1 Introduction
In this paper, we study the long-time behavior of solutions to the nonlinear heat equation with absorption,
(1.1)
u
t
-
Δ
u
+
|
u
|
α
u
=
0
,
where u=u(t,x)∈ℝ, (t,x)∈(0,∞)×ℝN and α>0, which are antisymmetric with respect to the variables x1,x2,…,xm for some m∈{1,2,…,N}. Our goal is to see how some well-known results [1, 3, 5, 6] for the long-time behavior of solutions to (1.1) carry over with the additional hypothesis of antisymmetry. For example, some of the results in the cited works concern positive solutions. We will see that these results have analogues for antisymmetric solutions which are positive on an appropriate sector in ℝN. In particular, these solutions are not positive on ℝN. Moreover, in many cases, the range of allowable powers α>0 will be larger with the additional hypothesis of antisymmetry than without. Also, the condition of antisymmetry allows consideration of a class of highly singular initial values.
Our previous paper [9] considered the linear heat equation on ℝN with antisymmetric solutions. The results and the theoretical framework from [9] were applied to the nonlinear heat equation with source term
(1.2)
u
t
-
Δ
u
-
|
u
|
α
u
=
0
in [12]. In the current paper, these ideas are applied to (1.1). We mention that this approach was earlier developed in [11], where solutions to (1.2) with antisymmetric initial values of the form u0=(-1)m∂1∂2⋯∂mδ were studied. In the current paper, as in [9, 12], initial values of the form u0=(-1)m∂1∂2⋯∂m|⋅|-γ for some 0<γ<N are considered.
In order to state our results precisely, we begin by recalling the definition of an antisymmetric function.
Definition 1.1.
Let m∈{1,2,…,N}. A function f:ℝN→ℝ is antisymmetric with respect to x1,…,xm if it satisfies
(1.3)
T
1
f
=
T
2
f
=
⋯
=
T
m
f
=
-
f
,
where Ti, i∈{1,2,…,N}, denotes the operator
[
T
i
f
]
(
x
1
,
…
,
x
i
-
1
,
x
i
,
x
i
+
1
,
…
,
x
N
)
=
f
(
x
1
,
…
,
x
i
-
1
,
-
x
i
,
x
i
+
1
,
…
,
x
N
)
.
We denote the set of functions antisymmetric with respect to x1,…,xm by
𝒜
=
𝒜
m
=
{
f
:
ℝ
N
→
ℝ
;
f
satisfies (1.3)
}
.
A function on ℝN which is antisymmetric with respect to x1,x2,…,xm for some m∈{1,2,…,N} is determined by its values on Ωm, the sector of ℝN defined by
Ω
m
=
{
(
x
1
,
x
2
,
…
,
x
N
)
∈
ℝ
N
;
x
1
>
0
,
x
2
>
0
,
…
,
x
m
>
0
}
.
Note that, by definition, an antisymmetric function must take the value 0 on the boundary ∂Ωm. Since the operators Ti defined above commute with the operations in equation (1.1), the study of antisymmetric solutions to (1.1) reduces to the study of solutions on Ωm with Dirichlet boundary conditions. This point is discussed in detail in [12, Section 3], and that discussion applies as well to the heat equation with absorption. Moreover, as in [12], we will construct certain classes of antisymmetric solutions to (1.1) on ℝN by constructing solutions on Ωm and extending them to ℝN by antisymmetry.
Since both the present paper and [12] are based on the framework developed in [9], we need to recall some definitions and notation used in [9]. Let ρm be the weight function defined on Ωm by
ρ
m
(
x
)
=
|
x
|
γ
+
2
m
x
1
⋯
x
m
for all
x
∈
Ω
m
,
where 0<γ<N. We consider the Banach space
𝒳
m
,
γ
=
{
ψ
:
Ω
m
→
ℝ
,
ρ
m
ψ
∈
L
∞
(
Ω
m
)
}
,
endowed with the norm ∥ψ∥𝒳m,γ=∥ρmψ∥L∞(Ωm) for all ψ∈𝒳m,γ. The closed ball of radius M on 𝒳m,γ is denoted by
ℬ
m
,
γ
,
M
=
{
ψ
∈
𝒳
m
,
γ
such that
∥
ψ
∥
𝒳
m
,
γ
≤
M
}
.
As observed in [9, p. 344], ℬm,γ,M⋆ the closed ball ℬm,γ,M endowed with the weak * topology of 𝒳m,γ is a compact metric space (hence complete and separable).
Let σ>0. For each λ>0, we let Dλσ denote the dilation operator defined by
(1.4)
D
λ
σ
u
(
x
)
=
λ
σ
u
(
λ
x
)
,
where u is a function defined on Ωm, or on ℝN. A function ψ:Ωm→ℝ is homogeneous of degree -σ if Dλσψ=ψ for all λ>0. The operators Dλσ, λ>0, act on the spaces 𝒳m,γ, but leave the norm invariant, i.e. leave the ball ℬm,γ,M invariant, if and only if σ=γ+m. In fact, we have
(1.5)
∥
D
λ
σ
ψ
∥
𝒳
m
,
γ
=
λ
σ
∥
ρ
m
ψ
(
λ
⋅
)
∥
L
∞
(
Ω
m
)
=
λ
σ
-
(
γ
+
m
)
∥
ρ
m
(
λ
⋅
)
ψ
(
λ
⋅
)
∥
L
∞
(
Ω
m
)
=
λ
σ
-
(
γ
+
m
)
∥
ψ
∥
𝒳
m
,
γ
for all λ>0 and ψ∈𝒳m,γ.
The function ψ0 defined on Ωm by
(1.6)
ψ
0
(
x
)
=
c
m
,
γ
(
ρ
m
(
x
)
)
-
1
=
c
m
,
γ
x
1
⋯
x
m
|
x
|
-
γ
-
2
m
,
x
∈
Ω
m
,
where cm,γ=γ(γ+2)⋯(γ+2m-2), will play a central role. It is homogeneous of degree -(γ+m), belongs to 𝒳m,γ, and satisfies ∥ψ0∥𝒳m,γ=cm,γ. Moreover, we have Dλσψ0(x)=λσ-(γ+m)ψ0(x) for all σ,λ>0, and so ∥Dλσψ0∥𝒳m,γ=λσ-(γ+m)cm,γ. Its interest lies in the fact that
ψ
0
(
x
)
=
(
-
1
)
m
∂
1
∂
2
⋯
∂
m
(
|
x
|
-
γ
)
,
x
∈
Ω
m
.
The heat semigroup on Ωm, denoted etΔm, is given by
(1.7)
e
t
Δ
m
ψ
(
x
)
=
∫
Ω
m
K
t
(
x
,
y
)
ψ
(
y
)
d
y
for all t>0, where
(1.8)
K
t
(
x
,
y
)
=
(
4
π
t
)
-
N
2
∏
j
=
m
+
1
N
e
-
|
x
j
-
y
j
|
2
4
t
∏
i
=
1
m
[
e
-
|
x
i
-
y
i
|
2
4
t
-
e
-
|
x
i
+
y
i
|
2
4
t
]
.
See, for example, [11, Proposition 3.1, p. 514]. It is well known that etΔm is a C0 semigroup on C0(Ωm), the space of continuous functions f:Ω¯m→ℝ such that f≡0 on the boundary ∂Ωm and f(x)→0 as |x|→∞ in Ωm. It is also well defined on 𝒳m,γ, and etΔm:𝒳m,γ→C0(Ωm)∩𝒳m,γ is continuous for all t>0. See [9, Theorem 1.1, p. 343]. We recall the commutation relation between etΔm and the operators Dλσ,
(1.9)
D
λ
σ
e
λ
2
t
Δ
m
=
e
t
Δ
m
D
λ
σ
for all λ>0 and σ>0, and for future use, we note the following identity, which is immediate to verify:
(1.10)
∫
Ω
m
K
t
(
x
,
y
)
y
1
⋯
y
m
d
y
=
x
1
⋯
x
m
for all t>0 and all x∈Ωm.
In terms of behavior on the sectors Ωm, our goal is to study the well-posedness of equation (1.1) on the space 𝒳m,γ and to obtain results on the large time behavior of solutions in the three cases α=2γ+m, α>2γ+m, and α<2γ+m. By interpreting these results for antisymmetric solutions on ℝN, we will extend some known results, [3, Theorem 1.3, Theorem 1.4] and [6], in the case m=0. We now describe these results in detail.
In Section 2, we consider the Cauchy problem
(1.11)
{
u
t
-
Δ
u
+
|
u
|
α
u
=
0
,
u
(
0
)
=
u
0
∈
𝒳
m
,
γ
.
It is well known that, given any u0∈C0(ℝN), there exists a unique function u∈C([0,∞),C0(ℝN)) which is a classical solution of (1.1) on ℝN for t>0 and such that u(0)=u0, which we denote by
(1.12)
u
(
t
)
=
𝒮
(
t
)
u
0
,
where u(t)=u(t,⋅). Likewise, for any u0∈C0(Ωm), there exists a unique function u∈C([0,∞),C0(Ωm)) which is a classical solution of (1.1) for t>0 and such that u(0)=u0. This defines a global semiflow 𝒮m(t) on C0(Ωm). In other words,
(1.13)
𝒮
m
(
t
)
u
0
=
u
(
t
)
,
where u(t)=u(t,⋅) is the solution of (1.1) with initial value u0∈C0(Ωm). In fact, existence and uniqueness of solutions in C0(Ωm) follows from the existence and uniqueness of solutions in u0∈C0(ℝN) since 𝒮(t) preserves antisymmetry; it suffices to consider the antisymmetric extension of u0∈C0(Ωm) to an element of C0(ℝN)∩𝒜.
Similarly, given any u0∈Lq(Ωm), 1≤q<∞, we deduce by Kato’s parabolic inequality (see Lemma A.1 and Corollary A.2 in the appendix) and the fact that 𝒟(Ωm) is dense in Lq(Ωm) that there exists a unique u∈C([0,∞),Lq(Ωm)) which is a classical solution of (1.1) for t>0 and such that u(0)=u0. Alternatively, see [6, Proposition 1.1, p. 261] for a proof using accretive operators. Again by preservation of antisymmetry, the result of [6], valid for ℝN, holds also on Ωm. Thus, the semiflow 𝒮m(t) extends to Lq(Ωm), and formula (1.13) is valid also for u0∈Lq(Ωm).
Here we consider initial data u0∈𝒳m,γ. Our first main result is the following.
Theorem 1.2.
Let m∈{1,2,…,N}, 0<γ<N and α>0. If u0∈Xm,γ, then there exists a unique solution u∈C((0,∞),C0(Ωm)) of equation (1.1) such that
u
(
t
)
→
u
0
in
L
loc
1
(
Ω
m
)
as
t
→
0
,
there exists
C
>
0
, independent of
u
0
, such that
∥
u
(
t
)
∥
𝒳
m
,
γ
≤
C
∥
u
0
∥
𝒳
m
,
γ
for all
t
>
0
.
In addition, the following properties hold.
For all
v
0
∈
𝒳
m
,
γ
, |u(t)-v(t)|≤etΔm|u0-v0|, where v is the solution of (1.1) with initial value v0 satisfying (i) and (ii).
