# American Institute of Mathematical Sciences

November  2019, 24(11): 5981-5988. doi: 10.3934/dcdsb.2019116

## Stable solution induced by domain geometry in the heat equation with nonlinear boundary conditions on surfaces of revolution

 Instituto de Matemática e Computação, Universidade Federal de Itajubá, MG, Brazil

Received  October 2018 Published  November 2019 Early access  June 2019

In this paper we consider the heat equation on surfaces of revolution subject to nonlinear Neumann boundary conditions. We provide a sufficient condition on the geometry of the surface in order to ensure the existence of an asymptotically stable nonconstant solution.

Citation: Maicon Sônego. Stable solution induced by domain geometry in the heat equation with nonlinear boundary conditions on surfaces of revolution. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5981-5988. doi: 10.3934/dcdsb.2019116
##### References:
 [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propation, Partial Differential Equations and Related Topic (A. Dold and B. Eckmann, Eds.), pp. 5–49, Lecture Notes in Mathematics, 446, Springer-Verlag, Berlin/Heidelberg/New York, 1975. [2] C. Bandle, F. Punzo and A. Tesei, Existence and non-existence of patterns on Riemannian manifolds, J. Math. Anal. Appl., 387 (2012), 33-47.  doi: 10.1016/j.jmaa.2011.08.060. [3] C. Bandle, P. Mastrolia, D. D. Monticelli and F. Punzo, On the stability of solutions of semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds, SIAM Journal on Mathematical Analysis, 48 (2016), 122-151.  doi: 10.1137/15M102647X. [4] R. G. Casten and C. J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. of Diff. Eqns., 27 (1978), 266-273.  doi: 10.1016/0022-0396(78)90033-5. [5] D. Castorina and M. Sanchón, Regularity of stable solutions to semilinear elliptic equations on Riemannian models, Adv. Nonlinear Anal., 4 (2015), 295-309.  doi: 10.1515/anona-2015-0047. [6] M. Chipot and J. Hale, Stable equilibria with variable diffusion, Contemp. Math., 17 (1983), 209-213.  doi: 10.1090/conm/017/706100. [7] N. Cònsul, On equilibrium solutions of diffusion equations with nonlinear boundary conditions, Z. Angew Math. Phys., 47 (1996), 194-209.  doi: 10.1007/BF00916824. [8] S. Dipierro, A. Pinamonti and E. Valdinoci, Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature, Advances in Nonlinear Analysis, 8 (2019), 1035-1042.  doi: 10.1515/anona-2018-0013. [9] S. Dipierro, A Pinamonti and E. Valdinoci, Rigidity results in diffusion Markov triples, J. Funct. Anal, 276 (2019), 785-814.  doi: 10.1016/j.jfa.2018.06.005. [10] A. S. do Nascimento and M. Sônego, Patterns on surfaces of revolution in a diffusion problem with variable diffusivity, Electron. J. Differential Equations, 2014 (2014), 1-13. [11] A. S. do Nascimento, J. Crema and M. Sônego, Sufficient conditions on diffusivity for the existence and nonexistence of stable equilibria with nonlinear flux on the boundary, Electronic Journal of Differential Equations, 2012 (2012), 1-14. [12] A. S. do Nascimento and M. Sônego, The roles of diffusivity and curvature in patterns on surfaces of revolution, J. Math. Anal. Appl., 412 (2014), 1084-1096.  doi: 10.1016/j.jmaa.2013.10.058. [13] A. C. Gonçalves and A. S. do Nascimento, Instability of elliptic equations on compact Riemannian manifolds with non-negative Ricci curvature, Electronic Journal of Differential Equations, 2010 (2010), 1-18. [14] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Mathematics, 840, 1981. [15] S. Jimbo, On a semilinear diffusion equation on a Riemannian manifold and its stable equilibrium solutions, Proc. Japan Acad. Ser. A Math. Sci., 60 (1984), 349-352.  doi: 10.3792/pjaa.60.349. [16] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. RIMS, Kyoto Univ., 15 (1979), 401-454.  doi: 10.2977/prims/1195188180. [17] M. Mimura, Stationary pattern of some density-dependent diffusive system with competing dynamics, Hiroshima Mad J., 11 (1981), 621-635.  doi: 10.32917/hmj/1206133994. [18] F. Punzo, The existence of patterns on surfaces of revolution without boundary, Nonlinear Analysis: Theory, Methods & Applications, 77 (2013), 94-102.  doi: 10.1016/j.na.2012.09.003. [19] J. Rubinstein and G. Wolansky, Instability results for reaction diffusion equations over surfaces of revolution, J. Math. Anal. Appl., 187 (1994), 485-489.  doi: 10.1006/jmaa.1994.1368. [20] M. Sônego, A note on existence of patterns on surfaces of revolution with nonlinear flux on the boundary, (submitted). [21] M. Sônego, Patterns in a balanced bistable equation with heterogeneous environments on surfaces of revolution, Differ. Equ. Appl., 8 (2016), 521-533.  doi: 10.7153/dea-08-29.

