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Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian

  • Qing Guo EMAIL logo
Published/Copyright: January 11, 2024
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 17 Issue 4

Abstract

We obtain the one-dimensional symmetry and monotonicity of the entire positive solutions to some reaction-diffusion equations involving fractional p-Laplacian by virtue of the sliding method. More precisely, we consider the following problem

{ ∂ ⁡ u ∂ ⁡ t ⁢ ( x , t ) + ( - Δ ) p s ⁢ u ⁢ ( x , t ) = f ⁢ ( t , u ⁢ ( x , t ) ) , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) > 0 , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) = 0 , ( x , t ) ∈ Ω c × ℝ ,

where s ∈ ( 0 , 1 ) , p ≥ 2 , ( - Δ ) p s is the fractional p-Laplacian, f ⁢ ( t , u ) is some continuous function, the domain Ω ⊂ ℝ n is unbounded and Ω c = ℝ n ∖ Ω . Firstly, we establish a maximum principle involving the parabolic p-Laplacian operator. Then, under certain conditions of f, we prove the asymptotic behavior of solutions far away from the boundary uniformly in t ∈ ℝ . Finally, the sliding method is implemented to derive the monotonicity and uniqueness of the bounded positive entire solutions. To our best knowledge, there has not been any results on the symmetry and monotonicity properties of solutions to the parabolic fractional p-Laplacian equations before.

Keywords: Sliding methods; entire solutions; maximum principles; monotonicity
MSC 2020: 35B40; 35B50

Communicated by Luis Silvestre

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11771469

Award Identifier / Grant number: 12271539

Funding statement: Qing Guo was supported by NNSF of China (No. 11771469, No. 12271539).

A Some essential estimates

Lemma A.1 (cf. [27]).

For G ⁢ ( r ) = | r | p - 2 ⁢ r , p ≥ 2 , if r 1 + r 2 > 0 , then

G ⁢ ( r 1 + r 2 ) ≤ e - ( p - 2 ) ⁢ ( G ⁢ ( r 1 ) + G ⁢ ( r 2 ) ) .

Lemma A.2 (cf. [27]).

For the bounded solution u of (1.3) and η ∈ C 0 ∞ ⁢ ( R n ) , and any 0 < δ < 1 , one has

| ( - Δ ) p s ⁢ ( u + ϵ ⁢ η ) ⁢ ( x , t ) - ( - Δ ) p s ⁢ u ⁢ ( x , t ) | < ϵ ⁢ C δ + C ⁢ δ p ⁢ ( 1 - s ) .

Lemma A.3.

For a bounded domain Γ ⊂ R n , if u , v ∈ C 1 ⁢ ( R ; C loc 1 , 1 ⁢ ( Γ ) ∩ L s , p ) and u - v is lower semi-continuous on Γ ¯ satisfying

(A.1) { ∂ ⁡ u ∂ ⁡ t ⁢ ( x , t ) + ( - Δ ) p s ⁢ u ⁢ ( x , t ) ≥ ∂ ⁡ v ∂ ⁡ t ⁢ ( x , t ) + ( - Δ ) p s ⁢ v ⁢ ( x , t ) , ( x , t ) ∈ Γ × ℝ , u ⁢ ( x , t ) ≥ v ⁢ ( x , t ) , ( x , t ) ∈ Γ c × ℝ ,

then u ⁢ ( x , t ) ≥ v ⁢ ( x , t ) , for ( x , t ) ∈ Γ × R .

Moreover, if u ⁢ ( x 0 , t 0 ) = v ⁢ ( x 0 , t 0 ) at some ( x 0 , t 0 ) ∈ Γ × R , then u ⁢ ( x , t 0 ) = v ⁢ ( x , t 0 ) almost everywhere in x ∈ R n .

Proof.