There exists
C
>
0
such that
|
u
(
t
,
x
)
|
≤
C
x
1
⋯
x
m
(
t
+
|
x
|
2
)
-
γ
+
2
m
2
∥
u
0
∥
𝒳
m
,
γ
for all
t
>
0
and for all
u
0
∈
𝒳
m
,
γ
.
The solution
u
(
t
)
satisfies the integral equation
(1.14)
u
(
t
)
=
e
t
Δ
m
u
0
-
∫
0
t
e
(
t
-
s
)
Δ
m
(
|
u
(
s
)
|
α
u
(
s
)
)
d
s
for all
t
>
0
, where the integrand is in
L
1
(
(
0
,
t
)
;
C
0
(
Ω
m
)
)
.
If
v
0
∈
𝒳
m
,
γ
, v0≥0, and |u0|≤v0, then |u(t)|≤v(t), where v is the solution of (1.1) with initial value v0 satisfying (i) and (ii).
In other words, the nonlinear operators 𝒮m(t), t>0, extend in a natural way to 𝒳m,γ. We remark that, in the case α<2γ+m, this well-posedness result was established in [12, Theorems 2.3 and 2.6] by a different method and with plus and minus sign in the term of the nonlinearity. Furthermore, the analogous results on the whole space ℝN follow from [1, Theorem 8.8, p. 536].
Definition 1.3.
Let m∈{1,2,…,N}, 0<γ<N and α>0. Given u0∈𝒳m,γ we set 𝒮m(t)u0=u(t) for all t>0, where u∈C((0,∞),C0(Ωm)) is the unique solution of (1.1) satisfying (i) and (ii) of Theorem 1.2.
We also establish the continuous dependence properties of solutions of equation (1.1) with initial values in 𝒳m,γ.
Theorem 1.4.
Let m∈{1,2,…,N}, 0<γ<N and M>0. It follows that Sm(t) is continuous Bm,γ,M⋆→C0(Ωm) for all t>0, where Bm,γ,M⋆ denotes the compact metric space topology induced by the weak * topology on Bm,γ,M.
It is well known that any solution u(t) of (1.1), for example, as constructed in Theorem 1.2, is always bounded by the spatially independent solution; more precisely,
(1.15)
|
u
(
t
,
x
)
|
≤
(
1
α
t
)
1
/
α
for all t>0, throughout the spatial domain of existence. See, for example, [6, p. 261]. In addition, it is clear from Theorem 1.2 that if u is the solution of (1.1) with positive initial data u0≥0, then
(1.16)
u
(
t
)
≤
e
t
Δ
m
u
0
for any t>0. We have the following upper estimate for solutions of (1.1) that combines (1.16) and (1.15) into one estimate which implies them both. Its proof is given in Section 3.
Proposition 1.5.
Let N≥1, m∈{1,…,N}, 0<γ<N and α>0. Let u0∈Xm,γ, u0≥0. Then the solution u of (1.1) with initial data u(0)=u0 satisfies the upper estimate
u
(
t
,
x
)
≤
e
t
Δ
m
u
0
(
x
)
(
1
+
α
t
(
e
t
Δ
m
u
0
(
x
)
)
α
)
1
/
α
for all t>0 and all x∈Ωm.
After proving global well-posedness of Cauchy problem (1.11), i.e. Theorems 1.2 and 1.4, we seek to describe the large time behavior of solutions of (1.1) on Ωm with initial values in 𝒳m,γ. Our basic approach is to study the effect of certain space-time dilations on such a solution, and to relate the resulting behavior to the effect of related spatial dilations on the initial value. In particular, we consider the space-time dilation operators Γλσ, λ>0, defined by
(1.17)
Γ
λ
σ
u
(
t
,
x
)
=
λ
σ
u
(
λ
2
t
,
λ
x
)
=
D
λ
σ
[
u
(
λ
2
t
)
]
(
x
)
for all λ,σ>0, where the spatial dilation operator Dλσ is given by (1.4). If u∈C((0,∞),C0(Ωm)) is solution of equation (1.1), then Γλσu is a solution of (1.1) if and only if σ=2α. Moreover, if a solution u has initial value u0, either in the sense of C0(Ωm) or in some more general sense, then Γλ2/αu has initial value Dλ2/αu0. If u0∈𝒳m,γ, the function Dλ2/αu0 belongs to 𝒳m,γ for all λ>0, and the uniqueness of solutions of (1.1) implies that Γλ2/αu coincides with 𝒮m(⋅)Dλ2/αu0. Thus, we have the relation
(1.18)
Γ
λ
2
/
α
[
𝒮
m
(
⋅
)
u
0
]
=
𝒮
m
(
⋅
)
[
D
λ
2
/
α
u
0
]
for all u0∈𝒳m,γ. We emphasize that, at this point, there is no assumed relationship between α and m. Formula (1.18) holds for any semiflow generated by (1.1) in place of 𝒮m(⋅), as long as the space of initial values is invariant under the dilations Dλ2/α and initial values give rise to unique solutions.
A solution u of (1.1) is self-similar if Γλ2/αu=u for all λ>0 or, equivalently, if
u
(
t
,
x
)
=
t
-
1
/
α
f
(
x
t
)
=
D
1
t
2
/
α
f
(
x
)
,
where f(x)=u(1,x) is called the profile of u. It follows that if a self-similar solution u of (1.1) has initial value u0, then Dλ2/αu0=u0 for all λ>0, i.e. u0 is homogeneous of degree -2α. Conversely, if u0 is homogeneous of degree -2α and u(t) is a solution with initial value u0 in some appropriate sense, then Γλ2/αu has the same initial value for all λ>0. Assuming that uniqueness of solutions having a given initial value has been proved in the appropriate class of functions, one then concludes that u=Γλ2/αu for all λ>0, i.e. that u is a self-similar solution.
More generally, we say that a solution u of (1.1) is asymptotically self-similar if
(1.19)
lim
λ
→
∞
Γ
λ
2
/
α
u
=
U
,
in some appropriate sense, and that U is also a solution to (1.1). If so, the limit is necessarily a self-similar solution. See [5, Section 3] for a discussion of several equivalent definitions of asymptotically self-similar solutions. Formally, if we put t=0 in (1.19), we obtain
(1.20)
lim
λ
→
∞
D
λ
2
/
α
u
0
=
φ
,
where φ=U(0) is homogeneous of degree -2α. In Section 4, we study the long-time asymptotic behavior of solutions to (1.1) with initial values in 𝒳m,γ in the case α=2γ+m. The first result shows that (1.20) implies (1.19).
Theorem 1.6.
Let m∈{1,2,…,N}, 0<γ<N and ψ∈Bm,γ,M. Let α>0 be such that α=2γ+m. Suppose that there exists φ∈Bm,γ,M such that limλ→∞Dλγ+mψ=φ in Bm,γ,M⋆. It follows that φ is homogeneous of degree -(γ+m) and that the solution u(t)=Sm(t)ψ is asymptotically self-similar to the self-similar solution U(t)=Sm(t)φ.
As is by now well established [4, 3, 5], the notion of asymptotically self-similar solution can be naturally extended by allowing different limits in (1.20) and (1.19) along different sequences (λn)n≥0 with λn→∞. The next step in our analysis is to generalize Theorem 1.6 in this fashion. To accomplish this, for u0∈𝒳m,γ and M≥∥u0∥𝒳m,γ, we consider the set of all accumulation points of Dλγ+mu0 as λ→∞, given by
(1.21)
𝒵
γ
(
u
0
)
=
{
z
∈
ℬ
m
,
γ
,
M
;
there exists
λ
n
→
∞
such that
lim
n
→
∞
D
λ
n
γ
+
m
u
0
=
z
in
ℬ
m
,
γ
,
M
⋆
}
.
Since ℬm,γ,M⋆ is a compact metric space, 𝒵γ(u0) is a nonempty compact subset for all u0∈𝒳m,γ, and independent of M≥∥u0∥𝒳m,γ by [9, Proposition 3.1, p. 356]. In particular, if u0 is homogeneous of degree -(γ+m), then 𝒵γ(u0)={u0}. We set u(t)=𝒮m(t)u0, and we also define the omega-limit set of all accumulation points of Γtγ+mu(1,⋅)=tγ+m2u(t,t⋅) as t→∞ by
(1.22)
𝒬
γ
(
u
0
)
=
{
f
∈
C
0
(
Ω
m
)
;
there exists
t
n
→
∞
such that
lim
n
→
∞
∥
Γ
t
n
γ
+
m
𝒮
m
(
1
)
u
0
-
f
∥
L
∞
(
Ω
m
)
=
0
}
.
Relation (1.18) and Theorem 1.4 are the essential elements needed to investigate the relationship between 𝒬γ(u0) and 𝒵γ(u0), which is given by our next main result.
Theorem 1.7.
Let m∈{1,2,…,N}, 0<γ<N and α>0 such that α=2γ+m. If u0∈Xm,γ, then
𝒬
γ
(
u
0
)
=
𝒮
m
(
1
)
𝒵
γ
(
u
0
)
.
In particular, Qγ(u0)⊂Sm(1)Bm,γ,M⋆ and is therefore a compact subset of C0(Ωm).
The last relation shows that, in the case α=2γ+m, the complexity in the large time behavior of a solution, as expressed in 𝒬γ(u0), is determined by the complexity in the spatial asymptotic behavior of its initial value as expressed in 𝒵γ(u0). Furthermore, Theorem 1.7 above is inspired from [3, Theorem 1.3, p. 83] which requires α≥2N, and we observe that, in Theorem 1.7, if γ+m>N, then α<2N. Since ℬm,γ,M⋆ is separable and 𝒵γ(u0) can contain any countable subset of ℬm,γ,M⋆, we show that 𝒵γ(U0)=ℬm,γ,M⋆ for some choice of U0∈ℬm,γ,M⋆.
Using [9, Theorem 1.4, p. 345], we obtain the following result.
Corollary 1.8.
Let m∈{1,2,…,N}, 0<γ<N and M>0. Let α>0 be such that α=2γ+m. Then there exists
U
0
∈
ℬ
m
,
γ
,
M
∩
C
∞
(
Ω
m
)
∩
C
0
(
Ω
m
)
such that Qγ(U0)=Sm(1)Bm,γ,M.
Remark 1.9.
If u0 belongs to 𝒳m,γ∩𝒳m,γ′ with γ<γ′<N, then 𝒵γ(u0)={0}. In fact, for all λ>0,
|
D
λ
γ
+
m
u
0
(
x
)
|
=
λ
γ
+
m
|
u
0
(
λ
x
)
|
≤
C
λ
γ
-
γ
′
|
x
|
-
(
γ
′
+
m
)
→
0
as λ→∞ uniformly on {x∈Ωm;|x|≥ε} for all ε>0. Thus, 𝒵γ(u0)={0}. For example, if α=2γ+m>2γ′+m, the function φ(x)=x1⋯xm|x|-γ′-2m𝟙{|x|>1}∈𝒳m,γ′∩𝒳m,γ. It follows from Theorem 1.7 that 𝒬γ(φ)={0}. However, we might have 𝒬γ′(φ)≠{0}.
