λ
≫
1
,
0
≤
t
≤
T
,
(2.7)
ψ
T
(
x
)
=
F
(
x
)
Φ
ω
(
|
x
|
2
T
2
r
)
,
r
>
0
,
ω
≫
1
,
x
∈
D
c
,
(2.8)
Ψ
T
(
x
)
=
Φ
ω
(
|
x
|
2
T
2
r
)
,
r
>
0
,
ω
≫
1
,
x
∈
D
c
.
Using (2.2) and (2.3), we obtain the following formula.
Lemma 2.1.
We have
Δ
ψ
T
(
x
)
=
F
(
x
)
Δ
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
+
2
(
N
-
2
)
|
x
|
N
〈
x
,
∇
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
〉
,
x
∈
D
c
,
where 〈⋅,⋅〉 denotes the scalar product in RN.
Using Lemma 2.1 and the Cauchy–Schwarz inequality, we obtain the following estimate.
Lemma 2.2.
We have
|
Δ
ψ
T
(
x
)
|
≤
F
(
x
)
|
Δ
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
|
+
2
(
N
-
2
)
|
x
|
N
-
1
|
∇
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
|
,
x
∈
D
c
.
Using Lemma 2.2 and the inequality
(
a
+
b
)
m
≤
2
m
-
1
(
a
m
+
b
m
)
,
a
≥
0
,
b
≥
0
,
m
>
1
,
we obtain the following estimate.
Lemma 2.3.
Let m>1. We have
|
Δ
ψ
T
(
x
)
|
m
≤
C
m
(
F
m
(
x
)
|
Δ
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
|
m
+
1
|
x
|
(
N
-
1
)
m
|
∇
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
|
m
)
,
x
∈
D
c
,
where Cm=22m-1(N-2)m.
Lemma 2.4.
Let m>1 and m′=mm-1. We have
∫
Q
T
[
φ
(
t
,
x
)
]
-
1
m
-
1
|
Δ
φ
(
t
,
x
)
|
m
′
𝑑
x
𝑑
t
=
O
(
T
1
+
N
r
-
2
m
r
m
-
1
)
as
T
→
∞
.
Proof.
Using (2.4), we obtain
[
φ
(
t
,
x
)
]
-
1
m
-
1
|
Δ
φ
(
t
,
x
)
|
m
′
=
ζ
T
(
t
)
[
ψ
T
(
x
)
]
-
1
m
-
1
|
Δ
ψ
T
(
x
)
|
m
′
,
(
t
,
x
)
∈
Q
T
.
Therefore, we have
(2.9)
∫
Q
T
[
φ
(
t
,
x
)
]
-
1
m
-
1
|
Δ
φ
(
t
,
x
)
|
m
′
𝑑
x
𝑑
t
=
(
∫
0
T
ζ
T
(
t
)
𝑑
t
)
(
∫
D
c
[
ψ
T
(
x
)
]
-
1
m
-
1
|
Δ
ψ
T
(
x
)
|
m
′
𝑑
x
)
.
On the other hand, using (2.6), we obtain
(2.10)
∫
0
T
ζ
T
(
t
)
𝑑
t
=
T
λ
+
1
.
Further, using (2.7) and Lemma 2.3, for x∈Dc we obtain
[
ψ
T
(
x
)
]
-
1
m
-
1
|
Δ
ψ
T
(
x
)
|
m
′
≤
C
m
′
F
(
x
)
|
Δ
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
|
m
m
-
1
Φ
-
ω
m
-
1
(
|
x
|
2
T
2
r
)
+
C
m
′
|
x
|
-
(
N
-
1
)
m
m
-
1
[
F
(
x
)
]
-
1
m
-
1
Φ
-
ω
m
-
1
(
|
x
|
2
T
2
r
)
|
∇
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
|
m
m
-
1
.
