Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator

Guangwei Du; Xinjing Wang

doi:10.1515/anona-2023-0135

Article Open Access

Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator

Guangwei Du and Xinjing Wang

Published/Copyright: February 16, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

In this article, we consider the parabolic equations with nonlocal Monge-Ampère operators

∂ u ∂ t ( x , t ) − D s θ u ( x , t ) = f ( u ( x , t ) ) , ( x , t ) ∈ R + n × R .

We first prove the narrow region principle and maximal principle for antisymmetric functions, under the condition that u is uniformly bounded, which weaken the general decay condition u → 0 at infinity. Then, the monotonicity of positive solutions is established using the method of moving planes.

Keywords: parabolic equations; nonlocal Monge-Ampère operator; method of moving planes; monotonicity

MSC 2010: 35R11; 35K58

1 Introduction

In this article, we study the monotonicity of positive solutions of the following nonlocal parabolic equation

(1.1) ∂ u ∂ t ( x , t ) − D s θ u ( x , t ) = f ( u ( x , t ) ) , ( x , t ) ∈ R + n × R ,

where D s θ is the so-called Monge-Ampère operator. For fixed t > 0 , it is given by

(1.2) D s θ u ( x , t ) = inf A ∈ A P.V. ∫ R n u ( x , t ) − u ( y , t ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y ,

where 0 < s < 1 , P.V. is the Cauchy principal value, and A = { A ∣ A i s n × n symmetric positive definite matrix, det A = 1 , λ min ( A ) ≥ θ > 0 } . Here, λ min ( A ) is the smallest eigenvalue of matrix A . In order to ensure that the integral in (1.2) is well defined, we suppose that u ∈ C loc 1 , 1 ( Ω ) ∩ ℒ 2 s with

ℒ 2 s = u ∈ L loc 1 ( R n ) ∣ ∫ R n ∣ u ( x ) ∣ 1 + ∣ x ∣ n + 2 s d x < ∞ .

The classical Monge-Ampère equation

(1.3) det D 2 u = f ( x , u , D u )

has many important applications in the fields of affine geometry and mass transformation (see [1,18,20, 21] and references therein). Here, det D 2 u is the determinant of the Hessian matrix D 2 u . It can be expressed as another form

( det D 2 u ( x ) ) 1 n = inf { a i j ∂ i j u ( x ) , det { a i j } = 1 , { a i j } > 0 } ,

which is equivalent to the operator

( det D 2 u ( x ) ) 1 n = inf { Δ [ u ∘ A ] ( x ) , det A = 1 , A > 0 } ,

where u ∘ A denotes the composition of the linear transformation x → A x of u .

Caffarelli and Charro [1] extended this definition to the fractional order case for any s ∈ ( 0 , 1 ) by

(1.4) F [ u ] ( x ) = inf { ( − Δ ) s [ u ∘ A ] ( x ) , det A = 1 , A > 0 } .

They showed that the fractional Monge-Ampère operator is strictly elliptic, which allowed us to apply the well-known regularity results for uniformly elliptic operators and also deduced that solutions are classical. For more literature studies about the Monge-Ampère equations, please refer to [2,7,14].

In [19], Niu proved the monotonicity of positive solutions of the following nonlinear equations

(1.5) D s θ u ( x ) = f ( u ( x ) ) , x ∈ Ω

in infinite slab and upper half space. Using the same method, Chen et al. [3] derived the monotonicity of solutions of (1.5) in both bounded domains and the whole space R n . Later on, in [4], they obtained the symmetry of solutions of (1.5) in an ellipsoid region and the whole space R n and further proved the non-existence of positive solutions on the upper half space R + n . For more information about equations with nonlocal Monge-Ampère operators, see [5,16] and references therein.

By (1.2), we can observe that the operator D s θ is closely related to the fractional Laplacian

( − Δ ) s u ( x , t ) = C n , s P.V. ∫ R n u ( x , t ) − u ( y , t ) ∣ y − x ∣ n + 2 s d y ,

where C n , s is a normalization constant. Obviously, the fractional Laplacian is a nonlocal operator, and we have the following comparison (see [19])

(1.6) D s θ u ( x , t ) ≤ − C n , s − 1 ( − Δ ) s u ( x , t ) .

Due to the nonlocality of fractional Laplacian, the methods of handling elliptic operators do not work anymore. Instead of using the extension method introduced by Caffarelli and Silvestre [6], Chen et al. [10] introduced a direct method of moving planes to deal with the fractional Laplacian problems, using which one can obtain the monotonicity and symmetry of positive solutions for various nonlocal problems (please refer to [8,9] and references therein).

In [15], the existence of solutions to the admissible boundary control problems for parabolic-type equations is proved by the Laplace transform method. Xu [22] obtained the decay estimate and the finite time blow-up of solutions for the degenerate parabolic equations constructed by Hörmander’s vector fields. With the help of the Nehari flow and Levine’s concavity method, the global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity is studied in [13]. For the nonlocal parabolic equation (1.1), the operator is a concave envelope of fractional linear operators (see (1.2)), where the set of operators is a degenerate class that corresponds to all affine transformations of determinant one of a given multiple of the fractional Laplacian. The monotonicity of positive solutions can be established using the method of moving planes.

Chen et al. [11,12] developed the method of moving planes for fractional parabolic problem and obtained the monotonicity of positive solutions to the following fractional parabolic problem

∂ u ∂ t ( x , t ) + ( − Δ ) s u ( x , t ) = f ( u ( x , t ) ) , ( x , t ) ∈ R + n × R , u ( x , t ) = 0 , ( x , t ) ∉ R + n × R .

Inspired by the methods in [11], we try to generalize the results in [19] to equations involving nonlocal Monge-Ampère operator to the parabolic equations (1.1) with the nonlocal Monge-Ampère operator. It is worth noting that, in this article, the condition of u is uniformly bounded, which weakens the general decay condition u → 0 at infinity. Before stating our main results, we give some notations.

Denote x = ( x 1 , x ′ ) ∈ R n , x λ = ( 2 λ − x 1 , x ′ ) , where x ∈ R n and x ′ ∈ R n − 1 . Let

Σ λ ≔ { x ∈ R n ∣ x 1 < λ }

and

T λ ≔ { x ∈ R n ∣ x 1 = λ } .

Set u λ ( x , t ) = u ( x λ , t ) and w λ ( x , t ) = u ( x λ , t ) − u ( x , t ) .

