Document Type : Research Paper


Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.


In this paper, we consider positive solutions for the Allen-Cahn equation
\Delta u+\left(1-u^{2}\right)u=0,
on an almost Ricci soliton   without a boundary. Firstly, using volume comparison Theorem and Sobolev inequality, we estimate the upper bound of $\vert \nabla u\vert^{2}$. As one of the applications, we extend this result to a gradient Ricci almost soliton. Finally, we obtain a Liouville-type theorem for almost Ricci solitons.


Main Subjects

[1] A. Abolarinwa, S.O. Edeki, N.K. Oladejo and O.P. Ogundile, Gradient estimates for bounded solutions of semilinear elliptic equations and the Allen-Cahn equation on manifolds, Journal of physics: Conf. ser. 2199012002, (2021).
[2] N. Alibaud, F. del Teso, J. Endal and E.R. Jakobsen, The Liouville theorem and linear operators satisfying the maximum principle, J. Math. Pures Appl., 142 (2020), pp. 229-242.
[3] S. Azami and S. Hajiaghasi, New volume comparison with almost Ricci soliton, Commun. Korean Math. Soc., 37(3)(2022), pp. 839-849.
[4] S. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27(1979), pp. 1085-1096 .
[5] M. Bailesteanu, A Harnack inequality for the parabolic Allen-Cahn equation, Ann. Glob. Anal. Geom., 51 (4) (2017), pp. 367-378.
[6] M. Bailesteanu, X.D. Cao and A. Palematov, Gradient estimates for the heat equation under the Ricci flow, J. Funct. Anal., 258(2010), pp. 3517-3542. 
[7] R.H. Bamler, Entropy and heat kernel bounds on a Ricci flow background, arXiv: 2008.07093v3 [math.DG] (2021).
[8] M. Benes, V. Chalupecky and K. Mikula, Geometrical image segmentation by the Allen-Cahn equation, Appl. Numer. Math., 51 (2)(2004), pp.187-205. 
[9] E. Calabi, An extension of E.Hopf's maximum principle with application to Riemannian geometry, Duck Math. J., 25(1957), pp. 45-46.
[10] X. Cao, B.F. Ljungberg and B. Liu, Differential Harnack estimates for a nonlinear heat equation, J. Funct. Anal. 265(2013), pp. 312-330.
[11] J. Cheeger and T.H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math., (2) 144 (1)(1996), pp. 189-237.
[12] S.Y. Cheng and S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., 28 (3)(1975), pp. 333-354.
[13] X. Dai, G. Wei and Z. Zhang, Local sobolev constant estimate for integral Ricci curvature bounds, Adv. Math., 325 (2018), pp. 1-33.
[14] E.B. Davies, Heat kernel and spectral theory, Cambridge Tracts in Math, Vol. 92. Cambridge University Press., Cambridge, UK (1989).
[15] S. Hou, Gradient estimates for the Allen-Cahn equation on Riemannian manifolds, Proc. Amer. Math. Soc., 147(2019), pp. 619-628.
[16] J. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal., 100(1991), pp. 233-256.
[17] P. Li and S.T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math., 156(1986), pp. 153-201.
[18] L. Ma, Gradient estimates for a simple elliptic equation on non-compact Riemannian manifolds, J. Funct. Anal., 241 (2006), pp. 374-382.
[19] O. Munteanu and N. Sesum, The Poisson equation on complete manifolds with positive spectrum and applications, Adv. Math., 223(2010), pp. 198-219.
[20] E. Negrin, Gradient estimates and a Liouville type theorem for the Schrodinger operator, J. Funct. Anal., 127(1995), pp. 198-203.
[21] G. Perelman, The entropy formula for thr Ricci flow and its geometric applications, Math. ArXiv, Math., DG/0211159., 2002
[22] P. Petersen and G. Wei, Analysis and geometry on manifolds with integral Ricci curvature bounds, Trans. Amer. Math. Soc., 353(2001), pp. 457-478.
[23] A. Shah, M. Sabir and P. Bastian, An efficient time-stepping scheme for numerical simulation of dendritic crystal growth, Eue. J. Comput. Mech., 25 (6)(2017), pp. 475-488.
[24] A. Shah, M. Sabir, M. Qasim and P. Bastian, Efficient numerical scheme for solving the Allen-Cahn equation, Numer. Methods Partial. Differ. Equ., 34 (2018), pp. 1820-1833.
[25] A. Shah and L. Yuan, Numerical solution of a phase-field model for incompressible two-phase flows based on artificial compressibility, Comput. Fluids, 42 (1)(2011), pp. 54-61 .
[26] P. Souplet and Q.S. Zhang, Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc., 38(2006), pp. 1045-1053.
[27] J. Wang, Global heat kernel estimates, Pac. J. Math., 178 (2) (1997), pp. 377-398.
[28] Y. Yang, Gradient estimates for a nolinear parabolic equation on Riemannian manifolds, Proc. Amer. Math. Soc., 136 (2008), pp. 4095-4102.
[29] Y. Yang, Gradient estimates for the equation $\Delta u+cu^{-\alpha}=0$ on Riemannian manifolds, Acta Math. Sin. (Engl. Ser.), 26 (2010), pp. 1177-1182.
[30] S.T. Yau, Harmonic function on complete Riemannian manifolds, Comnn. Pure Appl. Math., 28(1975), pp. 201-228.
[31] Qi. S. Zhang and M. Zhu, New volume comparison results and applications to degeneration of Riemannian metrics, Adv. Math., 352 (2019), pp. 1096-1154.