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##### References:
 [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propation, Partial Differential Equations and Related Topic (A. Dold and B. Eckmann, Eds.), pp. 5–49, Lecture Notes in Mathematics, 446, Springer-Verlag, Berlin/Heidelberg/New York, 1975. [2] C. Bandle, F. Punzo and A. Tesei, Existence and non-existence of patterns on Riemannian manifolds, J. Math. Anal. Appl., 387 (2012), 33-47.  doi: 10.1016/j.jmaa.2011.08.060. [3] C. Bandle, P. Mastrolia, D. D. Monticelli and F. Punzo, On the stability of solutions of semilinear elliptic equations with Robin boundary conditions on Riemannian manifolds, SIAM Journal on Mathematical Analysis, 48 (2016), 122-151.  doi: 10.1137/15M102647X. [4] R. G. Casten and C. J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. of Diff. Eqns., 27 (1978), 266-273.  doi: 10.1016/0022-0396(78)90033-5. [5] D. Castorina and M. Sanchón, Regularity of stable solutions to semilinear elliptic equations on Riemannian models, Adv. Nonlinear Anal., 4 (2015), 295-309.  doi: 10.1515/anona-2015-0047. [6] M. Chipot and J. Hale, Stable equilibria with variable diffusion, Contemp. Math., 17 (1983), 209-213.  doi: 10.1090/conm/017/706100. [7] N. Cònsul, On equilibrium solutions of diffusion equations with nonlinear boundary conditions, Z. Angew Math. Phys., 47 (1996), 194-209.  doi: 10.1007/BF00916824. [8] S. Dipierro, A. Pinamonti and E. Valdinoci, Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature, Advances in Nonlinear Analysis, 8 (2019), 1035-1042.  doi: 10.1515/anona-2018-0013. [9] S. Dipierro, A Pinamonti and E. Valdinoci, Rigidity results in diffusion Markov triples, J. Funct. Anal, 276 (2019), 785-814.  doi: 10.1016/j.jfa.2018.06.005. [10] A. S. do Nascimento and M. Sônego, Patterns on surfaces of revolution in a diffusion problem with variable diffusivity, Electron. J. Differential Equations, 2014 (2014), 1-13. [11] A. S. do Nascimento, J. Crema and M. Sônego, Sufficient conditions on diffusivity for the existence and nonexistence of stable equilibria with nonlinear flux on the boundary, Electronic Journal of Differential Equations, 2012 (2012), 1-14. [12] A. S. do Nascimento and M. Sônego, The roles of diffusivity and curvature in patterns on surfaces of revolution, J. Math. Anal. Appl., 412 (2014), 1084-1096.  doi: 10.1016/j.jmaa.2013.10.058. [13] A. C. Gonçalves and A. S. do Nascimento, Instability of elliptic equations on compact Riemannian manifolds with non-negative Ricci curvature, Electronic Journal of Differential Equations, 2010 (2010), 1-18. [14] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Mathematics, 840, 1981. [15] S. Jimbo, On a semilinear diffusion equation on a Riemannian manifold and its stable equilibrium solutions, Proc. Japan Acad. Ser. A Math. Sci., 60 (1984), 349-352.  doi: 10.3792/pjaa.60.349. [16] H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. RIMS, Kyoto Univ., 15 (1979), 401-454.  doi: 10.2977/prims/1195188180. [17] M. Mimura, Stationary pattern of some density-dependent diffusive system with competing dynamics, Hiroshima Mad J., 11 (1981), 621-635.  doi: 10.32917/hmj/1206133994. [18] F. Punzo, The existence of patterns on surfaces of revolution without boundary, Nonlinear Analysis: Theory, Methods & Applications, 77 (2013), 94-102.  doi: 10.1016/j.na.2012.09.003. [19] J. Rubinstein and G. Wolansky, Instability results for reaction diffusion equations over surfaces of revolution, J. Math. Anal. Appl., 187 (1994), 485-489.  doi: 10.1006/jmaa.1994.1368. [20] M. Sônego, A note on existence of patterns on surfaces of revolution with nonlinear flux on the boundary, (submitted). [21] M. Sônego, Patterns in a balanced bistable equation with heterogeneous environments on surfaces of revolution, Differ. Equ. Appl., 8 (2016), 521-533.  doi: 10.7153/dea-08-29.
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