Theorem 1.1

(Narrow region principle) Let Ω be a bounded or unbounded narrow region in Σ λ such that it is contained in { x ∣ λ − 2 l < x 1 < λ } with small l. Assume that w ( x , t ) ∈ ( C loc 1 , 1 ( Ω ) ∩ ℒ 2 s ) × C 1 ( R ) is uniformly bounded with respect to variable t and lower semi-continuous in x on Ω ¯ , and it satisfies

(1.7) ∣ w ( x , t ) ∣ ≤ o ( 1 ) ∣ x ∣ γ a s ∣ x ∣ → + ∞ ,

for any 0 < γ < 2 s , and

(1.8) ∂ w ∂ t ( x , t ) − D s θ w ( x , t ) ≥ c ( x , t ) w ( x , t ) , ( x , t ) ∈ Ω × R , w ( x , t ) ≥ 0 , ( x , t ) ∈ ( Σ λ \ Ω ) × R , w ( x λ , t ) = − w ( x , t ) , ( x , t ) ∈ Σ λ × R .

If c ( x , t ) is bounded from above, then for sufficiently small l, we have

(1.9) w ( x , t ) ≥ 0 , ( x , t ) ∈ Σ λ × R .

Furthermore, the following strong maximum principle holds

either w ( x , t ) > 0 , ∀ ( x , t ) ∈ R + n × R ; o r ∃ t 0 , s u c h t h a t w ( x , t 0 ) ≡ 0 , ∀ x ∈ R n .

Theorem 1.2

Let Ω be a bounded or unbounded narrow region in Σ λ , and assume that the width of Ω in x 1 direction is bounded. Suppose that w ( x , t ) ∈ ( C loc 1 , 1 ( Ω ) ∩ ℒ 2 s ) × C 1 ( R ) is uniformly bounded with respect to variable t and lower semi-continuous in x on Ω ¯ , and

(1.10) ∂ w ∂ t ( x , t ) − D s θ w ( x , t ) ≥ c ( x , t ) w ( x , t ) , ( x , t ) ∈ Ω × R , w ( x , t ) ≥ 0 , ( x , t ) ∈ ( Σ λ \ Ω ) × R , w ( x λ , t ) = − w ( x , t ) , ( x , t ) ∈ Σ λ × R .

(1.11) c ( x , t ) ≤ 0 o r c ( x , t ) > 0 i s s m a l l , ( x , t ) ∈ Ω × R ,

then

w ( x , t ) ≥ 0 , ( x , t ) ∈ Σ λ × R .

Furthermore, the following strong maximum principle holds

e i t h e r w ( x , t ) > 0 o r w ( x , t ) ≡ 0 , ( x , t ) ∈ Ω × R .

Based on Theorems 1.1 and 1.2, we can obtain the strict monotonicity of solutions to the following parabolic problem

(1.12) ∂ u ∂ t ( x , t ) − D s θ u ( x , t ) = f ( u ( x , t ) ) , ( x , t ) ∈ R + n × R , u ( x , t ) = 0 , ( x , t ) ∉ R + n × R .

Theorem 1.3

Assume that f : [ 0 , ∞ ) → R is a C 1 function with f ( 0 ) = 0 , f ′ ( 0 ) ≤ 0 , and u ( x , t ) ∈ ( C loc 1 , 1 ( Ω ) ∩ ℒ 2 s ) × C 1 ( R ) . If u is a uniformly bounded positive solution of (1.12), then it is increaing in x 1 and

(1.13) ∂ u ∂ x 1 ( x , t ) > 0 , ( x , t ) ∈ R + n × R .

This article is organized as follows. In Section 2, we prove the narrow region principle and the maximum principle for anti-symmetric functions. In Section 3, we establish the monotonicity of positive solutions of (1.12) in half spaces.

2 Maximum principles

In this section, we prove Theorems 1.1 and 1.2. We first prove the maximum principle and the Hopf’s lemma for antisymmetric functions.

Lemma 2.1

(Lemma 2.1, [11]) Let x = ( x 1 , x ′ ) and h ( x ) = h 1 ( x 1 ) h 2 ( x ′ ) , γ < β < 2 s , where

h 1 ( x 1 ) = 1 − x 1 2 l 2 + s + 1 a n d h 2 ( x ′ ) = ( 1 + ∣ x ′ ∣ 2 ) β 2 .

Then, there exists constant C 1 > 0 such that for any sufficiently small l ,

(2.1) ( − Δ ) s h ( x ) h ( x ) ≥ C 1 l 2 s , f o r a l l ∣ x 1 ∣ < l .

Now, we present a similar evaluation of − D s θ h ( x ) h ( x ) , which will play a key role in the proof of Theorem 1.1.

Lemma 2.2

Let x = ( x 1 , x ′ ) and h ( x ) = h 1 ( x 1 ) h 2 ( x ′ ) , γ < β < 2 s , where

h 1 ( x 1 ) = 1 − x 1 2 l 2 + s + 1 a n d h 2 ( x ′ ) = ( 1 + ∣ x ′ ∣ 2 ) β 2 .

Then, there exists C 0 > 0 such that, for any sufficiently small l, we have

(2.2) − D s θ h ( x ) h ( x ) ≥ C 0 l 2 s , f o r a l l ∣ x 1 ∣ < l .

Proof

By Lemma 2.1 and Inequality (1.6), it is easy to check that

□ − D s θ h ( x ) h ( x ) ≥ C n , s − 1 ( − Δ ) s h ( x ) h ( x ) ≥ C 0 l 2 s .

Lemma 2.3

Let Ω be a bounded domain in Σ λ . Suppose that w λ ( x , t ) ∈ ( C loc 1 , 1 ( Ω ) ∩ ℒ 2 s ) × C 1 ( [ 0 , + ∞ ) ) is lower semi-continuous in variable x on Ω ¯ and satisfies

(2.3) ∂ w λ ∂ t ( x , t ) − D s θ w λ ( x , t ) ≥ c λ ( x , t ) w λ ( x , t ) , ( x , t ) ∈ Ω × [ 0 , ∞ ) , w λ ( x λ , t ) = − w λ ( x , t ) , ( x , t ) ∈ Σ λ × [ 0 , ∞ ) , w λ ( x , t ) ≥ 0 , ( x , t ) ∈ ( Σ λ \ Ω ) × [ 0 , ∞ ) , w λ ( x , 0 ) ≥ 0 , x ∈ Ω .

If c λ ( x , t ) is bounded from above, then w λ ( x , t ) ≥ 0 , ( x , t ) ∈ Ω × [ 0 , T ] , for all T > 0 .

Proof

Since c λ ( x , t ) is bounded from above, we can choose m < 0 such that m + c λ ( x , t ) < 0 . Set

w λ ˜ ( x , t ) = e m t w λ ( x , t ) ,

which satisfies the inequality

(2.4) ∂ w λ ˜ ∂ t ( x , t ) − D s θ w λ ˜ ( x , t ) ≥ ( m + c λ ( x , t ) ) w λ ˜ ( x , t ) , ( x , t ) ∈ Ω × [ 0 , ∞ ) .

Then, we claim

(2.5) w λ ˜ ( x , t ) ≥ min Ω w λ ˜ ( x , 0 ) ≥ 0 , ( x , t ) ∈ Ω × [ 0 , T ] .

