Naito, Yuki

EPrint Series of Department of Mathematics, Hokkaido University

A variational approach to self-similar solutions
for semilinear heat equations
Yu¯ki Naito
Department of Applied Mathematics, Faculty of Engineering, Kobe University
Nada, Kobe 657-8501, Japan (e-mail: naito@cs.kobe-u.ac.jp)
1. Introduction
We consider the Cauchy problem for semilinear heat equations with singular
initial data:
(1) wt = ∆w + wp in RN × (0,∞),
(2)λ w(x, 0) = λa (x/|x|) |x|−2/(p−1) in RN \ {0},
where N > 2, p > 1, a : SN−1 → R, and λ > 0 is a parameter. We assume that
a ∈ L∞(SN−1) and a ≥ 0, a ≡ 0. A typical case is a ≡ 1.
The equation (1) is invariant under the similarity transformation
w(x, t) → wµ(x, t) = µ2/(p−1)w(µx, µ2t) for all µ > 0.
In particular, a solution w is said to be self-similar, when w = wµ for all µ > 0,
that is,
(3) w(x, t) = µ2/(p−1)w(µx, µ2t) for all µ > 0.
Such self-similar solutions are global in time and often used to describe the
large time behavior of global solutions to (1), see, e.g., [14, 15, 5, 21].
If w(x, t) is a self-similar solution of (1.1) and has an initial value A(x), then
we easily see that A has the form A(x) = A(x/|x|)|x|−2/(p−1). Then the problem
of existence of self-similar solutions is essentially depend on the solvablity of the
Cauchy problem (1)-(2)λ. In this talk we consider the existence of self-similar
solutions of the problem (1)-(2)λ. The idea of constructing self-similar solutions
by solving the initial value problem for homogeneous initial data goes back to
the study by Giga and Miyakawa [12] for the Navier-Stokes equation in vorticity
form.
It is well known by Fujita [9] that if 1 < p ≤ (N+2)/N then (1) has no time
global solution w such that w ≥ 0 and w ≡ 0. (See also [25, 14].) Then the
condition p > (N + 2)/N is necessary for the existence of positive self-similar
solutions of (1).
1
We brieﬂy review some results concerning the Cauchy problem for (1) with
initial date in Lq(RN ). Weissler [23, 24] showed that the IVP (1) with w(x, 0) =
A ∈ Lq(RN ) admits a unique time-local solution if q ≥ N(p − 1)/2. He also
showed in [25] that the solution exists time-globally if q = N(p − 1)/2 and if
‖A‖Lq(RN ) is suﬃciently small. Giga [11] has constructed a unique local regular
solution in Lα(0, T : Lβ), where α and β are chosen so that the norm of Lα(0, T :
Lβ) is invariant under scaling. On the other hand, for 1 ≤ q < N(p − 1)/2,
Haraux and Weissler [13] constructed a solution w0 ∈ C([0,∞);Lq(RN )) of
(1) satisfying w0(x, t) > 0 for t > 0 and ‖w0(·, t)‖Lq(RN ) → 0 as t → 0 when
(N + 2)/N < p < (N + 2)/(N − 2) by seeking solutions of self-similar form.
Therefore, the Cauchy problem
(4) wt = ∆w + wp in RN × (0,∞) and w(x, 0) = 0 in RN
admits a non-unique solution in C([0,∞);Lq(RN )) for 1 ≤ q < N(p − 1)/2
when (N + 2)/N < p < (N + 2)/(N − 2).
Kozono and Yamazaki [16] constructed Besov-type function spaces based
on the Morrey spaces, and then obtained global existence results for the equa-
tion (1) and the Navier-Stokes system with small initial data in these spaces.
Cazenave and Weissler [5] proved the existence of global solutions, including
self-similar solutions, to the nonlinear Schro¨dinger equations and the equation
(1) with small initial data by using the weighted norms. By [16, 5] the problem
(1)-(2)λ admits a time-global solution for suﬃciently small λ > 0.
We note here that the equation (1) with p > N/(N − 2) has a positive
singular stationary solution W (x) = L|x|−2/(p−1), where
L =
[
2
p− 1
(
N − 2− 2
p− 1
)]1/(p−1)
.
Galaktionov and Vazquez [10] investigated the uniqueness of solutions to the
problem (1)-(2)λ in the case where a ≡ 1 and λ = L, and showed that the
problem has a classical self-similar solution for t > 0 with certain values of p.
In [10, p. 41] they also conjectured that the problem (1)-(2)λ has exactly two
solutions (the minimal and maximal) when N/(N − 2) < p ≤ (N +2)/(N − 2).
Letting µ = t−1/2 in (3), we see that the self-similar solution w of (1) has
the form
(5) w(x, t) = t−1/(p−1)u(x/
√
t),
where u satisﬁes the elliptic equation
(6) ∆u +
1
2
x · ∇u + 1
p− 1u + u
p = 0 in RN .
2
In addition, if w satisﬁes (2)λ in the sense of L1loc(R
N ), then u satisﬁes
(7)λ lim
r→∞ r
2/(p−1)u(rω) = λa(ω) for a.e. ω ∈ SN−1.
Conversely, if u ∈ C2(RN ) is a solution of (6) satisfying (7)λ, then the function
w deﬁned by (5) satisﬁes (1)-(2)λ in the sense of L1loc(R
N ). (See Lemma B.1 in
[17].)
In this talk we investigate the problem (6)-(7)λ by making use of the meth-
ods for semilinear elliptic equations to derive the results for the Cauchy problem
(1)-(2)λ. First, we show the existence of the minimal solution by employing the
comparison results based on the maximum principle. Next we apply the varia-
tional method due to [1, 6, 4] to show the existence of the second solution of the
problem (6)-(7)λ, which implies the non-uniqueness of solutions to the problem
(1)-(2)λ.
2. Existence of the minimal solution [17, Sec. 4]
For simplicity, we deﬁne Lu by
Lu = ∆u + 1
2
x · ∇u + 1
p− 1u
for u ∈ C2(RN ). First we obtain the following results.
Lemma 1. Let p > (N + 2)/N . Assume that −Lu ≥ 0 in RN , and that
lim inf
|x|→∞
|x|2/(p−1)u(x) ≥ 0.
Then u > 0 or u ≡ 0 in RN . In particular, if −Lu ≥ 0 and u ≥ 0 in RN then
u > 0 or u ≡ 0 in RN .
Lemma 2. Assume that p > (N +2)/N , and that α, β ∈ L∞(SN−1) satisfy
0 ≤ α(ω) ≤ β(ω) for a.e. ω ∈ SN−1. Suppose that there exists a positive
function v satisfying
−Lv ≥ vp in RN and lim
r→∞ r
2/(p−1)v(rω) = β(ω), a.e. ω ∈ SN−1.
Then there exists a positive solution u of the problem
−Lu = up in RN and lim
r→∞ r
2/(p−1)u(rω) = α(ω), a.e. ω ∈ SN−1.
By using of Lemmas 1 and 2 we obtain the following:
3
Theorem 1. Assume that p > (N + 2)/N . Then there exists a constant
λ > 0 such that
(i) for 0 < λ < λ, (6)-(7)λ has a positive minimal solution uλ ∈ C2(RN );
the solution uλ is increasing with respect to λ and satisﬁes ‖uλ‖L∞(RN ) → 0 as
λ → 0;
(ii) for λ > λ, there are no positive solutions u ∈ C2(RN ) of (6)-(7)λ.
3. Weighted Sobolev space
Put ρ(x) = e|x|2/4. Then the equation (6) can be written as
∇ · (ρ∇u) + ρ
(
1
p− 1u + u
p
)
= 0.
Escobedo-Kavian [8] investigated the corresponding functional
I0(u) =
1
2
∫
RN
(
|∇u|2 − 1
p− 1u
2
)
ρdx− 1
p + 1
∫
RN
up+1ρdx
on the weighted functional spaces
Lqρ(R
N ) =
{
u ∈ Lq(RN ) :
∫
RN
uqρdx < ∞
}
for 1 ≤ q < ∞
and
H1ρ(R
N ) =
{
u ∈ H1(RN ) :
∫
RN
(|∇u|2 + u2)ρdx < ∞
}
.
We recall here some results about the weighted Sobolev space H1ρ (RN ).
Lemma 3 [8, 14]. (i) For every u ∈ H1ρ(RN ),
N
2
∫
RN
u2ρdx ≤
∫
RN
|∇u|2ρdx.
(ii) The embedding H1ρ(RN ) ⊂ Lp+1ρ (RN ) is continuous for 1 ≤ p ≤ (N +
2)/(N − 2), and is compact for 1 ≤ p < (N + 2)/(N − 2).
It was shown by [8, 24] that there exists a solution u0 of the problem
(8)