Consequently, we obtain
(2.11)
∫
D
c
[
ψ
T
(
x
)
]
-
1
m
-
1
|
Δ
ψ
T
(
x
)
|
m
′
≤
I
1
+
I
2
,
where
(2.12)
I
1
=
C
m
′
∫
D
c
F
(
x
)
|
Δ
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
|
m
m
-
1
Φ
-
ω
m
-
1
(
|
x
|
2
T
2
r
)
𝑑
x
and
I
2
=
C
m
′
∫
D
c
|
x
|
-
(
N
-
1
)
m
m
-
1
[
F
(
x
)
]
-
1
m
-
1
Φ
-
ω
m
-
1
(
|
x
|
2
T
2
r
)
|
∇
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
|
m
m
-
1
𝑑
x
.
Further, we shall estimate Ii, i=1,2. Using (2.2), the definition of the function Φ and the change of variable y=xTr, for T large enough we obtain
I
1
≤
C
m
′
∫
D
c
|
Δ
[
Φ
ω
(
|
x
|
2
T
2
r
)
]
|
m
m
-
1
Φ
-
ω
m
-
1
(
|
x
|
2
T
2
r
)
𝑑
x
=
C
m
′
T
N
r
-
2
m
r
m
-
1
∫
ℝ
N
∖
B
(
0
,
T
-
r
)
¯
|
Δ
[
Φ
ω
(
|
y
|
2
)
]
|
m
m
-
1
Φ
-
ω
m
-
1
(
|
y
|
2
)
𝑑
y
=
C
m
′
T
N
r
-
2
m
r
m
-
1
∫
B
(
0
,
2
)
¯
∖
B
(
0
,
1
)
|
Δ
[
Φ
ω
(
|
y
|
2
)
]
|
m
m
-
1
Φ
-
ω
m
-
1
(
|
y
|
2
)
𝑑
y
.
Then, for T large enough, we have
(2.13)
I
1
≤
C
T
N
r
-
2
m
r
m
-
1
,
where
C
=
C
m
′
∫
B
(
0
,
2
)
¯
∖
B
(
0
,
1
)
|
Δ
[
Φ
ω
(
|
y
|
2
)
]
|
m
m
-
1
Φ
-
ω
m
-
1
(
|
y
|
2
)
𝑑
y
.
Similarly, for T large enough, we obtain
(2.14)
I
2
≤
C
′
T
N
r
-
m
r
N
m
-
1
,
where C′>0 is a certain constant. Next, combining (2.11), (2.13) and (2.14), for T large enough we obtain
(2.15)
∫
Ω
[
ψ
T
(
x
)
]
-
1
m
-
1
|
Δ
ψ
T
(
x
)
|
m
′
≤
2
max
{
C
,
C
′
}
T
N
r
-
2
m
r
m
-
1
.
Using (2.9), (2.10) and (2.15), the desired result follows. ∎
Lemma 2.5.
Let m>1 and m′=mm-1. We have
∫
Q
T
[
𝒩
(
t
,
x
)
]
-
1
m
-
1
|
Δ
𝒩
(
t
,
x
)
|
m
′
𝑑
x
𝑑
t
=
O
(
T
1
+
N
r
-
2
m
r
m
-
1
)
as
T
→
∞
.
Proof.
Using (2.5), we obtain
[
𝒩
(
t
,
x
)
]
-
1
m
-
1
|
Δ
𝒩
(
t
,
x
)
|
m
′
=
ζ
T
(
t
)
[
Ψ
T
(
x
)
]
-
1
m
-
1
|
Δ
Ψ
T
(
x
)
|
m
′
,
(
t
,
x
)
∈
Q
T
.
Therefore, we have
(2.16)
∫
Q
T
[
𝒩
(
t
,
x
)
]
-
1
m
-
1
|
Δ
𝒩
(
t
,
x
)
|
m
′
𝑑
x
𝑑
t
=
(
∫
0
T
ζ
T
(
t
)
𝑑
t
)
(
∫
D
c
[
Ψ
T
(
x
)
]
-
1
m
-
1
|
Δ
Ψ
T
(
x
)
|
m
′
𝑑
x
)
.