In Section 5 of this paper, we consider the case α>2γ+m. Since α≠2γ+m, there is a disconnect between the transformations which preserve the set of solutions to (1.1), i.e. Γλ2/α, and those which leave invariant the norm of the space 𝒳m,γ where the solutions live, i.e. Γλγ+m. Indeed, by (1.17) and (1.5), it follows that, for u0∈𝒳m,γ,
(1.23)
∥
Γ
λ
σ
𝒮
m
(
t
)
u
0
∥
𝒳
m
,
γ
=
∥
D
λ
σ
𝒮
m
(
λ
2
t
)
u
0
∥
𝒳
m
,
γ
=
λ
σ
-
(
γ
+
m
)
∥
𝒮
m
(
λ
2
t
)
u
0
∥
𝒳
m
,
γ
.
Since ∥𝒮m(λ2t)u0∥𝒳m,γ≤C∥u0∥𝒳m,γ for some C>0, by Theorem 1.2, it follows, setting σ=2α in (1.23), that if 2α<γ+m, then ∥Γλ2/α𝒮m(t)u0∥𝒳m,γ→0 as λ→∞, uniformly for all u0 in a bounded set of 𝒳m,γ and all t>0.
It is clear from (1.23) that, for u0∈𝒳m,γ, the transformations most likely to yield some nontrivial asymptotic behavior are Γλγ+m. In other words, we still need to study 𝒬γ(u0) as given by (1.22), and likewise 𝒵γ(u0) as given by (1.21). However, we cannot expect the relationship between these two objects to be given as in Theorem 1.7 since the transformations do not preserve solutions of (1.1).
If u is a solution of (1.1), then v=Γλγ+mu is the solution of vt-Δv+λ2-(γ+m)α|v|αv=0. If α>2γ+m, it follows that, as λ→∞, the function v satisfies an equation which approaches the linear heat equation. Hence we should not be surprised if, in this case, 𝒬γ(u0) and 𝒵γ(u0) are related by the linear heat equation. The next theorem makes this idea precise, both in the asymptotically self-similar case, and the more general case of arbitrary u0∈𝒳m,γ. It is analogous to [3, Lemma 5.1, p. 110].
Theorem 1.10.
Let m∈{1,…,N}, 0<γ<N and M>0. Let α be such that α>2γ+m. We then have the following conclusions.
If
u
0
,
φ
∈
ℬ
m
,
γ
,
M
is such that
lim
λ
→
∞
D
λ
γ
+
m
u
0
=
φ
in
ℬ
m
,
γ
,
M
⋆
, then φ is homogeneous of degree -(γ+m), and u(t)=𝒮m(t)u0 is asymptotically self-similar to U(t)=etΔmφ.
𝒬
γ
(
u
0
)
=
e
Δ
m
𝒵
γ
(
u
0
)
for all
u
0
∈
𝒳
m
,
γ
.
There exists
U
0
∈
ℬ
m
,
γ
,
M
∩
C
∞
(
Ω
m
)
∩
C
0
(
Ω
m
)
such that
𝒬
γ
(
U
0
)
=
e
Δ
m
ℬ
m
,
γ
,
M
.
In Section 6 of this paper, we consider the case α<2γ+m. As in the case of Theorem 1.10, the transformations which leave solutions invariant, i.e. Γλ2/α, do not leave invariant the norm of 𝒳m,γ, which is the space where the solution lives. Nonetheless, unlike in the case α>2γ+m, the transformations Γλ2/α reveal nontrivial asymptotic behavior. Because of (1.23), to study this asymptotic behavior, we need to leave the context of the space 𝒳m,γ.
This is best illustrated by the result of Gmira and Véron [6] in the case of ℝN. If we express the upper bound (1.15) in terms more suggestive of the long-time asymptotic behavior of the solution, we see that, considering only positive solutions,
(1.24)
(
Γ
t
2
/
α
u
)
(
1
,
x
)
=
t
1
/
α
u
(
t
,
x
t
)
≤
(
1
α
)
1
/
α
.
The main result of [6] can be stated as follows. Suppose α<2N. Let u0∈Lq(ℝN) for some 1≤q<∞, or C0(ℝN), with u0≥0 be such that, for every k>0, there exists R0>0 such that u0(x)≥k|x|-2/α, |x|≥R0, i.e. lim inf|x|→∞|x|2/αu0(x)=∞. It follows that if u(t,x) is the resulting solution of (1.1), then
(1.25)
t
1
/
α
u
(
t
,
x
t
)
→
(
1
α
)
1
/
α
uniformly on compact subsets of ℝN. In light of the upper bound (1.24), the result (1.25) is rather sharp.
In the case of the sector Ωm, we have the following result, where C0b,u(Ωm) denotes the space of bounded uniformly continuous functions on Ωm which are zero on ∂Ωm.
Theorem 1.11.
Let m∈{1,…,N}, 0<γ<N and α>0 such that α<2γ+m. Let u0∈Xm,γ with u0≥0, and let u(t)=Sm(t)u0 be the resulting solution of (1.1) as given by Theorem 1.2. Suppose that there exist R0>0 and c0>0 such that
(1.26)
u
0
(
x
)
≥
c
0
ψ
0
(
x
)
,
x
∈
Ω
m
,
|
x
|
≥
R
0
,
where ψ0 is given by (1.6). Then
(1.27)
lim
t
→
∞
t
1
/
α
u
(
t
,
x
t
)
=
g
(
x
)
uniformly on compact subsets of Ω¯m, where g∈C0b,u(Ωm) is the profile of the self-similar solution of (1.1) given by Proposition 6.3.
Remark 1.12.
Condition (1.26) implies that, for any c>0,
lim
|
x
|
→
∞
,
x
1
⋯
x
m
|
x
|
-
m
≥
c
|
x
|
2
/
α
u
0
(
x
)
=
∞
since 2α>γ+m.
Remark 1.13.
Using (6.9) below and (1.8), we have that g in (1.27) satisfies the explicit bound
α
-
1
/
α
I
m
(
1
,
x
)
≤
g
(
x
)
≤
(
α
ε
)
-
1
/
α
I
m
(
(
1
-
ε
)
,
x
)
,
x
∈
Ω
m
,
for all 0<ε<1, where
I
m
(
δ
,
x
)
=
∏
i
=
1
m
(
1
π
∫
-
x
i
2
δ
x
i
2
δ
e
-
y
2
d
y
)
.
In Section 7 of this paper, we reinterpret the results of the previous sections on the global well-posedness and the asymptotic behavior of 𝒮m(t)u0, u0∈𝒳m,γ, in the case of antisymmetric functions defined on the whole space ℝN. Recall that the heat semigroup on ℝN is given by etΔφ=Gt⋆φ for all φ∈𝒮′(ℝN), where Gt is the Gauss kernel on ℝN,
G
t
(
x
)
=
(
4
π
t
)
-
N
2
e
-
|
x
|
2
4
t
for all t>0 and x∈ℝN. The heat semigroup etΔ was studied in [4] on the space
(1.28)
𝒲
σ
=
{
u
∈
L
loc
1
(
ℝ
N
\
{
0
}
)
;
|
x
|
σ
u
(
x
)
∈
L
∞
(
ℝ
N
)
}
with
0
<
σ
<
N
.
It was observed in [9] that we can consider the case N≤σ<2N for some class of antisymmetric initial values in 𝒲σ. See [9, Corollary 1.7, p. 346] and the discussion just after.
If ψ:Ωm→ℝ, we denote by ψ~ its pointwise extension to ℝN which is antisymmetric with respect to x1,x2,…,xm. If ψ∈𝒳m,γ, ψ~ has a natural interpretation as an element of 𝒮′(ℝN). See [9, Definition 1.6, p. 346]. We also define the space
𝒳
m
,
γ
~
=
{
ψ
~
;
ψ
∈
𝒳
m
,
γ
}
⊂
𝒮
′
(
ℝ
N
)
with the norm ∥φ∥𝒳m,γ~=∥φ∥|Ωm𝒳m,γ for all φ∈𝒳m,γ~. We also consider
ℬ
m
,
γ
,
M
~
=
{
ψ
~
;
ψ
∈
ℬ
m
,
γ
,
M
}
.
We denote by ℬm,γ,M⋆~ the ball ℬm,γ,M~ endowed with the weak * topology. ℬm,γ,M⋆~ inherits the metric space structure from ℬm,γ,M⋆. In addition, we observe that 𝒳m,γ~⊂𝒲γ+m with continuous injection. However the two norms are not equivalent. On the other hand, ℬm,γ,M⋆~⊂(ℬMγ+m)⋆ where (ℬMγ+m)⋆ denote the closed ball of radius M on 𝒲γ+m endowed with the weak * topology, but here the metric on ℬm,γ,M⋆~ is equivalent to the one it inherits from the metric space (ℬMγ+m)⋆. See Proposition 7.1 below.
The heat semigroup etΔ is well-defined on 𝒳m,γ~, and etΔmψ~=etΔψ~. See [9, Proposition 5.1, p. 361]. The last formula is the key to the study equation (1.1) in the space 𝒳m,γ~. The following result is essentially a reformulation of Theorem 1.2 for antisymmetric functions on ℝN.
Theorem 1.14.
Let m∈{1,2,…,N}, 0<γ<N and α>0. If v0∈Xm,γ~, then there exists a unique solution v∈C((0,∞),C0(RN)∩A) of equation (1.1) such that
v
(
t
)
→
v
0
in
L
loc
1
(
ℝ
N
\
{
0
}
)
as
t
→
0
,
there exists
C
>
0
, independent of
v
0
, such that
∥
v
(
t
)
∥
𝒳
m
,
γ
~
≤
C
∥
v
0
∥
𝒳
m
,
γ
~
for all
t
>
0
.
In addition, the following properties hold.
For all
w
0
∈
𝒳
m
,
γ
~
, |v(t)-w(t)|≤etΔ|v0-w0|, where w is the solution of (1.1) with initial value w0 satisfying (i) and (ii).
v
(
t
)
satisfies the integral equation
v
(
t
)
=
e
t
Δ
v
0
-
∫
0
t
e
(
t
-
s
)
Δ
(
|
v
(
s
)
|
α
v
(
s
)
)
d
s
for all
t
>
0
.
Since 𝒳m,γ~⊂𝒲γ+m, where 𝒲γ+m is given by (1.28), the last result gives a new class of initial values for which we have global well-posedness of solutions in the case α<2N (when γ+m>N). See [1] and [3, Section 4] for information about non-uniqueness of solutions in the case α<2N.
The semiflow 𝒮(t) defined by (1.12) extends to 𝒳m,γ~ as the following.
Definition 1.15.
Let m∈{1,2,…,N}, 0<γ<N and α>0. Given v0∈𝒳m,γ~, we set 𝒮(t)v0=v(t) for all t>0, where v∈C((0,∞),C0(ℝN)∩𝒜) is the unique solution of (1.1) given by Theorem 1.14.
From the construction in Theorem 1.14 and the uniqueness part, we have 𝒮(t)u0~=𝒮m(t)u0~ for all t>0 and u0∈𝒳m,γ. As in the case of the sectors Ωm, the flow 𝒮(t) depends continuously on the initial values. The following is an adaptation of Theorem 1.4.
Theorem 1.16.
Let m∈{1,2,…,N}, 0<γ<N and M>0. Then S(t) is continuous, Bm,γ,M⋆~→C0(RN), for all t>0.