If (2.5) is not valid, then there exists ( x 0 , t 0 ) ∈ Ω × [ 0 , T ] , such that

w λ ˜ ( x 0 , t 0 ) = min Σ λ × [ 0 , T ] w λ ˜ ( x , t ) < 0 .

Then,

(2.6) ( m + c λ ( x 0 , t 0 ) ) w λ ˜ ( x 0 , t 0 ) > 0 and ∂ w λ ˜ ∂ t ( x 0 , t 0 ) ≤ 0 .

For A ∈ A , since det A = 1 , λ min A ≥ θ > 0 , we deduce that λ min A ≤ θ 1 − n . Thus, by the definition of D s θ , for any sequences ε j → 0 , there exists A j ∈ A such that

− D s θ w λ ˜ ( x 0 , t 0 ) = − inf A ∈ A P.V. ∫ R n w λ ˜ ( y , t 0 ) − w λ ˜ ( x 0 , t 0 ) ∣ A − 1 ( y − x 0 ) ∣ n + 2 s d y ≤ − P.V. ∫ R n w λ ˜ ( y , t 0 ) − w λ ˜ ( x 0 , t 0 ) ∣ A j − 1 ( y − x 0 ) ∣ n + 2 s d y + ε j ≤ − C θ P.V. ∫ R n w λ ˜ ( y , t 0 ) − w λ ˜ ( x 0 , t 0 ) ∣ y − x 0 ∣ n + 2 s d y + ε j = C θ P.V. ∫ R n w λ ˜ ( x 0 , t 0 ) − w λ ˜ ( y , t 0 ) ∣ y − x 0 ∣ n + 2 s d y + ε j ≤ ε j ,

where C θ is a constant with respect to the eigenvalue of A j ∈ A .

Letting j → ∞ , we have

(2.7) − D s θ w λ ˜ ( x 0 , t 0 ) ≤ 0 .

Therefore,

0 ≥ ∂ w λ ˜ ∂ t ( x 0 , t 0 ) − D s θ w λ ˜ ( x 0 , t 0 ) ≥ ( m + c λ ( x 0 , t 0 ) ) w λ ˜ ( x 0 , t 0 ) > 0 ,

which is a contradiction, so (2.5) holds.□

Lemma 2.4

(Hopf’s lemma) Assume that w λ ( x , t ) ∈ ( C loc 1 , 1 ( Ω ) ∩ ℒ 2 s ) × C 1 ( R ) is bounded and satisfies

(2.8) ∂ w λ ∂ t ( x , t ) − D s θ w λ ( x , t ) ≥ c λ ( x , t ) w λ ( x , t ) , ( x , t ) ∈ Σ λ × R , w λ ( x , t ) ≥ 0 , ( x , t ) ∈ Σ λ × R , w λ ( x λ , t ) = − w λ ( x , t ) , ( x , t ) ∈ Σ λ × R ,

where c λ ( x , t ) is bounded from below. If there exists a point x ∈ Σ λ such that

(2.9) w λ ( x , t ) > 0 , ( x , t ) ∈ Σ λ × R ,

then

∂ w λ ∂ x 1 ( x 0 , t 0 ) < 0 , f o r a l l ( x 0 , t 0 ) ∈ T λ × R .

Proof

Without loss of generality, we only prove the case that λ = 0 . If λ = 0 , for convenience, denote Σ λ = Σ 0 , T λ = T 0 , x λ = x 0 = ( − x 1 , x ′ ) , y λ = y 0 = ( − y 1 , y ′ ) , w λ ( x , t ) = w 0 ( x , t ) , and c λ ( x , t ) = c 0 ( x , t ) . Set w ˜ ( x , t ) = e m t w 0 ( x , t ) , m > 0 . Due to the boundedness of c 0 ( x , t ) , we can take m large enough such that m + c 0 ( x , t ) ≥ 0 . For fixed t ˜ , w ˜ ( x , t ) satisfies

(2.10) ∂ w ˜ ∂ t ( x , t ) = m e m t w 0 ( x , t ) + e m t ∂ w 0 ∂ t ( x , t ) , D s θ w ˜ ( x , t ) = e m t D s θ w 0 ( x , t ) .

Combining (2.8) with (2.10), we have

(2.11) ∂ w ˜ ∂ t ( x , t ) − D s θ w ˜ ( x , t ) ≥ ( m + c 0 ( x , t ) ) w ˜ ( x , t ) ≥ 0 , ( x , t ) ∈ Σ 0 × [ t ˜ − 1 , t ˜ + 1 ] .

By (2.9) and w 0 ∈ C 1 ( R ) , there exist a set D ⊂ ⊂ Σ 0 and a constant c > 0 such that

(2.12) w 0 ( x , t ) > c , ( x , t ) ∈ D × [ t ˜ − 1 , t ˜ + 1 ] .

Let D 0 be the reflection of D about the plane T 0 for any time t . Set g ( x ) = x 1 ζ ( x ) , where

ζ ( x ) = ζ ( ∣ x ∣ ) = 1 , ∣ x ∣ < ε , 0 , ∣ x ∣ ≥ 2 ε ,

and 0 ≤ ζ ( x ) ≤ 1 , ζ ( x ) ∈ C 0 ∞ ( B 2 ε ( 0 ) ) . Obviously, g ( x ) satisfies the following equality

g ( − x 1 , x 2 , … , x n ) = g ( x 1 , x 2 , … , x n ) .

Let w ̲ ( x , t ) = χ D ∪ D 0 ( x ) w ˜ ( x , t ) + δ η ( t ) g ( x ) , where

χ D ∪ D 0 ( x ) = 1 , x ∈ D ∪ D 0 , 0 , x ∉ D ∪ D 0 ,

and η ( t ) ∈ C 0 ∞ ( [ t ˜ − 1 , t ˜ + 1 ] ) satisfies

η ( t ) = 1 , t ∈ t ˜ − 1 2 , t ˜ + 1 2 , 0 , t ∉ [ t ˜ − 1 , t ˜ + 1 ] .

Since g ( x ) ∈ C 0 ∞ ( B 2 ε ( 0 ) ) , by the definition of D s θ , for any sequences ε j → 0 , there exists A j ∈ A such that

(2.13) − D s θ g ( x ) = − inf A ∈ A P.V. ∫ R n g ( y ) − g ( x ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y ≤ − P.V. ∫ R n g ( y ) − g ( x ) ∣ A j − 1 ( y − x ) ∣ n + 2 s d y + ε j ≤ − C θ P.V. ∫ R n g ( y ) − g ( x ) ∣ y − x ∣ n + 2 s d y + ε j = C θ P.V. ∫ R n g ( x ) − g ( y ) ∣ y − x ∣ n + 2 s d y + ε j = C θ C n , s ( − Δ ) s g ( x ) + ε j ≤ C 0 x 1 + ε j ,

where C θ is a positive constant with respect to the eigenvalue of A ∈ A and the last inequality can be find in Lemma A.1 of [11].