Further, by using (2.8), it can be easily seen that
∫
D
c
[
Ψ
T
(
x
)
]
-
1
m
-
1
|
Δ
Ψ
T
(
x
)
|
m
′
𝑑
x
has the same behavior as I1 as T→∞, where I1 is given by (2.12). Consequently, by combining (2.10), (2.13) and (2.16), the desired estimate follows. ∎
Lemma 2.6.
Let m>1 and m′=mm-1. We have
∫
Q
T
[
φ
(
t
,
x
)
]
-
1
m
-
1
|
∂
k
φ
∂
t
k
(
t
,
x
)
|
m
′
𝑑
x
𝑑
t
=
O
(
T
1
+
N
r
-
k
m
m
-
1
)
as
T
→
∞
.
Proof.
Using (2.4), we obtain
[
φ
(
t
,
x
)
]
-
1
m
-
1
|
∂
k
φ
∂
t
k
(
t
,
x
)
|
m
′
=
ψ
T
(
x
)
[
ζ
T
(
t
)
]
-
1
m
-
1
|
ζ
T
(
k
)
(
t
)
|
m
′
,
(
t
,
x
)
∈
Q
T
.
Whereupon, we get
(2.17)
∫
Q
T
[
φ
(
t
,
x
)
]
-
1
m
-
1
|
∂
k
φ
∂
t
k
(
t
,
x
)
|
m
′
𝑑
x
𝑑
t
=
(
∫
D
c
ψ
T
(
x
)
𝑑
x
)
(
∫
0
T
[
ζ
T
(
t
)
]
-
1
m
-
1
|
ζ
T
(
k
)
(
t
)
|
m
′
𝑑
t
)
.
On the other hand, using (2.2), (2.7), the change of variable y=xTr and the definition of the function Φ, for T large enough we have
∫
D
c
ψ
T
(
x
)
𝑑
x
=
∫
D
c
F
(
x
)
Φ
ω
(
|
x
|
2
T
2
r
)
𝑑
x
≤
∫
D
c
Φ
ω
(
|
x
|
2
T
2
r
)
𝑑
x
=
T
N
r
∫
ℝ
N
∖
B
(
0
,
T
-
r
)
¯
Φ
ω
(
|
y
|
2
)
𝑑
y
=
T
N
r
∫
B
(
0
,
2
)
∖
B
(
0
,
T
-
r
)
¯
Φ
ω
(
|
y
|
2
)
𝑑
y
≤
meas
(
B
(
0
,
2
)
)
T
N
r
,
which yields
(2.18)
∫
D
c
ψ
T
(
x
)
𝑑
x
=
O
(
T
N
r
)
as
T
→
∞
.
On the other hand, using (2.6), for t∈[0,T] we have
[
ζ
T
(
t
)
]
-
1
m
-
1
|
ζ
T
(
k
)
(
t
)
|
m
′
=
C
λ
,
k
T
-
λ
(
T
-
t
)
λ
-
k
m
m
-
1
,
where
C
λ
,
k
=
[
λ
(
λ
-
1
)
⋯
(
λ
-
k
+
1
)
]
m
′
,
which yields
∫
0
T
[
ζ
T
(
t
)
]
-
1
m
-
1
|
ζ
T
(
k
)
(
t
)
|
m
′
𝑑
t
=
C
λ
,
k
λ
-
k
m
m
-
1
+
1
T
1
-
k
m
m
-
1
.
Therefore,
(2.19)
∫
0
T
[
ζ
T
(
t
)
]
-
1
m
-
1
|
ζ
T
(
k
)
(
t
)
|
m
′
𝑑
t
=
O
(
T
1
-
k
m
m
-
1
)
as
T
→
∞
.