We now consider the long-time asymptotic behavior of the solutions described in Theorem 1.14. In analogy with (1.21) and (1.22) above, and using a notation consistent with [4, formulas (1.17) and (1.18)] and [3, Definition 1.2], we make the following definitions. For v0∈𝒳m,γ~, we define the ω-limit set of possible asymptotic forms of v0 by
Ω
γ
+
m
(
v
0
)
=
{
z
∈
ℬ
m
,
γ
,
M
⋆
~
;
there exists
λ
n
→
∞
such that
lim
n
→
∞
D
λ
n
γ
+
m
v
0
=
z
in
ℬ
m
,
γ
,
M
⋆
~
}
,
and the ω-limit set of all limits of Γtγ+m𝒮(1)v0 as t→∞ by
ω
γ
+
m
(
v
0
)
=
{
f
∈
C
0
(
ℝ
N
)
;
there exists
t
n
→
∞
such that
lim
n
→
∞
∥
Γ
t
n
γ
+
m
𝒮
(
1
)
v
0
-
f
∥
L
∞
(
ℝ
N
)
=
0
}
.
The following three theorems are reformulations of the results above on the asymptotic behavior of solutions, adapted from the case of the sectors Ωm to the case of antisymmetric functions on ℝN, in the three cases: α equals, is greater than, and is less than 2γ+m.
Theorem 1.17.
Let m∈{1,2,…,N}, 0<γ<N and M>0. Let α>0 be such that α=2γ+m. It follows that
if
v
0
,
φ
∈
ℬ
m
,
γ
,
M
~
are such that
lim
λ
→
∞
D
λ
γ
+
m
v
0
=
φ
in
ℬ
m
,
γ
,
M
⋆
~
, then φ is homogeneous of degree -(γ+m), and the solution v(t)=𝒮(t)v0 of (1.1) is asymptotically self-similar to U(t)=𝒮(t)φ,
ω
γ
+
m
(
v
0
)
=
𝒮
(
1
)
Ω
γ
+
m
(
v
0
)
for all
v
0
∈
𝒳
m
,
γ
~
,
there exists
V
0
∈
ℬ
m
,
γ
,
M
~
∩
C
∞
(
ℝ
N
)
such that
ω
γ
+
m
(
V
0
)
=
𝒮
(
1
)
ℬ
m
,
γ
,
M
~
.
Theorem 1.18.
Let m∈{1,…,N}, 0<γ<N and M>0. Let α>0 be such that α>2γ+m. It follows that
if
v
0
,
φ
∈
ℬ
m
,
γ
,
M
~
are such that
lim
λ
→
∞
D
λ
γ
+
m
v
0
=
φ
in
ℬ
m
,
γ
,
M
⋆
~
, then φ is homogeneous of degree -(γ+m), and the solution v(t)=𝒮(t)v0 of (1.1) is asymptotic to the self-similar solution of the linear heat equation U(t)=etΔφ,
ω
γ
+
m
(
v
0
)
=
e
Δ
Ω
γ
+
m
(
v
0
)
for all
v
0
∈
ℬ
m
,
γ
,
M
~
,
there exists
V
0
∈
ℬ
m
,
γ
,
M
~
∩
C
∞
(
ℝ
N
)
such that
ω
γ
+
m
(
V
0
)
=
e
Δ
ℬ
m
,
γ
,
M
~
.
Theorem 1.19.
Let m∈{1,…,N}, 0<γ<N and α>0 such that α<2γ+m. Let v0∈X~m,γ with v0|Ωm≥0, and let v(t)=S(t)v0 be the resulting solution of (1.1) as given by Definition 1.15. Suppose that there exist R0>0 and c0>0 such that v0(x)≥c0ψ0(x), x∈Ωm, |x|≥R0, where ψ0 is given by (1.6). Then
lim
t
→
∞
t
1
/
α
v
(
t
,
x
t
)
=
g
(
x
)
uniformly on compact subsets of RN, where g∈Cb,u(RN) is the antisymmetric (bounded, uniformly continuous) profile of the self-similar solution of (1.1) given by Proposition 7.3.
Finally, in the appendix, for completeness, we give a proof of Kato’s parabolic inequality and the main application for which we use it. Also, we present some results which we found during the course of research for this article, which we feel have some independent interest, but which ultimately were not needed for the proofs of the main results. One of them concerns the lowest eigenvalue and corresponding eigenfunction for -Δ on B1={x∈Ωm:|x|<1} with Dirichlet boundary conditions.
2 Existence and Continuity Properties of Solutions
The purpose of this section is to study well-posedness of equation (1.1) with initial values in 𝒳m,γ and to give the proofs of Theorem 1.2 and Theorem 1.4. For this purpose, we need several results from [9], sometimes in a slightly stronger version. The first result below is a slight improvement of [9, Proposition 2.5, p. 353].
Proposition 2.1.
Let m∈{1,…,N}, 0<γ<N and ψ∈Xm,γ. Then etΔmψ→ψ as t→0 in Lloc1(Ωm). In particular, the convergence is also in D′(Ωm).
Proof.
Let ψ∈𝒳m,γ, and let K be a fixed compact in Ωm. Let ε=d(0,K)>0, and let η∈C∞(ℝN) denote a radial cut-off function satisfying
η
(
x
)
=
1
for all x∈ℝN with |x|≤ε4,
η
(
x
)
=
0
for all x∈ℝN with |x|≥ε2.
We write
(2.1)
e
t
Δ
m
ψ
=
e
t
Δ
m
[
η
ψ
]
+
e
t
Δ
m
[
(
1
-
η
)
ψ
]
.
Using the inequality
e
-
(
x
i
-
y
i
)
2
4
t
-
e
-
(
x
i
+
y
i
)
2
4
t
=
e
-
x
i
2
4
t
e
-
y
i
2
4
t
∫
-
x
i
y
i
2
t
x
i
y
i
2
t
e
s
d
s
≤
x
i
y
i
t
e
-
(
x
i
-
y
i
)
2
4
t
for all
i
∈
{
1
,
…
,
m
}
,
we deduce from (1.8) that, for all x,y∈Ωm,
(2.2)
K
t
(
x
,
y
)
≤
t
-
m
(
∏
i
=
1
m
x
i
y
i
)
G
t
(
x
-
y
)
.
Therefore,
|
e
t
Δ
m
[
η
ψ
]
(
x
)
|
≤
C
∫
Ω
m
K
t
(
x
,
y
)
y
1
⋯
y
m
η
(
y
)
|
y
|
-
γ
-
2
m
d
y
≤
C
t
-
m
x
1
⋯
x
m
∫
|
y
|
≤
ε
2
G
t
(
x
-
y
)
|
y
|
-
γ
d
y
.
Since, for x∈K (hence |x|≥ε) and |y|≤ε2, we have |x-y|≥|x|-|y|≥ε2, it follows that
|
e
t
Δ
m
[
η
ψ
]
(
x
)
|
≤
C
x
1
⋯
x
m
t
-
(
m
+
N
/
2
)
e
-
ε
2
16
t
∫
|
y
|
≤
ε
2
|
y
|
-
γ
d
y
for all
x
∈
K
.
This implies that etΔm[ηψ]→0 a.e. pointwise on K as t→0. Moreover, by Proposition [9, Theorem 1.1 (i), p. 343], we have |etΔm[ηψ]|≤Cψ0 for all t>0. Thus, by the dominated convergence theorem, etΔm[ηψ]→0 in L1(K) as t→0.
On the other hand, since (1-η)ψ∈Lp(Ωm) for p>max{1,Nγ+m}, it follows that etΔm[(1-η)ψ]→(1-η)ψ in Lp(Ωm) as t→0. In particular, since K⊂Ωm is compact, etΔm[(1-η)ψ]→ψ in L1(K) as t→0. Using (2.1), we obtain etΔmψ→ψ in Lloc1(Ωm) as t→0. This completes the proof. ∎
We also need to use a stronger version of [9, Lemma 2.6, p. 355], as follows.
Lemma 2.2.
Let m∈{1,…,N} and 0<γ<N. There exists C>0 such that
|
e
t
Δ
m
ψ
(
x
)
|
≤
C
x
1
⋯
x
m
(
t
+
|
x
|
2
)
-
γ
+
2
m
2
∥
ψ
∥
𝒳
m
,
γ
for all t>0, x∈Ωm and ψ∈Xm,γ.
Proof.
It suffices to prove the lemma for ψ=ψ0. Since ψ0 is homogeneous, we know that etΔmψ0 is self-similar, and so
(2.3)
e
t
Δ
m
ψ
0
(
x
)
=
t
-
γ
+
m
2
f
(
x
t
)
,
where f:=eΔmψ0. By [9, Proposition 2.2, p. 349], we have
f
(
x
)
=
e
Δ
m
ψ
0
(
x
)
≤
C
ψ
0
(
x
)
≤
C
x
1
⋯
x
m
|
x
|
-
γ
-
2
m
for all x∈Ωm. Therefore, there exists C>0 such that f(x)≤Cx1⋯xm(1+|x|2)-γ+2m2 for |x|≥1. On the other hand, for all x∈Ωm, we have
f
(
x
)
=
∫
Ω
m
K
1
(
x
,
y
)
ψ
0
(
y
)
d
y
.
Using inequality (2.2), we obtain
f
(
x
)
≤
C
x
1
⋯
x
m
∫
Ω
m
G
1
(
x
-
y
)
y
1
2
⋯
y
m
2
|
y
|
-
γ
-
2
m
d
y
≤
C
x
1
⋯
x
m
∫
ℝ
N
e
-
|
x
-
y
|
2
4
|
y
|
-
γ
d
y
≤
C
x
1
⋯
x
m
(
e
Δ
|
⋅
|
-
γ
)
(
x
)
≤
C
x
1
⋯
x
m
(
1
+
|
x
|
2
)
-
γ
2
by [1, Corollary 8.3, p. 531]. Hence, for |x|≤1, we have
f
(
x
)
≤
C
x
1
⋯
x
m
(
1
+
|
x
|
2
)
-
γ
+
2
m
2
.
Therefore, there exists C>0 such that
f
(
x
)
≤
C
x
1
⋯
x
m
(
1
+
|
x
|
2
)
-
γ
+
2
m
2
for all x∈Ωm. Using relation (2.3), we deduce that
e
t
Δ
m
ψ
0
(
x
)
≤
C
x
1
⋯
x
m
(
t
+
|
x
|
2
)
-
γ
+
2
m
2
.
This proves the result. ∎
The following is a version of [1, Corollary 8.3, p. 531] adapted from ℝN to Ωm.
Corollary 2.3.
Let m∈{1,…,N}, 0<γ<N and A>0. There exists C>0 such that if τ≥0 and u0∈Xm,γ is such that |u0(x)|≤Ax1⋯xm(τ+|x|2)-γ+2m2 for x∈Ωm, then
|
e
t
Δ
m
u
0
(
x
)
|
≤
C
x
1
⋯
x
m
(
τ
+
t
+
|
x
|
2
)
-
γ
+
2
m
2
for all t>0 and all x∈Ωm.
Proof.
From Lemma 2.2, the result is true for τ=0. Next, we consider the case τ=1. We put
g
(
x
)
=
x
1
⋯
x
m
(
1
+
|
x
|
2
)
-
γ
+
2
m
2
.
Using (2.2), we obtain
e
t
Δ
m
g
(
x
)
≤
t
-
m
x
1
⋯
x
m
∫
ℝ
N
G
t
(
x
-
y
)
y
1
2
⋯
y
m
2
(
1
+
|
y
|
2
)
-
γ
+
2
m
2
d
y
≤
t
-
m
x
1
⋯
x
m
∫
ℝ
N
G
t
(
x
-
y
)
(
1
+
|
y
|
2
)
-
γ
2
d
y
.