For ( x , t ) ∈ ( B 2 ε ( 0 ) ∩ Σ 0 ) × [ t ˜ − 1 , t ˜ + 1 ] , we have

(2.14) ∂ w ̲ ∂ t ( x , t ) = ∂ ( χ D ∪ D 0 ( x ) w ˜ ( x , t ) ) ∂ t + δ η ′ ( t ) g ( x )

and

(2.15) − D s θ w ̲ ( x , t ) ≤ − D s θ ( χ D ∪ D 0 ( x ) w ˜ ( x , t ) ) − D s θ ( δ η ( t ) g ( x ) ) .

For any sequences ε j → 0 , there exists A j ∈ A such that

(2.16) − D s θ ( χ D ∪ D 0 ( x ) w ˜ ( x , t ) ) = − inf A ∈ A P.V. ∫ R n χ D ∪ D 0 ( y ) w ˜ ( y , t ) − χ D ∪ D 0 ( x ) w ˜ ( x , t ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y = − inf A ∈ A P.V. ∫ R n χ D ∪ D 0 ( y ) w ˜ ( y , t ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y ≤ − P.V. ∫ R n χ D ∪ D 0 ( y ) w ˜ ( y , t ) ∣ A j − 1 ( y − x ) ∣ n + 2 s d y + ε j ≤ − C θ P.V. ∫ R n χ D ∪ D 0 ( y ) w ˜ ( y , t ) ∣ y − x ∣ n + 2 s d y + ε j = C θ P.V. ∫ R n − χ D ∪ D 0 ( y ) w ˜ ( y , t ) ∣ y − x ∣ n + 2 s d y + ε j = C θ P.V. ∫ D − w ˜ ( y , t ) ∣ y − x ∣ n + 2 s d y + C θ P.V. ∫ D 0 − w ˜ ( y , t ) ∣ y − x ∣ n + 2 s d y + ε j = C θ P.V. ∫ D − w ˜ ( y , t ) ∣ y − x ∣ n + 2 s d y + C θ P.V. ∫ D − w ˜ ( y 0 , t ) ∣ y 0 − x ∣ n + 2 s d y + ε j = C θ P.V. ∫ D 1 ∣ y 0 − x ∣ n + 2 s − 1 ∣ y − x ∣ n + 2 s w ˜ ( y , t ) d y + ε j = − C θ ∫ D 2 ( n + 2 s ) y 1 ζ ( y ) n + 2 s + 2 w ˜ ( y , t ) d y + ε j ≤ − C 2 + ε j ,

where ζ ( y ) is between ∣ y − x ∣ and ∣ y 0 − x ∣ , C 2 is a positive constant, and C θ is a positive constant with respect to the eigenvalue of A ∈ A . We used the mean value theorem to h ( z ) = z − n + 2 s 2 over [ z 1 , z 2 ] with z 1 = ∣ y − x ∣ 2 and z 2 = ∣ y 0 − x ∣ 2 in the second equality from the bottom of (2.16).

Thus, for ( x , t ) ∈ ( B 2 ε ( 0 ) ∩ Σ 0 ) × [ t ˜ − 1 , t ˜ + 1 ] , we obtain

∂ w ̲ ∂ t ( x , t ) − D s θ w ̲ ( x , t ) ≤ ∂ ( χ D ∪ D 0 ( x ) w ˜ ( x , t ) ) ∂ t + δ η ′ ( t ) g ( x ) − D s θ ( χ D ∪ D 0 ( x ) w ˜ ( x , t ) ) − D s θ ( δ η ( t ) g ( x ) ) = δ η ′ ( t ) g ( x ) − D s θ ( χ D ∪ D 0 ( x ) w ˜ ( x , t ) ) − δ η ( t ) D s θ g ( x ) ≤ δ η ′ ( t ) g ( x ) − C 2 + ε j + δ η ( t ) ( C 0 x 1 + ε j ) .

Furthermore, taking δ small enough, and letting j → ∞ , we obtain

(2.17) ∂ w ̲ ∂ t ( x , t ) − D s θ w ̲ ( x , t ) ≤ 0 , ( x , t ) ∈ ( B 2 ε ( 0 ) ∩ Σ 0 ) × [ t ˜ − 1 , t ˜ + 1 ] .

Let v ( x , t ) = w ˜ ( x , t ) − w ̲ ( x , t ) . It is easy to check that v ( x , t ) = − v ( x 0 , t ) . Combining (2.10) with (2.17), we have

∂ v ∂ t ( x , t ) − D s θ v ( x , t ) ≥ 0 , ( x , t ) ∈ ( Σ 0 \ ( B 2 ε ( 0 ) ∩ Σ 0 ) ) × [ t ˜ − 1 , t ˜ + 1 ] .

By the definition of w ̲ ( x , t ) , we know

v ( x , t ) ≥ 0 , ( x , t ) ∈ ( B 2 ε ( 0 ) ∩ Σ 0 ) × [ t ˜ − 1 , t ˜ + 1 ] ,

and v ( x , t ˜ − 1 ) ≥ 0 , x ∈ Σ 0 . Applying Lemma 2.3 to v ( x , t ) with c 0 ( x , t ) = 0 , we obtain

v ( x , t ) ≥ 0 , ( x , t ) ∈ ( B 2 ε ( 0 ) ∩ Σ 0 ) × [ t 0 − 1 , t 0 + 1 ] .

Thus, we have

e m t w 0 ( x , t ) − δ g ( x ) η ( t ) ≥ 0 , ( x , t ) ∈ ( B 2 ε ( 0 ) ∩ Σ 0 ) × [ t ˜ − 1 , t ˜ + 1 ] .

It follows that

w 0 ( x , t ) ≥ e − m t δ x 1 , ( x , t ) ∈ ( B ε ( 0 ) ∩ Σ 0 ) × t ˜ − 1 2 , t ˜ + 1 2 .

Since w 0 ( 0 , t ˜ ) = 0 , then

w 0 ( x , t ˜ ) − w 0 ( 0 , t ˜ ) x 1 − 0 ≥ δ e − m t ˜ > 0 , x ∈ B ε ( 0 ) ∩ Σ 0 .

Therefore,

□ ∂ w 0 ∂ x 1 ( 0 , t ˜ ) < 0 .

Next, we present the proof of Theorem 1.1.

Proof of Theorem 1.1

Let

h ( x ) = 1 − ( x 1 − λ ) 2 l 2 + s + 1 ( 1 + ∣ x ′ ∣ 2 ) β 2 , γ < β < 2 s ,

and

w ¯ ( x , t ) = e m t w ( x , t ) h ( x ) ,

where m is a positive constant.