∆u +
1
2
x · ∇u + 1
p− 1u + u
p = 0 in RN ,
u ∈ H1ρ(RN ) and u > 0 in RN ,
with (N + 2)/N < p < (N + 2)/(N − 2). Moreover, it was shown in [8] that
u0 ∈ C2(RN ) and u0(x) = O(e−|x|2/8) as |x| → ∞. The uniqueness of the
solution to the problem (8) was obtained by combining the results [7, 27, 19].
4
Now put
(9) w0(x, t) = t−1/(p−1)u0(x/
√
t),
where u0 is the solution of the problem (8). We note that u0 ∈ Lq(RN ) for all
q ≥ 1 and
‖w0(·, t)‖Lq(RN ) = t−1/(p−1)+N/2q‖u0‖Lq(RN ).
Then w0 solves the the Cauchy problem (4) in C([0,∞);Lq(RN )) for 1 ≤ q <
N(p − 1)/2. By the uniqueness result [19], we ﬁnd that w0 deﬁned by (9)
coincides with the non-unique solution of (4) constructed by [13].
4. Existence of the second solution: subcritical case [17, Sec. 5]
Let uλ be the positive minimal solution of (6)-(7)λ obtained in Theorem
1. In order to ﬁnd a second solution of (6)-(7)λ we introduce the following
problem:
(10)λ