By using (2.17), (2.18) and (2.19), the desired estimate follows. ∎
Using an argument similar to the one above, we obtain the following estimate.
Lemma 2.7.
Let m>1 and m′=mm-1. We have
∫
Q
T
[
𝒩
(
t
,
x
)
]
-
1
m
-
1
|
∂
k
𝒩
∂
t
k
(
t
,
x
)
|
m
′
𝑑
x
𝑑
t
=
O
(
T
1
+
N
r
-
k
m
m
-
1
)
as
T
→
∞
.
Further, let us consider the following semilinear exterior Dirichlet problem:
(2.20)
{
-
Δ
u
=
u
p
in
D
c
,
u
(
x
)
=
f
(
x
)
in
∂
D
,
where f∈L∞(∂D), f≥0 and f≢0. The following result was established in [18].
Lemma 2.8.
Suppose p>NN-2 and ∥f∥L∞(∂D) sufficiently small. Then (2.20) has a positive solution.
Similarly, let us consider the following semilinear exterior Neumann problem:
(2.21)
{
-
Δ
u
=
u
p
in
D
c
,
∂
u
(
x
)
∂
n
+
=
f
(
x
)
in
∂
D
,
where f∈L∞(∂D), f≥0 and f≢0. The following result was established in [18].
Lemma 2.9.
Suppose p>NN-2 and ∥f∥L∞(∂D) is sufficiently small. Then (2.21) has a positive solution.
Finally, the notation
f
(
T
)
=
O
(
g
(
T
)
)
as
T
→
∞
,
means that there exists a constant C>0 and T0>0 such that
|
f
(
T
)
|
≤
C
|
g
(
T
)
|
,
T
≥
T
0
.
3 Critical Exponents for Problems (1.1) and (1.2)
In this section, we give the proofs of Theorems 1.3 and 1.4.
3.1 Proof of Theorem 1.3
We start by proving part (a) of the theorem. We argue by contradiction. Suppose that u is a global weak solution to (1.1). Let 0<T<∞ be large enough. We take as a test function the non-negative function φ given by (2.4). It can be easily seen that the considered function belongs to Ct,xk,2(QT)∩C(QT¯), and that it satisfies conditions (i)–(iv) in Definition 1.1. Therefore, by (1.10), we have
I
T
-
∫
[
0
,
T
]
×
∂
D
∂
φ
∂
n
+
f
𝑑
ν
(
x
)
𝑑
t
+
∑
i
=
1
k
(
-
1
)
i
+
1
∫
D
c
u
k
-
i
(
x
)
∂
i
-
1
φ
∂
t
i
-
1
(
0
,
x
)
𝑑
x
(3.1)
≤
∫
Q
T
|
u
|
|
∂
k
φ
∂
t
k
|
𝑑
x
𝑑
t
+
∫
Q
T
|
u
|
|
Δ
φ
|
𝑑
x
𝑑
t
,
where
I
T
=
∫
Q
T
|
u
|
p
φ
𝑑
x
𝑑
t
.
On the other hand, using (2.4) and the fact that ui≥0, i=0,1,…,k-1, for x∈Dc we obtain
(
-
1
)
i
+
1
u
k
-
i
(
x
)
∂
i
-
1
φ
∂
t
i
-
1
(
0
,
x
)
=
u
k
-
1
(
x
)
φ
(
0
,
x
)
≥
0
if
i
=
1
,
and
(
-
1
)
i
+
1
u
k
-
i
(
x
)
∂
i
-
1
φ
∂
t
i
-
1
(
0
,
x
)
=
[
λ
(
λ
-
1
)
⋯
(
λ
-
i
+
2
)
]
T
1
-
i
u
k
-
i
(
x
)
ψ
T
(
x
)
≥
0
if
2
≤
i
≤
k
.