From the definition of the Monge-Ampère operator, we have

(2.18) D s θ ( h ( x ) w ¯ ( x , t ) ) = inf A ∈ A P.V. ∫ R n h ( y ) w ¯ ( y , t ) − h ( x ) w ¯ ( x , t ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y = inf A ∈ A P.V. ∫ R n h ( y ) w ¯ ( y , t ) − h ( y ) w ¯ ( x , t ) + h ( y ) w ¯ ( x , t ) − h ( x ) w ¯ ( x , t ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y ≥ inf A ∈ A P.V. ∫ R n h ( y ) ( w ¯ ( y , t ) − w ¯ ( x , t ) ) + ( h ( y ) − h ( x ) ) w ¯ ( x , t ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y ≥ inf A ∈ A P.V. ∫ R n h ( y ) ( w ¯ ( y , t ) − w ¯ ( x , t ) ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y + inf A ∈ A P.V. ∫ R n ( h ( y ) − h ( x ) ) w ¯ ( x , t ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y = inf A ∈ A P.V. ∫ R n h ( y ) ( w ¯ ( y , t ) − w ¯ ( x , t ) ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y + w ¯ ( x , t ) D s θ h ( x ) .

By (1.8), we obtain

(2.19) ∂ w ¯ ∂ t ( x , t ) − m w ¯ ( x , t ) − D s θ ( h ( x ) w ¯ ( x , t ) ) h ( x ) ≥ c ( x , t ) w ¯ ( x , t ) , ( x , t ) ∈ Ω × R .

Combining (1.8), (2.18), and (2.19), then w ¯ ( x , t ) satisfies

(2.20) ∂ w ¯ ∂ t ( x , t ) − 1 h ( x ) inf A ∈ A P.V. ∫ R n h ( y ) ( w ¯ ( y , t ) − w ¯ ( x , t ) ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y ≥ c ( x , t ) + m + D s θ ( h ( x ) ) h ( x ) w ¯ ( x , t ) , ( x , t ) ∈ Ω × R , w ¯ ( x , t ) ≥ 0 , ( x , t ) ∈ ( Σ λ \ Ω ) × R , w ¯ ( x , t ) → 0 , ( x , t ) ∈ Σ λ × R , ∣ x ′ ∣ → + ∞ .

We claim that

(2.21) w ¯ ( x , t ) ≥ min { 0 , inf Ω w ¯ ( x , t ¯ ) } , ( x , t ) ∈ Ω × [ t ¯ , T ] , for all [ t ¯ , T ] ⊂ R .

If (2.21) is not true, then there exists ( x 0 , t 0 ) ∈ [ t ¯ , T ] such that

w ¯ ( x 0 , t 0 ) = inf Σ λ × [ t ¯ , T ] w ¯ ( x , t ) < min { 0 , inf Ω w ¯ ( x , t ¯ ) } ,

and

∂ w ¯ ∂ t ( x 0 , t 0 ) ≤ 0 .

For any sequence ε j → 0 , there exists A j ∈ A such that

(2.22) inf A ∈ A P.V. ∫ R n h ( y ) ( w ¯ ( y , t 0 ) − w ¯ ( x 0 , t 0 ) ) ∣ A − 1 ( y − x 0 ) ∣ n + 2 s d y ≥ P.V. ∫ R n h ( y ) ( w ¯ ( y , t 0 ) − w ¯ ( x 0 , t 0 ) ) ∣ A j − 1 ( y − x 0 ) ∣ n + 2 s d y − ε j ≥ C θ P.V. ∫ R n h ( y ) ( w ¯ ( y , t 0 ) − w ¯ ( x 0 , t 0 ) ) ∣ y − x 0 ∣ n + 2 s d y − ε j = C θ P.V. ∫ Σ λ h ( y ) ( w ¯ ( y , t 0 ) − w ¯ ( x 0 , t 0 ) ) ∣ y − x 0 ∣ n + 2 s d y + C θ P.V. ∫ ( Σ λ ) c h ( y ) ( w ¯ ( y , t 0 ) − w ¯ ( x 0 , t 0 ) ) ∣ y − x 0 ∣ n + 2 s d y − ε j = C θ P.V. ∫ Σ λ h ( y ) ( w ¯ ( y , t 0 ) − w ¯ ( x 0 , t 0 ) ) ∣ y − x 0 ∣ n + 2 s d y + C θ P.V. ∫ Σ λ h ( y λ ) ( w ¯ ( y λ , t 0 ) − w ¯ ( x 0 , t 0 ) ) ∣ y λ − x 0 ∣ n + 2 s d y − ε j ≥ C θ P.V. ∫ Σ λ w ( y , t 0 ) e m t 0 − w ¯ ( x 0 , t 0 ) h ( y ) ∣ y λ − x 0 ∣ n + 2 s d y + C θ P.V. ∫ Σ λ w ( y λ , t 0 ) e m t 0 − w ¯ ( x 0 , t 0 ) h ( y λ ) ∣ y λ − x 0 ∣ n + 2 s d y − ε j = C θ P.V. ∫ Σ λ − w ¯ ( x 0 , t 0 ) ( h ( y ) + h ( y λ ) ) ∣ y λ − x 0 ∣ n + 2 s d y − ε j ,

where we have used the fact ∣ y − x 0 ∣ ≤ ∣ y λ − x 0 ∣ and antisymmetry of w ( x , t ) , and C θ is a constant with respect to the eigenvalue of A ∈ A . Combining (2.22) with h ( x ) > 0 and letting ε j → 0 , we have

inf A ∈ A P.V. ∫ R n h ( y ) ( w ¯ ( y , t 0 ) − w ¯ ( x 0 , t 0 ) ) ∣ A − 1 ( y − x 0 ) ∣ n + 2 s d y ≥ 0 .

Since c ( x , t ) is bounded from above, we apply Lemma 2.2 and choose m = C 0 2 l 2 s with l sufficiently small, then

c ( x 0 , t 0 ) + m + D s θ ( h ( x 0 ) ) h ( x 0 ) w ¯ ( x 0 , t 0 ) > 0 .

Therefore, by (2.20), we know

∂ w ¯ ∂ t ( x , t ) − 1 h ( x ) inf A ∈ A P.V. ∫ R n h ( y ) ( w ¯ ( y , t ) − w ¯ ( x , t ) ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y ≥ c ( x , t ) + m + D s θ ( h ( x ) ) h ( x ) w ¯ ( x , t ) > 0 ,

which is impossible, and this verifies (2.21). Hence, we obtain

w ¯ ( x , t ) ≥ min { 0 , inf Ω w ¯ ( x , t ¯ ) } = min 0 , e m t ¯ inf Ω w ( x , t ¯ ) h ( x ) ≥ − C e m t ¯ .

Thus,

w ( x , t ) ≥ − C e − m ( t − t ¯ ) h ( x ) .

Since t ¯ ( < T ) ∈ R and letting t ¯ → − ∞ , we have

(2.23) w ( x , t ) ≥ 0 , ( x , t ) ∈ Ω × R .