By [1, Corollary 8.3, p. 531], we have
e
t
Δ
m
g
(
x
)
≤
C
x
1
⋯
x
m
t
-
m
(
1
+
t
+
|
x
|
2
)
-
γ
2
≤
C
x
1
⋯
x
m
(
1
+
t
+
|
x
|
2
)
-
γ
+
2
m
2
for t≥1+|x|2, so that (2t)-m≤(1+t+|x|2)-m.
If t≤1+|x|2, we have (1+|x|2)-γ+2m2≤C(1+t+|x|2)-γ+2m2, so it suffices to prove that
e
t
Δ
m
g
(
x
)
≤
C
x
1
⋯
x
m
(
1
+
|
x
|
2
)
-
γ
+
2
m
2
.
Using (1.10), we obtain
e
t
Δ
m
g
(
x
)
=
∫
Ω
m
K
t
(
x
,
y
)
y
1
⋯
y
m
(
1
+
|
y
|
2
)
-
γ
+
2
m
2
d
y
≤
x
1
⋯
x
m
.
Hence, for |x|≤1,
e
t
Δ
m
g
(
x
)
≤
C
x
1
⋯
x
m
(
1
+
|
x
|
2
)
-
γ
+
2
m
2
.
In addition, g∈𝒳m,γ, so, by [9, Theorem 1.1 (i), p. 343],
e
t
Δ
m
g
(
x
)
≤
C
ψ
0
(
x
)
≤
C
x
1
⋯
x
m
|
x
|
-
γ
-
2
m
.
Therefore, if |x|>1 so that (1+|x|2)γ+2m2≤(2|x|)γ+2m, we have
e
t
Δ
m
g
(
x
)
≤
C
x
1
⋯
x
m
(
1
+
|
x
|
2
)
-
γ
+
2
m
2
.
It follows that
(2.4)
|
e
t
Δ
m
u
0
(
x
)
|
≤
e
t
Δ
m
g
(
x
)
≤
C
x
1
⋯
x
m
(
1
+
t
+
|
x
|
2
)
-
γ
+
2
m
2
for all x∈Ωm and all t>0. This proves the result for τ=1.
For the general case, we proceed by scaling and observe that
D
1
τ
γ
+
m
g
(
x
)
=
x
1
⋯
x
m
(
τ
+
|
x
|
2
)
-
γ
+
2
m
2
.
Using formula (1.9) and inequality (2.4), we obtain
|
e
t
Δ
m
u
0
(
x
)
|
≤
e
t
Δ
m
[
D
1
τ
γ
+
m
g
]
(
x
)
=
D
1
τ
γ
+
m
[
e
t
τ
Δ
m
g
]
(
x
)
≤
C
x
1
⋯
x
m
(
τ
+
t
+
|
x
|
2
)
-
γ
+
2
m
2
.
This completes the proof. ∎
We will also use the following lemma, which gives a property of convergence in Lloc1(Ωm) which is not shared by convergence in 𝒟′(Ωm).
Lemma 2.4.
Let (wk)k≥1⊂Bm,γ,M and w∈Bm,γ,M be such that wk→k→∞w in Lloc1(Ωm). Then
e
t
Δ
m
|
w
k
|
→
k
→
∞
e
t
Δ
m
|
w
|
𝑖𝑛
C
0
(
Ω
m
)
.
Proof.
Since wk→w in Lloc1(Ωm), then |wk|→|w| in Lloc1(Ωm); hence |wk|→|w| in 𝒟′(Ωm). From [9, Proposition 3.1 (i), p. 356], and since (|wk|)k≥1, |w|⊂ℬm,γ,M, we deduce that |wk|→|w| in ℬm,γ,M⋆. Since by [9, Proposition 4.1 (ii), p. 359], etΔm:ℬm,γ,M⋆→C0(Ωm) is continuous, it follows that etΔm|wk|→etΔm|w| in C0(Ωm) as k→∞. ∎
We now give the proof of Theorem 1.2.
Proof of Theorem 1.2.
Let u0∈𝒳m,γ and let (𝒦n)n≥1 be the sequence of nondecreasing compacts in Ωm defined by
𝒦
n
=
{
x
∈
Ω
m
such that
d
(
x
,
∂
Ω
m
)
≥
1
n
and
|
x
|
≤
n
}
.
We consider the function
(2.5)
u
0
,
n
=
ξ
n
u
0
,
where ξn is a cut-off function satisfying
0
≤
ξ
n
≤
1
for all x∈Ωm,
ξ
n
(
x
)
=
1
for all x∈𝒦n,
ξ
n
(
x
)
=
0
for all x∈Ωm\𝒦n+1.
Note that u0,n∈𝒳m,γ for all n≥1, and
∥
u
0
,
n
∥
𝒳
m
,
γ
≤
∥
u
0
,
n
+
1
∥
𝒳
m
,
γ
≤
∥
u
0
∥
𝒳
m
,
γ
;
u
0
,
n
→
u
0
pointwise and in Lloc1(Ωm) (hence in 𝒟′(Ωm)) as n→∞;
for a fixed compact K on Ωm, there exists n0 such that u0,n=u0 on K for all n≥n0.
Existence: The proof is motivated by the proof of [1, Theorem 8.8, p. 536]. Since u0,n∈Lp(Ωm),1≤p<∞, we consider the unique solution un∈C([0,∞),Lp(Ωm))∩C((0,∞),C0(Ωm)) of (1.1) with initial value u0,n∈Lp(Ωm). It follows from Kato’s parabolic inequality (see Corollary A.2 in the appendix) that, for all n, ℓ∈ℕ⋆,
(2.6)
|
u
n
(
t
)
-
u
ℓ
(
t
)
|
≤
e
t
Δ
m
|
u
0
,
n
-
u
0
,
ℓ
|
for all
t
>
0
.
Since |u0,n-u0,ℓ|≤|u0,n-u0| for all ℓ>n, we have
(2.7)
|
u
n
(
t
)
-
u
ℓ
(
t
)
|
≤
e
t
Δ
m
|
u
0
,
n
-
u
0
|
for all t>0 and ℓ>n. In addition, ∥u0,n-u0∥𝒳m,γ≤2∥u0∥𝒳m,γ for all n≥1, and (u0,n-u0)→0 in Lloc1(Ωm) as n→∞, and so it follows from Lemma 2.4 that etΔm|u0,n-u0|→0 in C0(Ωm) as n→∞. Therefore, from (2.7), un(t) is a Cauchy sequence in C0(Ωm) for all t>0, and so there exists a function u(t) such that un(t) converge to u(t) in C0(Ωm). Furthermore, by letting ℓ→∞ in (2.6), we obtain
|
u
n
(
t
)
-
u
(
t
)
|
≤
e
(
t
-
ε
)
Δ
m
[
e
ε
Δ
m
|
u
0
,
n
-
u
0
|
]
for all t>ε>0. Since eεΔm|u0,n-u0|→0 in C0(Ωm) as n→∞, and e(t-ε)Δm is C0 contraction on C0(Ωm), we deduce that un converges to u on L∞([ε,∞),C0(Ωm)) for all ε>0. The limit function u∈C((0,∞),C0(Ωm)) is clearly a solution of (1.1).
Again by Corollary A.2, we have
|
u
n
(
t
)
|
≤
e
t
Δ
m
|
u
0
,
n
|
≤
e
t
Δ
m
|
u
0
|
∈
𝒳
m
,
γ
for all n≥1. In addition, by letting n→∞, we obtain
(2.8)
|
u
(
t
)
|
≤
e
t
Δ
m
|
u
0
|
,
and so, by [9, Theorem 1.1 (i), p. 343], we deduce that ∥u(t)∥𝒳m,γ≤C∥u0∥𝒳m,γ for all t>0. This proves (ii).
It remains now to show that u(t)→u0 in Lloc1(Ωm) as t→0. We fix a compact subset K⊂Ωm and n such that u0,n=u0 on K. Thus,
∫
K
|
u
(
t
)
-
u
0
|
=
∫
K
|
u
(
t
)
-
u
0
,
n
|
≤
∫
K
|
u
(
t
)
-
u
n
(
t
)
|
+
∫
K
|
u
n
(
t
)
-
u
0
,
n
|
.
By letting ℓ→∞ in (2.6), we have
|
u
n
(
t
)
-
u
(
t
)
|
≤
e
t
Δ
m
|
u
0
,
n
-
u
0
|
.
Proposition 2.1 shows that etΔm|u0,n-u0|→|u0,n-u0| in Lloc1(Ωm) as t→0. Therefore,
∫
K
e
t
Δ
m
|
u
0
,
n
-
u
0
|
→
t
→
0
∫
K
|
u
0
,
n
-
u
0
|
=
0
,
and so ∫K|u(t)-un(t)|→0 as t→0. Since un∈C([0,∞),Lp(Ωm)), we have un(t)→u0,n in Lp(Ωm) as t→0 so that
∫
K
|
u
n
(
t
)
-
u
0
,
n
|
→
0
as
t
→
0
.
This proves that u(t) converges to u0 in Lloc1(Ωm) as t→0, and so (i) is proved.
Uniqueness: Let s>0, and let u,v be two solutions of (1.1) satisfying (i) and (ii). We have
|
u
(
t
+
s
)
-
v
(
t
+
s
)
|
≤
e
t
Δ
m
|
u
(
s
)
-
v
(
s
)
|
≤
e
t
Δ
m
|
u
(
s
)
-
u
0
|
+
e
t
Δ
m
|
v
(
s
)
-
u
0
|
for all t>s>0. Let M≥C∥u0∥𝒳m,γ. Since u(s),v(s)∈ℬm,γ,M⋆ for all s>0 and u(s),v(s)→u0 in Lloc1(Ωm) as s→0, it follows from Lemma 2.4 that the right-hand side of the last inequality tends to 0 in C0(Ωm) as s→0. This gives |u(t+s)-v(t+s)|→0 as s→0. But, since u,v∈C((0,∞),C0(Ωm)), we deduce that |u(t+s)-v(t+s)|→|u(t)-v(t)| as s→0 for every fixed t>0. By uniqueness of the limit, we have u(t)=v(t) for all t>0.
Additional properties: We next give the proof of statements (iii), (iv), and (vi). In fact, by (2.8), we have |u(t)|≤etΔm|u0|, and so, from Lemma 2.2, we obtain
|
u
(
t
,
x
)
|
≤
C
x
1
⋯
x
m
(
t
+
|
x
|
2
)
-
γ
+
2
m
2
∥
u
0
∥
𝒳
m
,
γ
for all t>0 and x∈Ωm. In addition, if u0,v0∈𝒳m,γ, we denote u(t) and v(t) the corresponding solutions. For all n≥1, we let u0,n=u0ξn and v0,n=v0ξn, where ξn is the cut-off function defined by (2.5). Then, for all n≥1,
|
u
n
(
t
)
-
v
n
(
t
)
|
≤
e
t
Δ
m
|
u
0
,
n
-
v
0
,
n
|
.
Letting n→∞ and using Lemma 2.4, we deduce that
|
u
(
t
)
-
v
(
t
)
|
≤
e
t
Δ
m
|
u
0
-
v
0
|
.
Finally, assertion (vi) is true since, under the same conditions, |un(t)|≤vn(t) by well-known comparison results.