Hence, we have
(3.2)
∑
i
=
1
k
(
-
1
)
i
+
1
∫
D
c
u
k
-
i
(
x
)
∂
i
-
1
φ
∂
t
i
-
1
(
0
,
x
)
𝑑
x
≥
0
.
Further, using (2.3) and (2.4), for (t,x)∈[0,T]×∂D we get
∂
φ
∂
n
+
(
t
,
x
)
=
-
ζ
T
(
t
)
Φ
ω
(
|
x
|
2
T
2
r
)
〈
∇
F
(
x
)
,
x
〉
=
-
(
N
-
2
)
Φ
ω
(
|
x
|
2
T
2
r
)
ζ
T
(
t
)
.
Hence, for T large enough, we get
-
∫
[
0
,
T
]
×
∂
D
∂
φ
∂
n
+
f
𝑑
ν
(
x
)
𝑑
t
=
(
N
-
2
)
(
∫
0
T
ζ
T
(
t
)
𝑑
t
)
(
∫
∂
D
f
(
x
)
𝑑
ν
(
x
)
)
.
Using (2.10), we obtain
(3.3)
-
∫
[
0
,
T
]
×
∂
D
∂
φ
∂
n
+
f
𝑑
ν
(
x
)
𝑑
t
=
C
T
,
where
C
=
(
N
-
2
)
λ
+
1
∫
∂
D
f
(
x
)
𝑑
ν
(
x
)
.
Note that as f∈C(∂D), f≥0 and f≢0, we have
0
<
C
<
∞
.
Further, combining (3.1), (3.2) and (3.3), for T large enough we obtain
(3.4)
I
T
+
C
T
≤
∫
Q
T
|
u
|
|
∂
k
φ
∂
t
k
|
𝑑
x
𝑑
t
+
∫
Q
T
|
u
|
|
Δ
φ
|
𝑑
x
𝑑
t
.
Writing
∫
Q
T
|
u
|
|
∂
k
φ
∂
t
k
|
𝑑
x
𝑑
t
=
∫
Q
T
|
u
|
φ
1
p
φ
-
1
p
|
∂
k
φ
∂
t
k
|
𝑑
x
𝑑
t
and using Hölder’s inequality with parameters p and p′=pp-1, we obtain
∫
Q
T
|
u
|
|
∂
k
φ
∂
t
k
|
𝑑
x
𝑑
t
≤
I
T
1
p
(
∫
Q
T
φ
-
1
p
-
1
|
∂
k
φ
∂
t
k
|
p
′
𝑑
x
𝑑
t
)
p
-
1
p
.
Using Lemma 2.6, we get
(3.5)
∫
Q
T
|
u
|
|
∂
k
φ
∂
t
k
|
𝑑
x
𝑑
t
≤
I
T
1
p
O
(
T
(
1
+
N
r
)
(
p
-
1
)
p
-
k
)
as
T
→
∞
.
Similarly, writing
∫
Q
T
|
u
|
|
Δ
φ
|
𝑑
x
𝑑
t
=
∫
Q
T
|
u
|
φ
1
p
φ
-
1
p
|
Δ
φ
|
𝑑
x
𝑑
t
and using Hölder’s inequality, we get
∫
Q
T
|
u
|
|
Δ
φ
|
𝑑
x
𝑑
t
≤
I
T
1
p
(
∫
Q
T
φ
-
1
p
-
1
|
Δ
φ
|
p
′
𝑑
x
𝑑
t
)
p
-
1
p
.
Using Lemma 2.4, we obtain
(3.6)
∫
Q
T
|
u
|
|
Δ
φ
|
𝑑
x
𝑑
t
≤
I
T
1
p
O
(
T
(
1
+
N
r
)
(
p
-
1
)
p
-
2
r
)
as
T
→
∞
.