Combining with (2.20) and (2.23), we conclude that

w ( x , t ) ≥ 0 , ( x , t ) ∈ Σ λ × R .

Furthermore, if there is a point ( x 0 , t 0 ) ∈ Ω × R such that w ( x 0 , t 0 ) = 0 , then

w ( x 0 , t 0 ) = inf Σ λ × R w ( x , t ) = 0 , ∂ w ∂ t ( x 0 , t 0 ) = 0 .

Therefore,

D s θ ( w ( x 0 , t 0 ) ) = inf A ∈ A P.V. ∫ R n w ( y , t 0 ) − w ( x 0 , t 0 ) ∣ A − 1 ( y − x 0 ) ∣ n + 2 s d y = inf A ∈ A P.V. ∫ R n w ( y , t 0 ) ∣ A − 1 ( y − x 0 ) ∣ n + 2 s d y ≥ P.V. ∫ R n w ( y , t 0 ) ∣ A j − 1 ( y − x 0 ) ∣ n + 2 s d y − ε j ≥ C θ P.V. ∫ R n w ( y , t 0 ) ∣ y − x 0 ∣ n + 2 s d y − ε j ,

and then letting ε j → 0 , we have D s θ ( w ( x 0 , t 0 ) ) > 0 .

By (1.8), we have

0 > ∂ w ∂ t ( x 0 , t 0 ) − D s θ w ( x 0 , t 0 ) ≥ c ( x 0 , t 0 ) w ( x 0 , t 0 ) = 0 .

Thus, we derive a contraction, and hence, the strong maximum principle holds. This completes the proof of Theorem 1.1.□

Next, we give the proof of Theorem 1.2.

Proof of Theorem 1.2

Denote a ˆ = sup { ∣ x 1 ∣ , x ∈ Ω } . Let x = ( x 1 , x ′ ) , a = 1 a ˆ + 1 , and

h ( x ) = [ ( 1 − a 2 x 1 2 ) + s + 1 ] ( 1 + ∣ b x ′ ∣ 2 ) β 2 , γ < β < 2 s .

By Lemma 2.2, there exists a constant b associated with a such that

(2.24) − D s θ h ( x ) h ( x ) ≥ C 0 a 2 s .

Denote

w ¯ ( x , t ) = e m t w ( x , t ) h ( x ) ,

where m is a positive constant to be chosen later. By (1.8), w ¯ ( x , t ) satisfies

(2.25) ∂ w ¯ ∂ t ( x , t ) − 1 h ( x ) inf A ∈ A P.V. ∫ R n h ( y ) ( w ¯ ( y , t ) − w ¯ ( x , t ) ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y ≥ c ( x , t ) + m + D s θ ( h ( x ) ) h ( x ) w ¯ ( x , t ) , ( x , t ) ∈ Ω × R , w ¯ ( x , t ) ≥ 0 , ( x , t ) ∈ ( Σ λ \ Ω ) × R , w ¯ ( x , t ) → 0 , ( x , t ) ∈ Σ λ × R , ∣ x ′ ∣ → + ∞ .

Next, we claim that

(2.26) w ¯ ( x , t ) ≥ min { 0 , inf Ω w ¯ ( x , t ¯ ) } , ( x , t ) ∈ Ω × [ t ¯ , T ] , for all [ t ¯ , T ] ⊂ R ,

which can be proved by a contraction argument. In fact, if (2.26) is not true, then there exists ( x 0 , t 0 ) ∈ [ t ¯ , T ] such that

(2.27) w ¯ ( x 0 , t 0 ) = inf Σ λ × [ t ¯ , T ] w ¯ ( x , t ) < min { 0 , inf Ω w ¯ ( x , t ¯ ) } ,

and hence,

∂ w ¯ ∂ t ( x 0 , t 0 ) ≤ 0 .

By the assumption on c ( x , t ) , we may assume that

c ( x 0 , t 0 ) < C 0 a 2 s 2 .

Choosing m = C 0 a 2 s 2 , by (2.24), we have

c ( x 0 , t 0 ) + m + D s θ h ( x ) h ( x ) < C 0 a 2 s 2 + C 0 a 2 s 2 − C 0 a 2 s = 0 .

By (2.24), we obtain

( c ( x 0 , t 0 ) + m + D s θ h ( x ) h ( x ) ) w ¯ ( x 0 , t 0 ) > 0 .

In addition, similar to Theorem 1.1, we have

inf A ∈ A P.V. ∫ R n h ( y ) ( w ¯ ( y , t 0 ) − w ¯ ( x 0 , t 0 ) ) ∣ A − 1 ( y − x 0 ) ∣ n + 2 s d y ≥ 0 .

Combining with h ( x ) > 0 , we obtain

∂ w ¯ ∂ t ( x , t ) − 1 h ( x ) inf A ∈ A P.V. ∫ R n h ( y ) ( w ¯ ( y , t ) − w ¯ ( x , t ) ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y ≤ 0 ,

which contradicts (2.25), and thus, (2.26) holds.

Hence,

w ¯ ( x , t ) ≥ min { 0 , inf Ω w ¯ ( x , t ¯ ) } = min 0 , e m t ¯ inf Ω w ( x , t ¯ ) h ( x ) ≥ − C e m t ¯ ,

and w ( x , t ) ≥ − C e m ( t − t ¯ ) h ( x ) .

Since t ¯ ( < T ) ∈ R , and letting t ¯ → − ∞ , we know

(2.28) w ( x , t ) ≥ 0 , ( x , t ) ∈ Ω × R .

Combining with (2.25) and (2.28), we derive

w ( x , t ) ≥ 0 , ( x , t ) ∈ Σ λ × R .

Furthermore, suppose that there is a point ( x 0 , t 0 ) ∈ Ω × R such that w ( x 0 , t 0 ) = 0 , then we must have

w ( x 0 , t 0 ) = inf Σ λ × R w ( x , t ) = 0 , and ∂ w ∂ t ( x 0 , t 0 ) = 0 , D s θ ( w ( x 0 , t 0 ) ) > 0 .

Hence, by (1.10), we derive

0 > ∂ w ∂ t ( x 0 , t 0 ) − D s θ w ( x 0 , t 0 ) ≥ c ( x 0 , t 0 ) w ( x 0 , t 0 ) = 0 .

This is impossible, and thus, the strong maximum principle holds. This completes the proof of Theorem 1.2.□

3 Monotonicity of solutions to (1.12) on an upper half space

In this section, using the method of moving planes, we obtain the monotonicity of solutions to the problem:

∂ u ∂ t ( x , t ) − D s θ u ( x , t ) = f ( u ( x , t ) ) , ( x , t ) ∈ R + n × R , u ( x , t ) = 0 , ( x , t ) ∉ R + n × R ,

where f is a C 1 function satisfying f ( 0 ) = 0 , f ′ ( 0 ) ≤ 0 .