∆u +
1
2
x · ∇u + 1
p− 1u + g(u, uλ) = 0 in R
N ,
u ∈ H1ρ(RN ) and u > 0 in RN ,
where g(t, s) = (t + s)p − sp. We easily see that, if (10)λ possesses a solution
uλ, then we can get another positive solution uλ = uλ + uλ of (6)-(7)λ.
In this section we will show the existence of solutions of (10)λ in the sub-
critical case (N +2)/N < p < (N +2)/(N−2) by using the variational method.
To this end we deﬁne the corresponding functional of (10)λ by
Iλ(u) =
1
2
∫
RN
(
|∇u|2 − 1
p− 1u
2
)
ρdx−
∫
RN
G(u, uλ)ρdx
with u ∈ H1ρ (RN ), where
G(t, s) =
1
p + 1
(t + s)p+1 − 1
p + 1
sp+1 − spt.
We see that the nontrivial critical point u ∈ H1ρ(RN ) of the functional Iλ is
a weak solution of the equation in (10)λ. Moreover, we have uλ ∈ C2(RN )
and uλ > 0 in RN by employing the bootstrap arguments and the maximum
principle.
We will verify the existence of nontrivial solution of (10)λ by means of the
Mountain Pass lemma ([1, 20]).
Lemma 4. For λ ∈ (0, λ) there exist some constants δ = δ(λ) > 0 and
η = η(λ) > 0 such that Iλ(u) ≥ η for all u ∈ H1ρ (RN ) with ‖∇u‖L2ρ = δ.
5
Lemma 5. For any v ∈ H1ρ (RN ) with v ≥ 0, v ≡ 0, we have Iλ(tv) → −∞
as t →∞.
Lemma 6. The functional Iλ satisﬁes the Palais-Smale condition, that is,
any sequence {uk} ⊂ H1ρ (RN ) such that
{Iλ(uk)} is bounded and I ′λ(uk)→ 0 as k →∞
contains a convergent subsequence.
In the proofs of Lemmas 4-6, the following results play a crucial role.
Lemma 7. Let uλ be the minimal solution obtained in Theorem 1 for
λ ∈ (0, λ). Then the linearized eigenvalue problem