Combining (3.4), (3.5) and (3.6), we obtain
(3.7)
I
T
+
C
T
≤
I
T
1
p
[
O
(
T
(
1
+
N
r
)
(
p
-
1
)
p
-
k
)
+
O
(
T
(
1
+
N
r
)
(
p
-
1
)
p
-
2
r
)
]
as
T
→
∞
.
Observe that for r=k2 we have
(
1
+
N
r
)
(
p
-
1
)
p
-
k
=
(
1
+
N
r
)
(
p
-
1
)
p
-
2
r
=
(
1
+
N
k
2
)
(
p
-
1
)
p
-
k
.
Therefore, taking r=k2 in (3.7), we deduce that, for T large enough,
(3.8)
I
T
+
C
T
≤
C
′
T
μ
I
T
1
p
,
where C′>0 and
(3.9)
μ
=
(
1
+
N
k
2
)
(
p
-
1
)
p
-
k
.
On the other hand, writing
C
′
T
μ
I
T
1
p
=
(
p
1
p
I
T
1
p
)
(
p
-
1
p
C
′
T
μ
)
and using Young’s inequality, we get
(3.10)
C
′
T
μ
I
T
1
p
≤
I
T
+
C
~
T
μ
p
′
,
where C~>0 is a constant (independent of T) and p′=pp-1. Further, combining (3.8) with (3.10), for T large enough we get
(3.11)
0
<
C
C
~
≤
T
μ
p
′
-
1
.
Observe that when
1
<
p
<
N
N
-
2
,
by (3.9) we have
μ
p
′
-
1
<
0
.
Hence, passing to the limit as T→∞ in (3.11), we get
0
<
C
C
~
≤
0
,
which is a contradiction. This proves part (a) of the theorem.
Now, we prove part (b). Suppose that p>NN-2. Given f∈L∞(∂D) and f≥0 such that ∥f∥L∞(∂D) is sufficiently small, from Lemma 2.8 we know that there exists a positive solution v to (2.20). Let
u
(
t
,
x
)
=
v
(
x
)
,
(
t
,
x
)
∈
(
0
,
∞
)
×
D
c
.
Then it can be easily seen that u is a solution to (1.1) with
u
0
(
x
)
=
v
(
x
)
,
u
i
(
x
)
=
0
,
i
=
1
,
2
,
…
,
k
-
1
.
This proves part (b) of the theorem.
3.2 Proof of Theorem 1.4
We start by proving part (a) of the theorem. We argue by contradiction. Suppose that u is a global weak solution to (1.2). Let 0<T<∞ be large enough. We take as a test function the non-negative function 𝒩 given by (2.5). It can be easily seen that the considered function belongs to Ct,xk,2(QT)∩C(QT¯) and satisfies conditions (i)–(iv) in Definition 1.2. Therefore, by (1.11), we have
J
T
+
∫
[
0
,
T
]
×
∂
D
f
𝒩
𝑑
ν
(
x
)
𝑑
t
+
∑
i
=
1
k
(
-
1
)
i
+
1
∫
D
c
u
k
-
i
(
x
)
∂
i
-
1
𝒩
∂
t
i
-
1
(
0
,
x
)
𝑑
x
(3.12)
≤
(
-
1
)
k
∫
Q
T
u
∂
k
𝒩
∂
t
k
𝑑
x
𝑑
t
-
∫
Q
T
u
Δ
𝒩
𝑑
x
𝑑
t
,
where
(3.13)
J
T
=
∫
Q
T
|
u
|
p
𝒩
𝑑
x
𝑑
t
.
On the other hand, using (2.5) and (2.10), for T large enough we have
(3.14)
∫
[
0
,
T
]
×
∂
D
f
𝒩
𝑑
ν
(
x
)
𝑑
t
=
(
∫
0
T
ζ
T
(
t
)
𝑑
t
)
(
∫
∂
D
f
(
x
)
𝑑
ν
(
x
)
)
=
C
T
,
where
C
=
1
λ
+
1
∫
∂
D
f
(
x
)
𝑑
ν
(
x
)
>
0
.