Proof of Theorem 1.3

In order to prove the strict monotonicity of u ( x , t ) in x 1 -direction, it is sufficient to prove w λ ( x , t ) > 0 , ( x , t ) ∈ Σ λ × R , for any λ > 0 .

Let Ω λ = { x ∈ R + n ∣ 0 < x 1 < λ } . Note that

D s θ u λ ( x , t ) − D s θ u ( x , t ) = inf A ∈ A P.V. ∫ R n u λ ( y , t ) − u λ ( x , t ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y − inf A ∈ A P.V. ∫ R n u ( y , t ) − u ( x , t ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y ≥ inf A ∈ A P.V. ∫ R n w λ ( y , t ) − w λ ( x , t ) ∣ A − 1 ( y − x ) ∣ n + 2 s d y = D s θ w λ ( x , t ) .

It follows that

(3.1) ∂ w λ ∂ t ( x , t ) − D s θ w λ ( x , t ) ≥ c λ ( x , t ) w λ ( x , t ) , ( x , t ) ∈ Ω λ × R , w λ ( x , t ) ≥ 0 , ( x , t ) ∈ ( Σ λ \ Ω λ ) × R , w λ ( x λ , t ) = − w λ ( x , t ) , ( x , t ) ∈ Σ λ × R ,

and

c λ ( x , t ) = ∫ 0 1 f ′ ( s u ( x , t ) + ( 1 − s ) u λ ( x , t ) ) d s

is bounded. Next, we divide the proof into three steps.

Step 1. We show that

(3.2) w λ ( x , t ) ≥ 0 , ( x , t ) ∈ Σ λ × R ,

for the sufficiently small λ . Since Ω λ is a narrow region when λ is sufficiently small, and w λ ( x , t ) satisfies (3.1), then by Theorem 1.1, we derive (3.2).

Step 2. Denote λ 0 = sup { λ ∣ w μ ( x , t ) ≥ 0 , ( x , t ) ∈ Ω μ × R , μ ≤ λ } . We will prove that λ 0 = + ∞ . If not, we assume 0 < λ 0 < + ∞ , and there exist sequences λ k ≥ λ 0 such that

(3.3) λ k → λ 0

and Σ λ k − = { ( x , t ) ∈ Ω λ k × R ∣ w λ k ( x , t ) < 0 } is nonempty. Set

q k = sup Σ λ k − c λ k ( x , t ) ,

where

c λ k ( x , t ) = ∫ 0 1 f ′ ( s u ( x , t ) + ( 1 − s ) u λ k ( x , t ) ) d s .

We consider the following two cases:

Case 1: Suppose that q k → 0 . By (1.12), we have

∂ w λ k ∂ t ( x , t ) − D s θ w λ k ( x , t ) ≥ c λ k ( x , t ) w λ k ( x , t ) , ( x , t ) ∈ Ω λ k × R , w λ k ( x , t ) ≥ 0 , ( x , t ) ∈ ( Σ λ k \ Ω λ k ) × R , w λ k ( x λ k , t ) = − w λ k ( x , t ) , ( x , t ) ∈ Σ λ k × R .

By the definition of c λ k , q k , and Theorem 1.2, we see that w λ k ( x , t ) ≥ 0 , ( x , t ) ∈ Ω λ k × R ; thus, Σ λ k − is empty. This contradicts the definition of Σ λ k − ; therefore, Case 1 is impossible.

Case 2: Passing to a subsequence, we have q k ≥ ε 0 for some ε 0 > 0 . Then, there exist subsequences ( x k , t k ) = ( x 1 k , x ′ k , t k ) , x 1 k ∈ ( 0 , λ k ) , such that

(3.4) u ( x k , t k ) ≥ ε 0 , ∇ w λ k ( x k , t k ) = o ( 1 ) ( k → ∞ ) .

Since 0 < x 1 k < λ , we may assume that

(3.5) x 1 k → a , for some a ∈ [ 0 , λ 0 ] .

Let

u k ( x , t ) ≔ u ( x 1 , x ′ + x ′ k , t + t k ) .

Since u k ( x , t ) is uniformly bounded, by the regularity estimates for fractional parabolic equations in [17], up to a sequence (still denoted by u k ), as k → + ∞ , we have

u k ( x , t ) → u ¯ ( x , t ) , D s θ u k ( x , t ) → D s θ u ¯ ( x , t ) .

By (1.12), we obtain

(3.6) ∂ u ¯ ∂ t ( x , t ) − D s θ u ¯ = f ( u ¯ ) , ( x , t ) ∈ Σ λ 0 × R .

Because of (3.4), we know

(3.7) u k ( x 1 k , 0 , 0 ) = u ( x 1 k , x ′ k , t k ) ≥ ε 0 , u ¯ ( a , 0 , 0 ) ≥ ε 0 .

Combining (3.6) with (3.7), by the strong maximum principle, we know

u ¯ ( x , t ) > 0 , ( x , t ) ∈ R + n × R .

Similarly, define

w k ( x 1 , x ′ , t ) ≔ w λ k ( x 1 , x ′ + x ′ k , t + t k ) ,

then, as k → + ∞ , we have

w k ( x , t ) → w ¯ ( x , t ) , D s θ w k ( x , t ) → D s θ w ¯ ( x , t ) .

Hence,

(3.8) ∂ w ¯ ∂ t ( x , t ) − D s θ w ¯ ≥ c ¯ ( x , t ) w ¯ , ( x , t ) ∈ Σ λ 0 × R ,

and

w λ k ( x 1 , x ′ k , t k ) = w k ( x 1 , 0 , 0 ) → w ¯ ( a , 0 , 0 ) .

Since λ k ≥ λ 0 , λ k → λ 0 , using the continuity of w λ with respect to λ , w λ 0 ( x , t ) ≥ 0 , ( x , t ) ∈ Σ λ 0 × R , we obtain

(3.9) w ¯ ( x , t ) ≥ 0 , ( x , t ) ∈ Σ λ 0 × R .

On the other hand, w λ k ( x k , t k ) < 0 , and we have w ¯ ( a , 0 , 0 ) ≤ 0 . Thus, w ¯ ( a , 0 , 0 ) = 0 . We claim that

(3.10) ( a , 0 ) ∈ T λ 0 .

If not, we have ( a , 0 ) ∈ Ω λ 0 and

∂ w ¯ ∂ t ( a , 0 ) = 0 , − D s θ w ¯ ( a , 0 ) > 0 .

Combining with (3.9), by the strong maximum principle, we know

w ¯ ( x , t ) ≡ 0 , ( x , t ) ∈ R + n × R .

