Chapter 4. Gradient estimates for a general type of nonlinear parabolic
4.3 Liouville type theorems and gradient estimates for some important
4.3.2 On the Einstein-scalar field Lichnerowicz type equations
The purpose of this subsection was to derive Liouville type results and gradient estimates for positive, smooth solutions of the following nonlinear elliptic equation on smooth metric measure spaces (M, g, e−fdà) of dimension n ≥ 3,
∆fu(x) + a(x)u(x) + buα(x) + cuβ(x) = 0 (4.61) and its parabolic counterpart
ut(x, t) = ∆fu(x, t) + a(x, t)u(x, t) + buα(x, t) + cuβ(x, t) (4.62) Here, α, β, b, c ∈ R, a(x) is a C1 function of x in (4.61), and a(x, t) is a function which is C2 in the x-variable and C1 in the t-variable in (4.62).
Assume that the Bakry-Émery Ricci Ricf is bounded below, we now apply Theorem 4.2 to derive a local gradient estimate for the equation (4.62) on the static smooth metric measure space (M, g, e−fdà). For F (u) = buα + cuβ, we
have
2F′(u) − 2F (u) u + 1
h2 F (u)
u
= 2[(α − 1)b]uα−1+ 1
h2buα−1+ [(2β − 1)c]uβ−1+ 1
h2 − 1
cuβ−1, (4.63) where h =
q
1 + lnBu ≥ 1. Note that 0 < h12 ≤ 1 and h12 − 1 ≤ 0. Thus, we get 2(α − 1)buα−1+ 1
h2buα−1
≤ 2[(α − 1)b]+uα−1+ b+uα−1 ≤ {2[(α − 1)b]++ b+} sup
QR,T
uα−1 , (4.64) and
(2β − 1)cuβ−1 + 1
h2 − 1
cuβ−1
≤ [(2β − 1)c]+uβ−1 + c−
h2uβ−1 − c−uβ−1 ≤ {[(2β − 1)c]+− c−} sup
QR,T
uβ−1 .
Plugging this and (4.64) into (4.63) , we imply that P ≤ {2[(α − 1)b]++ b+} sup
QR,T
uα−1 + {[(2β − 1)c]+− c−} sup
QR,T
uβ−1 = P2. From this and Theorem 4.3, we obtain the following gradient estimate result for positive solutions of the equation (4.62).
Theorem 4.10. Under the same assumption as in Theorem 4.2, if 0 < u (x, t) ≤ B for some constant B, is a smooth solution to the nonlinear parabolic equation (4.62) in QR,T, then there exists a constant c depending only n such that
|∇u|
u ≤ c
"√ A R +
rà+ R + 1
√ t + √
K + P2 + Γa
# r
1 + ln B
u (4.65)
in QR
2,T with t ̸= 0, where A = 1 + ln B − ln
QinfR,T
u
, Γa = supQR,T n(a+)
1
2 + |∇a|13o, and
P2 = q
2[(α − 1)b]++ b+sup
QR,T
n
uα−12 o+ q
[(2β − 1)c]+− c−sup
QR,T
n
uβ−12 o.
As an application of Theorem 4.10, we can get a Liouville type result for positive solutions of the Einstein-scalar field Lichnerowicz type equation (4.61).
Corollary 4.12. Let (M, g, e−fdà) be an n-dimensional complete smooth metric measure space with Ricf ≥ 0. Assume that δ ≤ u (x, t) ≤ B for some positive constants δ and B, is a smooth solution to the equation (4.61). If a, b, c, α, β are constants satisfying a ≤ 0, b ≤ 0, c ≥ 0, α ≥ 1 and β ≤ 1
2, then u is constant.
Proof of Corollary 4.12. Suppose that u is a positive solution of (4.44) with δ ≤ u ≤ B for some constants δ, B > 0. Since u does not depend on t, u is also a solution of the prabolic equation (4.62) in the case a, b, c, α, β ∈ R. Furthermore, since a ≤ 0, we have Γa = 0. From the assumption b ≤ 0, c ≥ 0, α ≥ 1 and β ≤ 12, we see that
2[(α − 1)b]++ b+ = 0, [(2β − 1)c]+ − c− = 0.
This shows that P2 = 0. Letting t → +∞ in (4.65) and note that K = Γa = P2 = 0, we get
|∇u|
u ≤ c
"√ A R +
rà+ R
# r
1 + ln B
u . (4.66)
Since δ ≤ u ≤ D, we deduce that A = 1 + ln B − ln infQR,T u ≤ 1 + ln B − ln δ.
Then, letting R → +∞ in (4.66), we obtain |∇u|u ≤ 0. Thus, u is a constant. We finish the proof.
Remark 4.12. It is worth noting that Corollary 4.12 is an improvement of Dung- Khanh-Ngo’s result (see [36], Corollary 2.5).
An immediate application of Corollary 4.8 is the following Liouville type result for positive solutions of Yamabe-type equations of the form (4.67) below.
Corollary 4.13. Let (M, g, e−fdà) be an n-dimensional complete smooth metric measure space with Ricf ≥ 0. Suppose that α, a, b are real numbers. Assume that δ ≤ u (x, t) ≤ B for some positive constants δ and B, is a smooth solution to the following equation
∆fu + au + buα = 0. (4.67)
(i) If α ≥ 1, a < 0, b < 0, then u does not exist.