Integral equation: Since u0,n∈Lp(Ωm) for all p>max[1,Nα2], the corresponding solution un(t) satisfies the integral equation
u
n
(
t
)
=
e
t
Δ
m
u
0
,
n
-
∫
0
t
e
(
t
-
s
)
Δ
m
(
|
u
n
(
s
)
|
α
u
n
(
s
)
)
d
s
for all t>0, where each term is in C([0,∞);Lp(Ωm)).
Since u0,n→u0 in ℬm,γ,M⋆ as n→∞, we know, for example by Lemma 2.4, that etΔmu0,n→etΔmu0 in C0(Ωm) as n→∞ for all t>0. On the other hand, for all 0<s<t,
e
(
t
-
s
)
Δ
m
(
|
u
n
(
s
)
|
α
u
n
(
s
)
)
→
e
(
t
-
s
)
Δ
m
(
|
u
(
s
)
|
α
u
(
s
)
)
on
C
0
(
Ω
m
)
as
n
→
∞
.
From property (iii) above and [6, inequality (1.8), p. 261], we have
|
u
n
(
s
)
|
≤
C
x
1
⋯
x
m
(
s
+
|
x
|
2
)
-
γ
+
2
m
2
and
|
u
n
(
s
)
|
≤
(
α
s
)
-
1
/
α
for all s>0. Therefore, for all 0<ε<1,
|
u
n
(
s
)
|
α
+
1
=
|
u
n
(
s
)
|
α
(
1
-
ε
)
|
u
n
(
s
)
|
1
+
α
ε
≤
C
|
u
n
(
s
)
|
α
(
1
-
ε
)
(
ω
(
x
)
)
1
+
α
ε
(
s
+
|
x
|
2
)
-
(
γ
+
m
)
2
(
1
+
α
ε
)
≤
C
s
1
-
ε
ω
(
x
)
(
s
+
|
x
|
2
)
-
(
γ
+
m
)
2
(
1
+
α
ε
)
,
where ω(x)=x1⋯xm(s+|x|2)-m2≤1. We then take ε<N-γα(γ+m) so that (γ+m)(1+αε)=γ′+m, 0<γ′<N and 0<γ<γ′. It follows that
(2.9)
|
u
n
(
s
)
|
α
+
1
≤
C
s
1
-
ε
x
1
⋯
x
m
(
s
+
|
x
|
2
)
-
γ
′
+
2
m
2
.
We deduce by Corollary 2.3 that
e
(
t
-
s
)
Δ
m
|
u
n
(
s
)
|
α
+
1
≤
C
s
1
-
ε
x
1
⋯
x
m
(
t
+
|
x
|
2
)
-
γ
′
+
2
m
2
for all t>s>0. Likewise, since un(s)→u(s) in C0(Ωm),
(2.10)
e
(
t
-
s
)
Δ
m
|
u
(
s
)
|
α
+
1
≤
C
s
1
-
ε
x
1
⋯
x
m
(
t
+
|
x
|
2
)
-
γ
′
+
2
m
2
for all t>s>0 so that s→e(t-s)Δm|u(s)|α+1 is in L1((0,t);C0(Ωm)).
We deduce, using the dominated convergence theorem,
∫
0
t
e
(
t
-
s
)
Δ
m
(
|
u
n
(
s
)
|
α
u
n
(
s
)
)
d
s
→
∫
0
t
e
(
t
-
s
)
Δ
m
(
|
u
(
s
)
|
α
u
(
s
)
)
d
s
as
n
→
∞
,
and so that the solution u(t) satisfies
u
(
t
)
=
e
t
Δ
m
u
0
-
∫
0
t
e
(
t
-
s
)
Δ
m
(
|
u
(
s
)
|
α
u
(
s
)
)
d
s
.
This proves (v).
Note that (2.10) implies that
|
∫
0
t
e
(
t
-
s
)
Δ
m
(
|
u
(
s
)
|
α
u
(
s
)
)
d
s
|
≤
C
t
ε
x
1
x
2
⋯
x
m
(
t
+
|
x
|
2
)
-
γ
′
+
2
m
2
,
with ε and γ′ as above. ∎
The following lemma is needed to establish Theorem 1.4.
Lemma 2.5.
Let (un)n≥0 be a sequence of solutions of (1.1), un∈C((0,∞),C0(Ωm)), satisfying
(2.11)
|
u
n
(
t
,
x
)
|
≤
C
x
1
⋯
x
m
(
t
+
|
x
|
2
)
-
γ
+
2
m
2
for all
x
∈
Ω
m
and all
t
>
0
.
There exists a subsequence (unk)k≥0 and a solution g∈C((0,∞),C0(Ωm)) of (1.1) such that unk→g as k→∞, in C([τ,∞),C0(Ωm)) for every τ>0.
Proof.
Fix τ>0. Using (2.11) with t=τ2, we deduce that the set {un(τ2),n≥1} is bounded in Lp(Ωm) for all p satisfying max(1,Nγ+m)<p≤∞. By standard smoothing effects, we see that the un(τ)=𝒮m(τ2)un(τ2) are uniformly bounded in W1,∞(Ωm). Thus, {un(τ)} is relatively compact in C(ΩmR¯) for all R>1, where ΩmR={x∈Ωm;|x|≤R}. Using the decay estimate (2.11), {un(τ)} is also relatively compact on C0(Ωm). By continuous dependence in C0(Ωm) of (1.1), it follows that {un(⋅),n≥1} is relatively compact in C([τ,T],C0(Ωm)) for all T>τ, the limit points being solutions of (1.1). Since, by (2.11), ∥un(t)∥L∞→0 as t→∞ uniformly in n≥1, we may let T=∞ in the previous property. By letting τ→0 and using a diagonal procedure, we see that there exists a solution g∈C((0,∞),C0(Ωm)) of (1.1) and a subsequence (unk)k≥0 such that unk→g as k→∞ in C([τ,∞),C0(Ωm)) for every τ>0. ∎
Proof of Theorem 1.4.
Let (u0,n)n≥0⊂ℬm,γ,M and u0∈ℬm,γ,M such that u0,n→u0 in ℬm,γ,M⋆ as n→∞. Let u(t)=𝒮m(t)u0 and un(t)=𝒮m(t)u0,n for all t>0 be the corresponding solutions of (1.1) as in Definition 1.3. By Theorem 1.2 (iv), we have
(2.12)
|
u
n
(
t
,
x
)
|
≤
C
x
1
⋯
x
m
(
t
+
|
x
|
2
)
-
γ
+
2
m
2
for all x∈Ωm and t>0. It follows from Lemma 2.5 that there exists a solution g∈C((0,∞),C0(Ωm)) of (1.1) and a subsequence (unk)k≥0 such that unk→g as k→∞ in C([τ,∞),C0(Ωm)) for every τ>0. To see that g(t)→u0 in Lloc1(Ωm) as t→0, we consider a compact K⊂Ωm and let 𝒪 be an open, bounded, and regular subset of Ωm with K⊂𝒪. By (2.12), we have |un(t,x)|≤C for all x∈𝒪, t>0, and n≥0. Since u0,(u0,n)n≥0⊂ℬm,γ,M⋆ and u0,n→u0 in ℬm,γ,M⋆ as n→∞, we have by [9, Proposition 3.1 (i), p. 356] that u0,n→u0 in 𝒟′(Ωm); we conclude using [3, Lemma 2.6, p. 89] that g(t)→u0 in L1(𝒪) as t→0. Therefore, g(t)→u0 in Lloc1(Ωm) as t→0, and from uniqueness of solutions of (1.1), we have g≡u so that the limit g is determined by u0. In particular, it does not depend on the subsequence (unk)k≥0 so that the whole sequence (un)n≥0 converges to u in C([τ,∞),C0(Ωm)) for every τ>0. This completes the proof. ∎
6 Nonlinear Asymptotic Behavior on Sectors
In this section, we consider equation (1.1) with nonnegative initial value u0∈𝒳m,γ in the case α<2γ+m, and our goal is to prove Theorem 1.11. First however, we need to show that the hypothesis on the initial condition u0, which gives a lower bound for large |x|, implies a lower bound on the resulting solution at any fixed positive time. The key point is the behavior near the boundary. We prove the following result.
Proposition 6.1.
Let u0∈Xm,γ with u0≥0, and suppose that there exist ρ>0 and c>0 such that, for all x∈Ωm with |x|≥ρ, u0(x)≥cψ0(x), where ψ0 is given by (1.6). Let u(t,⋅)=u(t)=Sm(t)u0 be the resulting solution of (1.1) as given by Theorem 1.2, and fix any t0>0. It follows that v0≡Sm(t0)u0 verifies the condition
(6.1)
v
0
(
x
)
≥
{
c
′
x
1
x
2
⋯
x
m
,
0
<
|
x
|
≤
1
,
c
′
x
1
x
2
⋯
x
m
|
x
|
-
(
γ
+
2
m
)
,
|
x
|
≥
1
,
for some c′>0, where the constant c′ may depend on t0.
We refer the reader to [8] for results of this type on a general domain. The present situation differs from that in [8] in that the sector Ωm does not have the required regularity, and also that here we include the possibility that u0 could be identically zero on a bounded subset of Ωm. Unlike [8], our proof makes use of the explicit form of the kernel for the heat semigroup on Ωm.
Proof.
We first note that it suffices by comparison to prove this for the specific initial value
(6.2)
u
0
(
x
)
=
{
0
,
x
∈
Ω
m
,
|
x
|
<
ρ
,
c
ψ
0
(
x
)
,
x
∈
Ω
m
,
|
x
|
≥
ρ
,
where ψ0 is given by (1.6), and ρ>0 is arbitrary. To accomplish this, we first prove that, for any fixed t0>0, v0=et0Δmu0 verifies (6.1), where u0 is given by (6.2). For this purpose, since the estimate is linear, the value of c>0 in (6.2) is of no importance.
Thus, we consider etΔmu0 on Ωm given by (1.7) and (1.8), where u0 is given by (6.2). Using the fact that es-e-s≥2s for all s≥0, we see that if x,y∈Ωm and 1≤i≤m (so that xi≥0 and yi≥0), then
(6.3)
e
-
|
x
i
-
y
i
|
2
4
t
-
e
-
|
x
i
+
y
i
|
2
4
t
=
e
-
|
x
i
|
2
4
t
e
-
|
y
i
|
2
4
t
[
e
x
i
y
i
2
t
-
e
-
x
i
y
i
2
t
]
≥
[
x
i
y
i
t
]
e
-
|
x
i
|
2
4
t
e
-
|
y
i
|
2
4
t
≥
[
x
i
y
i
t
]
e
-
|
x
i
|
2
2
t
e
-
|
y
i
|
2
2
t
.
In addition, for m+1≤j≤N, we have (since (s-r)2≤2s2+2r2)
e
-
|
x
j
-
y
j
|
2
4
t
≥
e
-
|
x
j
|
2
2
t
e
-
|
y
j
|
2
2
t
.