Next, using (3.12), (3.14) and the fact that
∑
i
=
1
k
(
-
1
)
i
+
1
∫
D
c
u
k
-
i
(
x
)
∂
i
-
1
𝒩
∂
t
i
-
1
(
0
,
x
)
𝑑
x
≥
0
,
for T large enough we obtain
(3.15)
J
T
+
C
T
≤
∫
Q
T
|
u
|
|
∂
k
𝒩
∂
t
k
|
𝑑
x
𝑑
t
+
∫
Q
T
|
u
|
|
Δ
𝒩
|
𝑑
x
𝑑
t
.
Further, using an argument similar to that of the proof of Theorem 1.3 (a) and using Lemmas 2.5 and 2.7, we obtain
(3.16)
∫
Q
T
|
u
|
|
∂
k
𝒩
∂
t
k
|
𝑑
x
𝑑
t
≤
J
T
1
p
O
(
T
(
1
+
N
r
)
(
p
-
1
)
p
-
k
)
as
T
→
∞
and
(3.17)
∫
Q
T
|
u
|
|
Δ
𝒩
|
𝑑
x
𝑑
t
≤
J
T
1
p
O
(
T
(
1
+
N
r
)
(
p
-
1
)
p
-
2
r
)
as
T
→
∞
.
Combining (3.15), (3.16) and (3.17) and taking r=k2, we obtain
J
T
+
C
T
≤
C
′
T
μ
J
T
1
p
,
where C′>0 and μ is given by (3.9). The rest of the proof is exactly the same as that of Theorem 1.3 (a).
Finally, part (b) of the theorem can be proved using Lemma 2.9 and an argument similar to that used in the proof of Theorem 1.3 (b).
4 The Influence of the Initial Data
In this section, we give the proof of Theorem 1.8 .
4.1 Proof of Theorem 1.8 (a)
Let j∈{1,2,…,k} be such that (1.13) is satisfied. Suppose that u is a global weak solution to (1.1). Using (3.1), we obtain
I
T
-
∫
[
0
,
T
]
×
∂
D
∂
φ
∂
n
+
f
𝑑
ν
(
x
)
𝑑
t
+
(
-
1
)
j
+
1
∫
D
c
u
k
-
j
(
x
)
∂
j
-
1
φ
∂
t
j
-
1
(
0
,
x
)
𝑑
x
≤
∫
Q
T
|
u
|
|
∂
k
φ
∂
t
k
|
𝑑
x
𝑑
t
+
∫
Q
T
|
u
|
|
Δ
φ
|
𝑑
x
𝑑
t
.
Since
-
∫
[
0
,
T
]
×
∂
D
∂
φ
∂
n
+
f
𝑑
ν
(
x
)
𝑑
t
≥
0
,
we obtain
(4.1)
I
T
+
(
-
1
)
j
+
1
∫
D
c
u
k
-
j
(
x
)
∂
j
-
1
φ
∂
t
j
-
1
(
0
,
x
)
𝑑
x
≤
∫
Q
T
|
u
|
|
∂
k
φ
∂
t
k
|
𝑑
x
𝑑
t
+
∫
Q
T
|
u
|
|
Δ
φ
|
𝑑
x
𝑑
t
.
On the other hand, using (2.4), we have
(
-
1
)
j
+
1
u
k
-
j
(
x
)
∂
j
-
1
φ
∂
t
j
-
1
(
0
,
x
)
=
C
j
T
1
-
j
u
k
-
j
(
x
)
ψ
T
(
x
)
,
x
∈
D
c
,
where Cj>0 is a certain constant (independent of T). Therefore, using (2.7), we obtain
(
-
1
)
j
+
1
∫
D
c
u
k
-
j
(
x
)
∂
j
-
1
φ
∂
t
j
-
1
(
0
,
x
)
𝑑
x
=
C
j
T
1
-
j
∫
D
c
u
k
-
j
(
x
)
F
(
x
)
Φ
ω
(
|
x
|
2
T
2
r
)
𝑑
x
.