(ii) If α ≤ 1
2, a < 0, b > 0, then u = α−1q−ab .
Using Theorem 4.5, we can derive a local parabolic gradient estimate for positive smooth solutions to the following nonlinear parabolic equation
ut(x, t) = ∆u(x, t) + a(x, t)u(x, t) + buα(x, t) + cuβ(x, t), (4.68) on Riemaniann manifolds along super Ricci flow (1.4).
Corollary 4.14. Under the same assumption as in Theorem 4.5, if u (x, t) ≤ B for some constant B > 0 in QR,T, then there exists a constant c depending only n such that
|∇u|
u ≤ c
"√ A R + 1
√ t + √
κ + P2+ Γa
# r
1 + ln B
u (4.69)
in QR
2,T with t ̸= 0, where
A = 1 + ln B − ln infQR,T u, Γa = supQ
R,T
n(a+)12 + |∇a|13o, and
P2 = q
2[(α − 1)b]++ b+sup
QR,T
n
uα−12 o+ q
[(2β − 1)c]+− c−sup
QR,T
n
uβ−12 o.
Moreover, applying Theorem 4.8, we obtain the following local gradient estimate for the equation (4.68) under Yamabe flow.
Corollary 4.15. Under the same assumption as in Theorem 4.3, if 0 < u (x, t) ≤ B for some constant B > 0 is a smooth solution to the equation (4.68) in QR,T, then there exists a constant c depending only n such that
|∇u|
u ≤ c
"√ A R +
rà+ R + 1
√ t + √4
K2 + H2 + P2+ Γa
# r
1 + ln B
u (4.70) in QR
2,T with t ̸= 0, where A, P2, Γa are the same as Corollary 4.14.
Inspired by the recent work due to Dung-Khanh-Ngo [36], in the last of this subsection, we will study the gradient estimate for solutions of the following general f-heat equation
ut = ∆fu + au + bu ln u + Auα+ Buβ (4.71) on complete smooth metric measure spaces (M, g, e−fdà) of dimension n ≥ 3, where a, b, A, B, α, and β be constants with A ≤ 0, B ≥ 0, α ≥ 1, β ≤ 12. We
first obtain the following result.
Theorem 4.11. Let (M, g, e−fdà) be an n-dimensional complete smooth metric measure space with Ricf ≥ − (n − 1) K for some constant K ≥ 0 in B (x0, R) . Assume that u ∈ (0, 1] is a smooth solution to the nonlinear heat equation (4.71) in QR,T. Then there exists a constant c depending only n such that
|∇u|
u ≤ c
q
1 − ln infQR,T u
R +
rà+ R + 1
√ t +
√ K +
√ Λ
√
1 − ln u,
for all (x, t) ∈ QR
2,T with t ̸= 0, where Λ = max{b + max{a + b, 0}, 0}.
Proof of Theorem 4.11. We will apply the Theorem 4.2 to the function F (u) = au + bu ln u + Auα + Buβ
to prove Theorem 4.11. Observe that 2F′(u) − 2F (u)
u + 1 h2
F (u) u
= 2b + 1
h2(a + b ln u) + 2[(α − 1)A]uα−1+ 1
h2Auα−1 + [(2β − 1)B]uβ−1+
1 h2 − 1
Buβ−1,
where h = √
1 − ln u ≥ 1. We notice that 2b + 1
h2(a + b ln u) = b + a + b
h2 ≤ b + max{a + b, 0}
≤ max{b + max{a + b, 0}, 0}.
Since A ≤ 0, B ≥ 0, α ≥ 1, β ≤ 12, we obtain 2[(α − 1)A]uα−1+ 1
h2Auα−1 + [(2β − 1)B]uβ−1+ 1
h2 − 1
Buβ−1 ≤ 0.
From the above results, we imply that
P ≤ max{b + max{a + b, 0}, 0} = Λ.
The proof is complete.
When f is constant, using Theorem 4.5, we can give a local gradient estimate
for the positive bounded solutions to the equation (4.71) under the super Ricci flow.
Theorem 4.12. Let (M, g(x, t))t∈[0,T] be a complete solution to the super Ricci flow (1.4) and u be a smooth positive solution to the nonlinear heat equation
ut = ∆u + au + bu ln u + Auα + Buβ (4.72) in the set QR,T, where a, b, A, B, α, and β be constants with A ≤ 0, B ≥ 0, α ≥ 1, β ≤ 12. Assume that |Ric(x, t)| ≤ κ for some constant κ ≥ 0 for all (x, t) ∈ QR,T. If u ∈ (0, 1], then there exists a constant c depending only n such that
|∇u|
u ≤ c
q
1 − ln infQR,T u
R + 1
√ t + √
κ + √ Λ
√ 1 − ln u (4.73)
for all (x, t) ∈ QR
2,T with t ̸= 0, where Λ = max{b + max{a + b, 0}, 0}.
Remark 4.13. In the caseb ≤ 0,Theorem 4.11 is better than Theorem 1.1 in [36].
Besides, Theorem 4.12 can be seen as an extension and improvement Theorem 1.2 in [36].