It follows that
(6.4)
e
t
Δ
m
u
0
(
x
)
=
∫
Ω
m
K
t
(
x
,
y
)
u
0
(
y
)
d
y
=
(
4
π
t
)
-
N
2
∫
Ω
m
∏
j
=
m
+
1
N
e
-
|
x
j
-
y
j
|
2
4
t
∏
i
=
1
m
[
e
-
|
x
i
-
y
i
|
2
4
t
-
e
-
|
x
i
+
y
i
|
2
4
t
]
u
0
(
y
)
d
y
≥
c
x
1
x
2
⋯
x
m
t
-
m
(
4
π
t
)
-
N
2
e
-
|
x
|
2
2
t
∫
Ω
m
y
1
y
2
⋯
y
m
e
-
|
y
|
2
2
t
u
0
(
y
)
d
y
=
c
x
1
x
2
⋯
x
m
t
-
m
(
4
π
t
)
-
N
2
e
-
|
x
|
2
2
t
∫
y
∈
Ω
m
|
y
|
≥
ρ
y
1
2
y
2
2
⋯
y
m
2
e
-
|
y
|
2
2
t
|
y
|
-
γ
-
2
m
d
y
.
This shows in particular that, for any t>0, etΔmu0 satisfies (6.1), but only on any given bounded set in Ωm.
We turn our attention to the case where |x| is large. For x∈Ωm, let
Ω
m
(
x
)
=
{
y
∈
Ω
m
:
0
<
x
i
≤
y
i
≤
max
[
2
x
i
,
2
]
,
1
≤
i
≤
m
,
0
<
|
x
i
|
≤
|
y
i
|
≤
max
[
2
|
x
i
|
,
2
]
,
m
<
i
≤
N
}
.
If y∈Ωm(x), then
|
y
|
2
=
∑
i
=
1
N
y
i
2
≤
∑
i
=
1
N
max
[
2
|
x
i
|
,
2
]
2
≤
∑
i
=
1
N
(
4
x
i
2
+
4
)
=
4
|
x
|
2
+
4
N
.
Since this calculation is for large |x|, we may suppose that |x|2≥N, and so we see that
y
∈
Ω
m
(
x
)
⟹
|
y
|
≤
2
2
|
x
|
≤
4
|
x
|
.
Also, we want to use the specific formula in (6.2), so we impose |x|≥ρ, where ρ is as in (6.2). Hence
y
∈
Ω
m
(
x
)
⟹
|
y
|
≥
ρ
.
We can now calculate
(6.5)
e
t
Δ
m
u
0
(
x
)
=
∫
Ω
m
K
t
(
x
,
y
)
u
0
(
y
)
d
y
≥
∫
Ω
m
(
x
)
K
t
(
x
,
y
)
u
0
(
y
)
d
y
=
∫
Ω
m
(
x
)
K
t
(
x
,
y
)
y
1
y
2
⋯
y
m
|
y
|
-
γ
-
2
m
d
y
≥
4
-
γ
-
2
m
|
x
|
-
γ
-
2
m
∫
Ω
m
(
x
)
K
t
(
x
,
y
)
y
1
y
2
⋯
y
m
d
y
=
4
-
γ
-
2
m
|
x
|
-
γ
-
2
m
(
4
π
t
)
-
N
2
(
∏
i
=
1
m
∫
x
i
max
[
2
x
i
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
-
e
-
|
x
i
+
y
i
|
2
4
t
]
y
i
d
y
i
)
×
(
∏
i
=
1
+
m
N
∫
|
x
i
|
≤
|
y
i
|
≤
max
[
2
|
x
i
|
,
2
]
e
-
|
x
i
-
y
i
|
2
4
t
d
y
i
)
=
4
-
γ
-
2
m
|
x
|
-
γ
-
2
m
(
4
π
t
)
-
N
2
(
∏
i
=
1
m
∫
x
i
max
[
2
x
i
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
-
e
-
|
x
i
+
y
i
|
2
4
t
]
y
i
d
y
i
)
×
(
∏
i
=
1
+
m
N
∫
|
x
i
|
max
[
2
|
x
i
|
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
+
e
-
|
x
i
+
y
i
|
2
4
t
]
d
y
i
)
.
We first need to examine the integral
∫
x
i
max
[
2
x
i
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
-
e
-
|
x
i
+
y
i
|
2
4
t
]
y
i
d
y
i
for 1≤i≤m, under two different circumstances, 0<xi<1 and xi≥1. Consider first the case xi≥1. We have, since yi≥xi,
∫
x
i
max
[
2
x
i
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
-
e
-
|
x
i
+
y
i
|
2
4
t
]
y
i
d
y
i
≥
x
i
∫
x
i
2
x
i
[
e
-
|
x
i
-
y
i
|
2
4
t
-
e
-
|
x
i
+
y
i
|
2
4
t
]
d
y
i
=
x
i
∫
0
x
i
[
e
-
|
y
i
|
2
4
t
-
e
-
|
2
x
i
+
y
i
|
2
4
t
]
d
y
i
≥
x
i
∫
0
x
i
[
e
-
|
y
i
|
2
4
t
-
e
-
|
2
x
i
|
2
4
t
e
-
|
y
i
|
2
4
t
]
d
y
i
=
x
i
∫
0
x
i
e
-
|
y
i
|
2
4
t
[
1
-
e
-
|
2
x
i
|
2
4
t
]
d
y
i
≥
x
i
[
1
-
e
-
1
t
]
∫
0
1
e
-
|
y
i
|
2
4
t
d
y
i
=
C
t
1
x
i
.
We next consider the case xi≤1. We have, by (6.3),
∫
x
i
max
[
2
x
i
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
-
e
-
|
x
i
+
y
i
|
2
4
t
]
y
i
d
y
i
≥
∫
1
2
[
x
i
y
i
t
]
e
-
|
x
i
|
2
2
t
e
-
|
y
i
|
2
2
t
y
i
d
y
i
=
x
i
e
-
|
x
i
|
2
2
t
∫
1
2
[
y
i
2
t
]
e
-
|
y
i
|
2
2
t
d
y
i
≥
x
i
e
-
1
2
t
∫
1
2
[
y
i
2
t
]
e
-
|
y
i
|
2
2
t
d
y
i
=
C
t
2
x
i
.
We second need to examine the integral
∫
|
x
i
|
max
[
2
|
x
i
|
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
+
e
-
|
x
i
+
y
i
|
2
4
t
]
d
y
i
for m+1≤i≤N, under two different circumstances, 0<|xi|<1 and |xi|≥1. Consider first the case |xi|≥1. We have, if in addition xi<0, that is -xi≥1,
∫
|
x
i
|
max
[
2
|
x
i
|
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
+
e
-
|
x
i
+
y
i
|
2
4
t
]
d
y
i
=
∫
-
x
i
-
2
x
i
[
e
-
|
x
i
-
y
i
|
2
4
t
+
e
-
|
x
i
+
y
i
|
2
4
t
]
d
y
i
≥
∫
-
x
i
-
2
x
i
[
e
-
|
x
i
+
y
i
|
2
4
t
]
d
y
i
=
∫
0
-
x
i
e
-
|
y
i
|
2
4
t
d
y
i
≥
∫
0
1
e
-
|
y
i
|
2
4
t
d
y
i
=
C
t
3
.
We have, if in addition xi>0, that is xi≥1,
∫
|
x
i
|
max
[
2
|
x
i
|
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
+
e
-
|
x
i
+
y
i
|
2
4
t
]
d
y
i
=
∫
x
i
2
x
i
[
e
-
|
x
i
-
y
i
|
2
4
t
+
e
-
|
x
i
+
y
i
|
2
4
t
]
d
y
i
≥
∫
x
i
2
x
i
[
e
-
|
x
i
-
y
i
|
2
4
t
]
d
y
i
=
∫
0
x
i
e
-
|
y
i
|
2
4
t
d
y
i
≥
∫
0
1
e
-
|
y
i
|
2
4
t
d
y
i
=
C
t
3
.
We next consider the case |xi|≤1. By the inequality e-|xj-yj|24t≥e-|xj|22te-|yj|22t, we have
∫
|
x
i
|
max
[
2
|
x
i
|
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
+
e
-
|
x
i
+
y
i
|
2
4
t
]
d
y
i
≥
∫
1
2
[
e
-
|
x
i
-
y
i
|
2
4
t
]
d
y
i
≥
∫
1
2
e
-
|
x
i
|
2
2
t
e
-
|
y
i
|
2
2
t
d
y
i
≥
e
-
1
2
t
∫
1
2
e
-
|
y
i
|
2
2
t
d
y
i
=
C
t
4
.
In all cases, we have
(6.6)
∫
x
i
max
[
2
x
i
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
-
e
-
|
x
i
+
y
i
|
2
4
t
]
y
i
d
y
i
≥
C
t
x
i
for
1
≤
i
≤
m
,
(6.7)
∫
|
x
i
|
max
[
2
|
x
i
|
,
2
]
[
e
-
|
x
i
-
y
i
|
2
4
t
+
e
-
|
x
i
+
y
i
|
2
4
t
]
d
y
i
≥
C
t
for
m
+
1
≤
i
≤
N
,
whenever x∈Ωm. It therefore follows from (6.5), (6.6), (6.7) that, if x∈Ωm, then
(6.8)
e
t
Δ
m
u
0
(
x
)
≥
C
t
x
1
x
2
⋯
x
m
|
x
|
-
γ
-
2
m
,
|
x
|
≥
max
[
N
,
ρ
]
.
Combining (6.4) and (6.8), we obtain that, for any fixed t>0, etΔmu0 satisfies (6.1).
We next show the same result for u(t,⋅)=u(t)=𝒮m(t)u0 being the resulting solution of (1.1), where u0 is given by (6.2). To do so, set w(t)=eμtu(t), where μ=[cργ+m]α≥∥u0∥L∞(Ωm)α. Since u(t)≤∥u0∥L∞(Ωm) for all t>0, we have u(t)α≤μ for all t>0. It follows that
w
′
(
t
)
=
e
μ
t
u
′
(
t
)
+
e
μ
t
μ
u
(
t
)
≥
e
μ
t
u
′
(
t
)
+
e
μ
t
u
(
t
)
α
u
(
t
)
=
e
μ
t
Δ
u
(
t
)
=
Δ
w
(
t
)
.
Hence w(t)≥etΔmw(0)=etΔmu0. In other words, u(t)≥e-μtetΔmu0, which implies the desired result. ∎
Remark 6.2.
In addition to being well-posed in C0(Ωm), in Lq(Ωm) for 1≤q<∞, as noted in the introduction, and in 𝒳m,γ, as per Theorem 1.2, equation (1.1) is globally well-posed in L∞(Ωm) in the following sense. For every u0∈L∞(Ωm), there is a unique solution u∈C((0,∞);C0b,u(Ωm)) of the integral equation (1.14), where C0b,u(Ωm) denotes the closed subspace of L∞(Ωm) of bounded, uniformly continuous functions on Ωm which are zero on ∂Ωm, but not necessarily as |x|→∞. This solution has the following additional properties: the function u is a classical solution of (1.1) on (0,∞)×Ωm, ∥u(t)-etΔmu0∥L∞(Ωm)→0 as t→0, and |u(t)|≤(αt)1/α for all t>0. One way to see this is first to establish the corresponding result on L∞(ℝN), but of course with Cb,u(ℝN) instead of C0b,u(Ωm), and then to restrict to antisymmetric functions on ℝN. The result on ℝN follows from standard arguments, i.e. contraction mapping, parabolic regularity, and comparison. We refer the reader to [2, Appendices B and C] for detailed information about etΔ on Cb,u(ℝN). In particular, [2, Lemma B.1] establishes that etΔh∈Cb,u(ℝN) for all h∈L∞(ℝN), and [2, Theorem C.1], which is still valid for the nonlinear heat equation with absorption, establishes the necessary regularity.