Using the change of variable y=xTr and the definition of Φ, we get
(
-
1
)
j
+
1
∫
D
c
u
k
-
j
(
x
)
∂
j
-
1
φ
∂
t
j
-
1
(
0
,
x
)
𝑑
x
=
C
j
T
1
-
j
+
N
r
∫
T
-
r
<
|
y
|
<
2
u
k
-
j
(
T
r
y
)
F
(
T
r
y
)
Φ
ω
(
|
y
|
2
)
𝑑
y
.
On the other hand, by (1.13), we know that
u
k
-
j
(
x
)
≥
A
|
x
|
γ
,
|
x
|
>
ℛ
,
for some ℛ>1. Hence, for T large enough, we get
(
-
1
)
j
+
1
∫
D
c
u
k
-
j
(
x
)
∂
j
-
1
φ
∂
t
j
-
1
(
0
,
x
)
𝑑
x
≥
A
C
j
T
1
-
j
+
N
r
+
r
γ
∫
T
-
r
ℛ
<
|
y
|
<
2
F
(
T
r
y
)
|
y
|
γ
Φ
ω
(
|
y
|
2
)
𝑑
y
.
Using the dominated convergence theorem and the fact that -N<γ, we obtain
lim
T
→
∞
∫
T
-
r
ℛ
<
|
y
|
<
2
F
(
T
r
y
)
|
y
|
γ
Φ
ω
(
|
y
|
2
)
𝑑
y
=
L
Φ
,
where
0
<
L
Φ
=
∫
0
<
|
y
|
<
2
|
y
|
γ
Φ
ω
(
|
y
|
2
)
𝑑
y
<
∞
.
Therefore, for T large enough, we have
(4.2)
(
-
1
)
j
+
1
∫
D
c
u
k
-
j
(
x
)
∂
j
-
1
φ
∂
t
j
-
1
(
0
,
x
)
𝑑
x
≥
C
T
1
-
j
+
N
r
+
r
γ
,
where C>0 is a constant. Further, combining (4.1) with (4.2), for T large enough we get
I
T
+
C
T
1
-
j
+
N
r
+
r
γ
≤
∫
Q
T
|
u
|
|
∂
k
φ
∂
t
k
|
𝑑
x
𝑑
t
+
∫
Q
T
|
u
|
|
Δ
φ
|
𝑑
x
𝑑
t
.
Next, taking r=k2 and using estimates (3.5) and (3.6), we obtain
(4.3)
I
T
+
C
T
ν
≤
C
′
T
μ
I
T
1
p
,
where μ is given by (3.9) and
(4.4)
ν
=
1
-
j
+
N
k
2
+
k
γ
2
.
Further, using Young’s inequality, we get
(4.5)
C
′
T
μ
I
T
1
p
≤
I
T
+
C
~
T
μ
p
′
,
where p′=pp-1. Combining (4.3) with (4.5), for T large enough we obtain
(4.6)
0
<
C
C
~
≤
T
μ
p
′
-
ν
.
Observe that under condition (1.14) we have
μ
p
′
-
ν
<
0
.
Hence, passing to the limit as T→∞ in (4.6), we get a contradiction. This proves Theorem 1.8 (a).
4.2 Proof of Theorem 1.8 (b)
Let j∈{1,2,…,k} be such that (1.13) is satisfied. Suppose that u is a global weak solution to (1.2). Using an argument similar to the one in the proof of Theorem 1.8 (a), for T large enough we obtain
J
T
+
C
T
ν
≤
C
′
T
μ
I
T
1
p
,
where JT is given by (3.13), μ is given by (3.9) and ν is given by (4.4). The rest of the proof is exactly the same as that of Theorem 1.8 (a).