−∆w − 1
2
x · ∇w − 1
p− 1w = µp[uλ]
p−1w in RN ,
w ∈ H1ρ (RN ),
has the ﬁrst eigenvalue µ = µ(λ) > 1. Moreover, µ(λ) is strictly decreasing in
λ ∈ (0, λ).
Lemma 7 follows from the fact that uλ is the positive minimal solution.
As a consequence of Lemmas 4-6 we obtain the following:
Theorem 2. Assume that (N + 2)/N < p < (N + 2)/(N − 2). Then, for
0 < λ < λ, there exists a positive solution uλ of (6)-(7)λ satisfying uλ > uλ,
uλ − uλ ∈ H1ρ (RN ), and uλ(x)− uλ(x) = O(e−|x|
2/4) as |x| → ∞.
Furthermore,
uλ − uλ → u0 in H1ρ (RN ) ∩ L∞(RN ) as λ → 0,
where u0 is the solution of the problem (8). In particular, uλ → u0 in L∞(RN )
as λ → 0.
Now we consider the Cauchy problem (1)-(2)λ. Recall that, if u is a solution
of (6)-(7)λ, then the function w deﬁned by (5) is a solution of (1)-(2)λ in the
sense of L1loc(R
N ), and that w0 deﬁned by (9) coincides with the non-unique
solution of (4) constructed by [13]. As a consequence of Theorems 1 and 2, we
obtain the following results.
6
Corollary 1. Assume that p > (N +2)/N . Then there exists a constant λ > 0
such that
(i) for 0 < λ < λ, (1)-(2)λ has a positive self-similar solution wλ; the
solution wλ(·, t) satisﬁes, for each ﬁxed t > 0,
‖wλ(·, t)‖L∞(RN ) → 0 as λ → 0;
(ii) for λ > λ, (1)-(2)λ has no positive self-similar solutions.
Assume, furthermore, that p < (N + 2)/(N − 2). Then (1)-(2)λ has a positive
self-similar solution wλ satisfying wλ > wλ in RN × (0,∞) for 0 < λ < λ. The
solution wλ satisﬁes, for each ﬁxed t > 0,
‖wλ(·, t)− w0(·, t)‖L∞(RN ) → 0 as λ → 0,
where w0 is the non-unique solution of (4) in C([0,∞);Lq(RN )) for 1 ≤ q <
N(p− 1)/2, which is constructed by [13].
5. Existence and nonexistence of second solutions: critical case [18]
In this section we consider the existence and nonexistence of second solutions
of the problem (6)-(7)λ in the critical case p = (N + 2)/(N − 2) by following
the argument due to Brezis-Nirenberg [4].
For the critical growth case, there are serious diﬃculties in obtaining solu-
tions by using variational methods because the Sobolev embedding H1 ⊂ Lp+1
is not compact. It is well known that this lack of compactness exhibits many
interesting existence and nonexistence phenomena. See, e.g., [4, 2].
Let us denote by S the best Sobolev constant of the embedding H1(RN ) ⊂
L2N/(N+2)(RN ), which is given by
S = inf
u∈H1(RN )\{0}
∫
RN
|∇u|2dx
(∫
RN
|u|2N/(N−2)dx
)(N−2)/N .
In the critical case, the functional Iλ satisﬁes the following local Palais-Smale
condition.
Lemma 8. Let p = (N +2)/(N −2). Then Iλ satisﬁes the (PS)c condition
for c ∈ (0, SN/2/N), that is, any sequence {uk} ⊂ H1ρ(RN ) such that
Iλ(uk) → c ∈
(
0,
1
N
SN/2
)
and I ′λ(uk) → 0 as k →∞
contains a convergent subsequence.
7
By Lemma 8 and the Mountain Pass lemma, we obtain the following exis-
tence result.
Lemma 9. Let p = (N+2)/(N−2). Assume that there exists v0 ∈ H1ρ (RN )
with v0 ≥ 0, v0 ≡ 0 such that
(11) sup
t>0
Iλ(tv0) <
1
N
SN/2.
Then there exists a positive solution uλ ∈ H1ρ (RN ) of (10)λ.
Moreover, we have uλ ∈ C2(RN ) by employing the estimate due to Brezis-
Kato [3], based on the Moser’s iteration technique.
In order to ﬁnd a positive function v0 ∈ H1ρ (RN ) satisfying (11), we set
uε(x) =
φ(x)
(ε + |x|2)(N−2)/2 ρ
−1/2 and vε(x) =
uε(x)
‖uε‖Lp+1ρ
for ε > 0, where φ ∈ C∞0 (RN ) is a cut oﬀ function. We remark that the
functional Iλ can be written as
Iλ(u) =
1
2
∫
RN
(
|∇u|2 − 1
p− 1u
2
)
ρdx− 1
p + 1
∫
RN
up+1ρdx
−
∫
RN
H(u, uλ)ρdx
≡ I0(u)−
∫
RN
H(u, uλ)ρdx,
where
H(t, s) = G(t, s)− 1
p + 1
tp+1.
Lemma 10. For suﬃcient small ε > 0, there exists tε > 0 such that
supt>0 Iλ(tvε) = Iλ(tεvε). Moreover, as ε → 0 we have
I0(tεvε) ≤ 1
N
SN/2 +


O(ε), N ≥ 5
O(ε| log ε|), N = 4
O(ε1/2), N = 3
∫
RN
H(tεvε, uλ)ρdx ≥


Cε3/4, N = 5
Cε1/2, N = 4
Cε1/4, N = 3
with some constant C > 0.
8
As a consequence, we obtain the following:
Theorem 3. Let p = (N+2)/(N−2) and N = 3, 4, 5. Then, for 0 < λ < λ,
the problem (6)-(7)λ has a positive solution uλ ∈ C2(RN ) satisfying uλ > uλ
and uλ − uλ ∈ H1ρ(RN ).
On the other hand, for the case N ≥ 6 we obtain the uniqueness result in
the radial class by employing the Pohozaev type identity.
Theorem 4. Let p = (N + 2)/(N − 2) and N ≥ 6. Assume that a ≡ 1 in
(7)λ. Then there exists a constant λ0 ∈ (0, λ) such that (6)-(7)λ has no positive
radial solutions u ∈ C2(RN ) with u ≡ uλ for λ ∈ (0, λ0), that is, (6)-(7)λ has a
unique positive radial solution uλ for 0 < λ < λ0.
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A variational approach to self-similar solutions for semilinear heat equations

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