Let w ⁢ ( x , t ) = u ⁢ ( x , t ) - v ⁢ ( x , t ) . Suppose on the contrary; since w is lower semi-continuous on Γ ¯ , there exists ( x 0 , t 0 ) ∈ Γ × ℝ such that w ⁢ ( x 0 , t 0 ) ≤ min Γ × ℝ ⁡ w ⁢ ( x , t ) < 0 . From the second inequality in (A.1), we have

( - Δ ) p s ⁢ u ⁢ ( x 0 , t 0 ) - ( - Δ ) p s ⁢ v ⁢ ( x 0 , t 0 ) = C n , s , p ⁢ P.V. ⁢ ∫ ℝ n G ⁢ ( u ⁢ ( x 0 , t 0 ) - u ⁢ ( y , t 0 ) ) - G ⁢ ( v ⁢ ( x 0 , t 0 ) - v ⁢ ( y , t 0 ) ) | x 0 - y | n + s ⁢ p ⁢ 𝑑 y ≤ C n , s , p ⁢ P.V. ⁢ ∫ Γ c G ⁢ ( u ⁢ ( x 0 , t 0 ) - u ⁢ ( y , t 0 ) ) - G ⁢ ( v ⁢ ( x 0 , t 0 ) - v ⁢ ( y , t 0 ) ) | x 0 - y | n + s ⁢ p ⁢ 𝑑 y < 0 ,

which contradicts the first inequality in (A.1). Hence u ≥ v , ( x , t ) ∈ Γ × ℝ .

If w ⁢ ( x 0 , t 0 ) = 0 at some ( x 0 , t 0 ) ∈ Γ × ℝ , then

(A.2) ( - Δ ) p s ⁢ u ⁢ ( x 0 , t 0 ) - ( - Δ ) p s ⁢ v ⁢ ( x 0 , t 0 ) = C n , s , p ⁢ P.V. ⁢ ∫ ℝ n G ⁢ ( u ⁢ ( x 0 , t 0 ) - u ⁢ ( y , t 0 ) ) - G ⁢ ( v ⁢ ( x 0 , t 0 ) - v ⁢ ( y , t 0 ) ) | x 0 - y | n + s ⁢ p ⁢ 𝑑 y ≤ 0 .

On the other hand, the first inequality in (A.1) gives that

( - Δ ) p s ⁢ u ⁢ ( x 0 , t 0 ) - ( - Δ ) p s ⁢ v ⁢ ( x 0 , t 0 ) ≥ 0 .

Then, in view of (A.2), due to w ≥ 0 , it holds that w ⁢ ( x , t 0 ) = 0 almost everywhere in ℝ n . ∎

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Received: 2022-12-20
Accepted: 2023-11-11
Published Online: 2024-01-11
Published in Print: 2024-10-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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  2. Another proof of the existence of homothetic solitons of the inverse mean curvature flow
  3. A Weierstrass extremal field theory for the fractional Laplacian
  4. Minimizing movements for anisotropic and inhomogeneous mean curvature flows
  5. A singular Yamabe problem on manifolds with solid cones
  6. Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball
  7. Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity
  8. Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian
  9. Hierarchy structures in finite index CMC surfaces
  10. No breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature
  11. Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data
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  15. Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: The limiting regimes
  16. On the area-preserving Willmore flow of small bubbles sliding on a domain’s boundary
  17. Sobolev contractivity of gradient flow maximal functions
  18. Discrete approximation of nonlocal-gradient energies
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  20. Flat flow solution to the mean curvature flow with volume constraint
https://doi.org/10.1515/acv-2022-0109
Keywords for this article
Sliding methods; entire solutions; maximum principles; monotonicity

Articles in the same Issue

  1. Frontmatter
  2. Another proof of the existence of homothetic solitons of the inverse mean curvature flow
  3. A Weierstrass extremal field theory for the fractional Laplacian
  4. Minimizing movements for anisotropic and inhomogeneous mean curvature flows
  5. A singular Yamabe problem on manifolds with solid cones
  6. Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball
  7. Limit of solutions for semilinear Hamilton–Jacobi equations with degenerate viscosity
  8. Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian
  9. Hierarchy structures in finite index CMC surfaces
  10. No breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature
  11. Wolff potentials and local behavior of solutions to elliptic problems with Orlicz growth and measure data
  12. Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity
  13. On the regularity of optimal potentials in control problems governed by elliptic equations
  14. Sobolev embeddings and distance functions
  15. Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: The limiting regimes
  16. On the area-preserving Willmore flow of small bubbles sliding on a domain’s boundary
  17. Sobolev contractivity of gradient flow maximal functions
  18. Discrete approximation of nonlocal-gradient energies
  19. Symmetry and monotonicity of singular solutions to p-Laplacian systems involving a first order term
  20. Flat flow solution to the mean curvature flow with volume constraint
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