This contradicts u ( x , t ) > 0 . So (3.10) holds. Moreover, w ¯ ( x , t ) satisfies

∂ w ¯ ∂ t ( x , t ) − D s θ w ¯ ( x , t ) ≥ c ¯ ( x , t ) w ¯ ( x , t ) , ( x , t ) ∈ Σ 0 × R , w ¯ ( x , t ) ≥ 0 , ( x , t ) ∈ Σ 0 × R , w ¯ ( x λ 0 , t ) = − w ¯ ( x , t ) , ( x , t ) ∈ Σ 0 × R ,

where

c ¯ ( x , t ) = ∫ 0 1 f ′ ( s u ¯ ( x , t ) + ( 1 − s ) u ¯ λ 0 ( x , t ) ) d s .

By Lemma 2.4, we see that

(3.11) ∂ w ¯ ∂ x 1 ( a , 0 ) < 0 .

In addition, by ∇ w λ k ( x k , t k ) → 0 , j → ∞ , we have ∇ w λ k ( a , 0 ) = 0 . This contradicts with (3.11). Therefore, we conclude that λ 0 = + ∞ .

Step 3. For any 0 < λ < ∞ , by Steps 1 and 2, we have

w λ ( x , t ) ≥ 0 , ( x , t ) ∈ Σ λ × R .

It follows that if there exists an point ( x 0 , t 0 ) ∈ Σ λ × R such that w λ ( x 0 , t 0 ) = 0 = inf R n × R w λ ( x , t ) , then

0 > ∂ w λ ∂ t ( x 0 , t 0 ) − D s θ w λ ( x 0 , t 0 ) ≥ f ( u λ ( x 0 , t 0 ) ) − f ( u ( x 0 , t 0 ) ) = 0 ,

which is impossible. Therefore, for any 0 < λ < ∞ , we obtain

w λ ( x , t ) > 0 , ( x , t ) ∈ Σ λ × R .

It follows that

∂ u ∂ x 1 ( x , t ) > 0 , ( x , t ) ∈ R + n × R .

This completes the proof of Theorem 1.3.□

Funding information: This work was supported by the Natural Science Foundation of Shandong Province of China (No. ZR2020QA017, No. ZR2019MA067), the Youth Backbone Teacher Funding Program in Huanghuai University, and the Key Specialized Research and Development Breakthrough Program in Henan Province (No. 222102310265).
Author contributions: Both authors contributed equally and significantly to this manuscript.
Conflict of interest: The authors state no conflict of interest.

References

[1] L. Caffarelli and F. Charro, On a fractional Monge-Ampère operator, Ann. PDE. 1 (2015), 47. 10.1007/s40818-015-0005-xSearch in Google Scholar

[2] L. Caffarelli and F. Charro, On a family of fully nonlinear integro-differential operators: From fractional Laplacian to nonlocal Monge-Ampère, 2021. arXiv: http://arXiv.org/abs/arXiv:2111.12781. Search in Google Scholar

[3] X. Chen, G. Bao, and G. Li, The sliding method for the nonlocal Monge-Ampère operator, Nonlinear Anal. 196 (2020), 111786. 10.1016/j.na.2020.111786Search in Google Scholar

[4] X. Chen, G. Bao, and G. Li, Symmetry of solutions for a class of nonlocal Monge-Ampère equations, Complex Var. Elliptic Equ. 67 (2022), 129–150. 10.1080/17476933.2020.1816986Search in Google Scholar

[5] X. Chen, G. Li, and S. Bao, Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems, Commun. Pure Appl. Anal. 21 (2022), no. 5, 1755–1772. 10.3934/cpaa.2022045Search in Google Scholar

[6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ. 32 (2007), 1245–1260. 10.1080/03605300600987306Search in Google Scholar

[7] L. Cafarelli and L. Silvestre, A nonlocal Monge-Ampère equation, Commun. Anal. Geom. 24 (2016), 307–335. 10.4310/CAG.2016.v24.n2.a4Search in Google Scholar

[8] W. Chen and Y. Hu, Monotonicity of positive solutions for nonlocal problems in unbounded domains, J. Func. Anal. 281 (2021), no.9, Paper No. 109187. 10.1016/j.jfa.2021.109187Search in Google Scholar

[9] W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differ. Equ. 56 (2017), Art. 29, 18pp. 10.1007/s00526-017-1110-3Search in Google Scholar

[10] W. Chen, C. Li, and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math. 308 (2017), 404–437. 10.1016/j.aim.2016.11.038Search in Google Scholar

[11] W. Chen and L. Wu, Liouville theorems for fractional parabolic equations, Adv. Nonl. Stud. 21 (2021), 939–958. 10.1515/ans-2021-2148Search in Google Scholar

[12] W. Chen, P. Wang, Y. Niu, and Y. Hu, Asymptotic method of moving planes for fractional parabolic equations, Adv. Math. 377(2021), Paper No. 107463. 10.1016/j.aim.2020.107463Search in Google Scholar

[13] Y. Chen, Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity, Commun. Anal. Mech. 15 (2023), no. 4, 658–694. 10.3934/cam.2023033Search in Google Scholar

[14] G. De Philippis and A. Figalli, W2,1 regularity for solutions of the Monge-Ampère equation, Invent. Math. 192 (2013), 55–69. 10.1007/s00222-012-0405-4Search in Google Scholar

[15] F. Dekhkonov, On a boundary control problem for a pseudo-parabolic equation, Commun. Anal. Mech. 15 (2023), no. 2, 289–299. 10.3934/cam.2023015Search in Google Scholar

[16] W. Dai and G. Qin, Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications, Ann. Mat. Pura Appl. 200 (2021), 1085–1134. 10.1007/s10231-020-01027-9Search in Google Scholar

[17] X. Fernández-Real and X. Ros-Oton, Regularity theory for general stable operators: Parabolic equations, J. Funct. Anal. 272 (2017), no. 10, 4165–4221. 10.1016/j.jfa.2017.02.015Search in Google Scholar

[18] H. Jian and X. Wang, Existence of entire solutions to the Monge-Ampère equation, Am. J. Math. 136 (2014), 1093–1106. 10.1353/ajm.2014.0029Search in Google Scholar

[19] Y. Niu, Monotonicity of solutions for a class of nonlocal Monge-Ampère problem, Comm. Pure Appl. Anal. 19 (2020), 5269–5283. 10.3934/cpaa.2020237Search in Google Scholar

[20] N. Trudinger and X. Wang, The Monge-Ampère equation and its geometric applications, Handbook of Geometric Analysis 1 (2008), 467–524. Search in Google Scholar

[21] N. Trudinger and X. Wang, Boundary regularity for the Monge-Ampère and affine maximal surface equations, Ann. Math. 167 (2008), 993–1028.10.4007/annals.2008.167.993Search in Google Scholar

[22] H. Xu, Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials, Commun. Anal. Mech. 15 (2023), no. 2, 132–161. 10.3934/cam.2023008Search in Google Scholar

Received: 2023-04-14

Revised: 2023-12-02

Accepted: 2024-01-03

Published Online: 2024-02-16

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2023-0135

Keywords for this article

parabolic equations; nonlocal Monge-Ampère operator; method of moving planes; monotonicity

Creative Commons

BY 4.0