Proposition 6.3.
Let m∈{1,2,…,N} and α>0. There exists a self-similar solution V(t,x)=t-1/αg(xt) of equation (1.1) such that g∈C0b,u(Ωm), the space of bounded uniformly continuous functions on Ωm which are zero on ∂Ωm, g≥0, and
(6.9)
α
-
1
/
α
e
Δ
m
h
≤
g
≤
(
α
ε
)
-
1
/
α
e
(
1
-
ε
)
Δ
m
h
for all 0<ε<1, where h(x)=1 is the constant function on Ωm.
The self-similar solution V is characterized by
(6.10)
V
=
lim
λ
→
∞
Γ
λ
2
/
α
v
,
where v is the solution to (1.1) with initial value v0=h, as described in Remark 6.2, the dilations Γλ2/α are defined by (1.17), and where the limit (6.10) is uniform on compact subsets of (0,∞)×Ω¯m.
We observe that, in the case m=0, the corresponding self-similar solution is (αt)-1/α.
Proof.
Throughout this proof, we let h∈L∞(Ωm) denote the specific function h(x)=1, x∈Ωm. It follows from (1.9) that
(6.11)
(
e
λ
2
t
Δ
m
h
)
(
λ
x
)
=
(
e
t
Δ
m
h
)
(
x
)
Next we let v=v(t,x) be the global solution of (1.1) or (1.14) with initial value v0=h, i.e. v0(x)=v(0,x)=1 for all x∈Ωm, as described in Remark 6.2. For all λ>0, vλ(t,x)=λ2/αv(λ2t,λx) is likewise a solution of (1.1) or (1.14), but with initial value v0,λ(x)=vλ(0,x)=λ2/αv0(λx)=λ2/α for all x∈Ωm. Since λ→v0,λ is an increasing function, by comparison so must be λ→vλ. Moreover, we know that
(6.12)
v
λ
(
t
,
x
)
≤
(
α
t
)
-
1
/
α
so that the vλ must converge to some function V(t,x)≤(αt)-1/α, and in particular, V(t)∈L∞(Ωm) for t>0. Since each vλ is a solution of the integral equation (1.14) on every interval [ε,T]⊂(0,∞), the same must be true for V by the monotone convergence theorem. Hence V is a solution of (1.1) and V∈C((0,∞);C0b,u(Ωm)) since initial values in L∞(Ωm) give rise to solutions of (1.14) in C((0,∞);C0b,u(Ωm)) as per Remark 6.2. Note that, by parabolic regularity and standard compactness arguments, the convergence of the vλ to V is uniform on compact subsets of (0,∞)×Ω¯m. Moreover, V a self-similar solution, being the limit of the dilated solutions vλ. Thus, we can write
V
(
t
,
x
)
=
t
-
1
/
α
g
(
x
t
)
,
where g=V(1)∈C0b,u(Ωm) is the profile of V.
As for the behavior of g, we first observe that, for t>0 and ε>0, by (6.12),
v
λ
(
t
+
ε
,
⋅
)
≤
e
t
Δ
m
(
v
λ
(
ε
)
)
≤
(
α
ε
)
-
1
/
α
e
t
Δ
m
h
.
Letting λ→∞, we see that, for t>0 and ε>0,
V
(
t
+
ε
)
≤
e
t
Δ
m
V
(
ε
)
≤
(
α
ε
)
-
1
/
α
e
t
Δ
m
h
so that g=V(1)≤(αε)-1/αe(1-ε)Δmh for all small 0<ε<1. Also, V(2t)≤(αt)-1/αetΔmh. On the other hand, we claim that, for all t>0,
(6.13)
V
(
t
)
≥
(
α
t
)
-
1
/
α
e
t
Δ
m
h
.
To see this, we first show that
(6.14)
v
(
t
)
≥
(
1
+
α
t
)
-
1
/
α
e
t
Δ
m
h
.
Indeed, if we set w(t)=(1+αt)1/αv(t) so that w(0)=v(0)=h, then, since v(t)≤(1+αt)-1/α (which follows in particular from Proposition 1.5 since |etΔmv0|≤1), we get that
w
′
(
t
)
=
(
1
+
α
t
)
1
/
α
-
1
v
(
t
)
+
(
1
+
α
t
)
1
/
α
v
′
(
t
)
≥
(
1
+
α
t
)
1
/
α
v
(
t
)
α
+
1
+
(
1
+
α
t
)
1
/
α
v
′
(
t
)
=
Δ
w
(
t
)
,
which implies that w(t)≥etΔmw(0)=etΔmh. This proves (6.14). By (6.11), it follows that
(6.15)
v
λ
(
t
,
x
)
≥
λ
2
/
α
(
1
+
α
λ
2
t
)
-
1
/
α
(
e
t
λ
2
Δ
m
h
)
(
λ
x
)
=
(
λ
-
2
+
α
t
)
-
1
/
α
e
t
Δ
m
h
.
The lower bound (6.13) now follows by letting λ→∞ in (6.15). Hence g=V(1)≥α-1/αeΔmh.
Finally, we note the perhaps curious result that V(2t)≤(αt)-1/αetΔmh≤V(t) for all t>0. ∎
Proof of Theorem 1.11.
By the hypotheses on u0 and by Proposition 6.1, we have that, for t0>0,
(6.16)
u
(
t
0
,
x
)
≥
c
x
1
⋯
x
m
min
[
1
,
|
x
|
-
γ
-
2
m
]
on Ωm, and we know that u(t0)∈C0(Ωm). Up to a translation in time and since we are concerned with the large time behavior, we may suppose that u0∈𝒳m,γ∩C0(Ωm), u0≥0, and it verifies (6.16).
In fact, it suffices to assume
(6.17)
u
0
(
x
)
=
c
x
1
⋯
x
m
min
[
1
,
|
x
|
-
γ
-
2
m
]
.
Indeed, suppose u0(x)=cx1⋯xmmin[1,|x|-γ-2m]≤v0(x)≤c′ for some c′>c, and that u(t,x), v(t,x), and w(t,x) are the solutions of (1.1) with initial values, respectively, u0, v0, and w0≡c′. We know by comparison that
t
1
/
α
u
(
t
,
x
t
)
≤
t
1
/
α
v
(
t
,
x
t
)
≤
t
1
/
α
w
(
t
,
x
t
)
.
Hence if we prove that
lim
t
→
∞
t
1
/
α
u
(
t
,
x
t
)
=
g
(
x
)
uniformly on compact subsets of Ω¯m, then clearly, since, by Proposition 6.3 along with a rescaling,
lim
t
→
∞
t
1
/
α
w
(
t
,
x
t
)
=
g
(
x
)
also uniformly on compact subsets of Ω¯m, it follows that
lim
t
→
∞
t
1
/
α
v
(
t
,
x
t
)
=
g
(
x
)
uniformly on compact subsets of Ω¯m. Thus, we now assume the initial value u0∈𝒳m,γ is given by (6.17), and we denote by u(t)=𝒮m(t)u0 the resulting solution of (1.1) given by Theorem 1.2.
We use a method introduced in [7]. Consider the space-time dilations defined by (1.17) with σ=2α,
u
λ
(
t
,
x
)
=
Γ
λ
2
/
α
u
(
t
,
x
)
=
λ
2
/
α
u
(
λ
2
t
,
λ
x
)
,
λ
>
0
,
t
>
0
,
x
∈
Ω
m
.
In particular, uλ is the solution of (1.1) with initial data
(6.18)
u
0
,
λ
(
x
)
=
D
λ
2
/
α
u
0
(
x
)
=
λ
2
/
α
u
0
(
λ
x
)
=
c
λ
2
/
α
x
1
⋯
x
m
min
[
λ
m
,
λ
-
γ
-
m
|
x
|
-
γ
-
2
m
]
,
x
∈
Ω
m
.
Since 2α>γ+m, it follows that u0,λ(x) is an increasing function in λ>0 for all x∈Ωm. (It is the minimum of two functions which are obviously increasing in λ.) Consequently, the solutions uλ(t,x) are likewise increasing in λ>0. We note also that the solutions wλ(t,x) are increasing in λ>0 (as in the proof of Proposition 6.3), where w is the solution with initial value w0≡c′ as above.
Since
u
λ
(
t
,
x
)
≤
w
λ
(
t
,
x
)
≤
V
(
t
,
x
)
in
(
0
,
∞
)
×
Ω
m
,
where V is the self-similar solution of Proposition 6.3, it follows that the limit
(6.19)
lim
λ
→
∞
u
λ
(
t
,
x
)
=
U
(
t
,
x
)
≤
V
(
t
,
x
)
exists and
(6.20)
u
λ
(
t
,
x
)
≤
U
(
t
,
x
)
for all λ>0. Moreover, by parabolic regularity and standard compactness arguments, the limit (6.19) is uniform on compact subsets of (0,∞)×Ω¯m.
We next wish to show that U(t,x)=V(t,x) on (0,∞)×Ωm. For this, we need to obtain a lower bound for U. Let A>0, and consider the family of truncated initial values
u
0
,
λ
A
(
x
)
=
min
[
u
0
,
λ
(
x
)
,
A
]
,
x
∈
Ω
m
.
Let zλA be the solution of (1.1) with initial data u0,λA. By comparison principle and (6.20),
z
λ
A
(
t
,
x
)
≤
u
λ
(
t
,
x
)
≤
U
(
t
,
x
)
in
[
0
,
∞
)
×
Ω
m
for every λ>0 and A>0. Moreover, it is clear from (6.18) that, for each fixed A>0, the initial values u0,λA(x) are an increasing function of λ>0, and so therefore must be the solutions zλA(t,x). Furthermore, the initial values satisfy the monotone limit
(6.21)
lim
λ
→
∞
u
0
,
λ
A
(
x
)
=
A
,
and the corresponding solutions converge in a monotone fashion to some function
(6.22)
lim
λ
→
∞
z
λ
A
(
t
,
x
)
=
Z
A
(
t
,
x
)
≤
U
(
t
,
x
)
.
We next consider the integral equation satisfied by zλA(t), i.e. equation (1.14) with initial value u0,λA. Using (6.21) and (6.22) along with the monotone convergence theorem, we see that ZA(t) satisfies
Z
A
(
t
)
=
e
t
Δ
m
A
-
∫
0
t
e
(
t
-
s
)
Δ
m
(
|
Z
A
(
s
)
|
α
Z
A
(
s
)
)
d
s
,
i.e. ZA is the solution of (1.14) with initial value ZA(0)≡A on Ωm.
We know by (the proof of) Proposition 6.3 that limA→∞ZA(t)=V(t), which implies, along with (6.22), that V(t,x)≤U(t,x). Thus, by (6.19), V(t,x)=U(t,x).
Thus, we have shown that
(6.23)
lim
λ
→
∞
u
λ
(
t
,
x
)
=
lim
λ
→
∞
λ
2
/
α
u
(
λ
2
t
,
λ
x
)
=
V
(
t
,
x
)
=
t
-
1
/
α
g
(
x
t
)
,
where the limit is uniform on compact subsets of (0,∞)×Ω¯m. The result now follows first by setting t=1 in (6.23), and then by replacing λ2 by τ. ∎
