March  2019, 24(3): 1033-1047. doi: 10.3934/dcdsb.2019005

On the exact number of monotone solutions of a simplified Budyko climate model and their different stability

1. 

Dynamical Systems and Applications Laboratory, Department of Mathematics, Faculty of Sciences, University of Tlemcen, B.P. 119, Tlemcen 13000, Algeria

2. 

Instituto de Matemática Interdisciplinar, Applied Math. and Math. Analysis Dept, Parque de Ciencias 3, 28040–Madrid, Spain

* Corresponding author: Jesús Ildefonso Díaz

Received  December 2017 Revised  May 2018 Published  January 2019

We consider a simplified version of the Budyko diffusive energy balance climate model. We obtain the exact number of monotone stationary solutions of the associated discontinuous nonlinear elliptic with absorption. We show that the bifurcation curve, in terms of the solar constant parameter, is S-shaped. We prove the instability of the decreasing part and the stability of the increasing part of the bifurcation curve. In terms of the Budyko climate problem the above results lead to an important qualitative information which is far to be evident and which seems to be new in the mathematical literature on climate models. We prove that if the solar constant is represented by $ \lambda \in (\lambda _{1}, \lambda _{2}), $ for suitable $ \lambda _{1}<\lambda _{2}, $ then there are exactly two stationary solutions giving rise to a free boundary (i.e. generating two symmetric polar ice caps: North and South ones) and a third solution corresponding to a totally ice covered Earth. Moreover, we prove that the solution with smaller polar ice caps is stable and the one with bigger ice caps is unstable.

Citation: Sabri Bensid, Jesús Ildefonso Díaz. On the exact number of monotone solutions of a simplified Budyko climate model and their different stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1033-1047. doi: 10.3934/dcdsb.2019005
References:
[1]

D. ArcoyaJ. I. Díaz and L. Tello, S-Shaped bifurcation branch in a quasilinear multivalued model arising in Climatology, Journal of Differential Equations, 150 (1998), 215-225. doi: 10.1006/jdeq.1998.3502. Google Scholar

[2]

J. ArrietaA. Rodríguez-Bernal and J. Valero, Dynamics of a reaction–diffusion equation with a discontinuous nonlinearity, International Journal of Bifurcation and Chaos, 16 (2006), 2965-2984. doi: 10.1142/S0218127406016586. Google Scholar

[3]

S. Bensid and J. I. Diaz, Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions, Discrete and Continuous Dynamical Systems, Series B, 22 (2017), 1757-1778. doi: 10.3934/dcdsb.2017105. Google Scholar

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M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

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H. Deguchi, Existence, uniqueness and non-uniqueness of weak solutions of parabolic initial-value problems with discontinuous nonlinearities, Proceedings of the Royal Society of Edinburgh, 135A (2005), 1139-1167. doi: 10.1017/S0308210500004315. Google Scholar

[6]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London, 1985. Google Scholar

[7]

J. I. Díaz, Mathematical analysis of some diffusive energy balance climate models, In Mathematics, Climate and Environment (J. Díaz and J.-L. Lions, eds.) Masson, Paris, 27 (1993), 28–56. Google Scholar

[8]

J. I. DíazA. C. FowlerA. I. Muñoz and E. Schiavi, Mathematical analysis of a model of river channel formation, Pure Appl. Geophys., 165 (2008), 1663-1682. Google Scholar

[9]

J. I. DíazJ. Hernandez and L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in Climatology, Mathematical Analysis and Applications, 216 (1997), 593-613. doi: 10.1006/jmaa.1997.5691. Google Scholar

[10]

J. I. Díaz and G. Hetzer, A Functional Quasilinear Reaction-Diffusion Equation Arising in Climatology, É Quations Aux Dérivées Partielles et Applications. Articles Dédi és à J.-L. Lions, Elsevier, Paris, (1998), 461–480. Google Scholar

[11]

J. I. DíazJ. A. Langa and J. Valero, On the asymptotic behaviour of solutions of a stochastic energy balance climate model, Physica D, 238 (2009), 880-887. doi: 10.1016/j.physd.2009.02.010. Google Scholar

[12]

J. I. Díaz and L. Tello, On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica, 50 (1999), 19-51. Google Scholar

[13]

J. I. Díaz and L. Tello, Infinitely many stationary solutions for a simple climate model via a shooting method, Mathematical Methods in the Applied Sciences, 25 (2002), 327-334. doi: 10.1002/mma.289. Google Scholar

[14]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. Google Scholar

[15]

R. Gianni and J. Hulshof, The semilinear heat equation with a Heaviside source term, Euro. J. of Applied Mathematics, 3 (1992), 367-379. doi: 10.1017/S0956792500000917. Google Scholar

[16]

N. V. GorbanO. V. KhomenkoL. S. Paliichuk and A. M. Tkachuk, Long-time behavior of state functions for climate energy balance model, Discrete and Continuous Dynamical Systems, Series B, 22 (2017), 1887-1897. doi: 10.3934/dcdsb.2017112. Google Scholar

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics No. 840, Springer-Verlag, New York, 1981. Google Scholar

[18]

G. Hetzer, The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston Journal of Mathematics, 16 (1990), 203-216. Google Scholar

[19]

G. Hetzer, The number of stationary solntions for one-dimensional Budyko-type climate models, Nonlinear Analysis: Real World Applications, 2 (2001), 259-272. doi: 10.1016/S0362-546X(00)00103-6. Google Scholar

[20]

F. Issard-Roch, Subsolutions, supersolutions, and asymptotic behavior of an evolution problem with unilateral constraint, J. Math. Anal. Appl, 140 (1989), 50-66. doi: 10.1016/0022-247X(89)90093-0. Google Scholar

[21]

X. LiangX. Lin and H. Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations, Transactions of the American Mathematical Society, 362 (2010), 5605-5633. doi: 10.1090/S0002-9947-2010-04931-1. Google Scholar

[22]

H. P. McKean, Nagumo's equation, Advances in Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X. Google Scholar

[23]

G. R. North and J. A. Coakley, Differences between seasonal and mean annual energy balance model calculations of climate and climate sensitivity, Journal of the Atmospheric Sciences, 36 (1979), 1189-1203. Google Scholar

[24]

B. Schmidt, Bifurcation of Stationary Solutions for Legendre-Type Boundary Value Problems Arising from Climate Modeling, PhD Thesis, Auburn Univ., 1994.Google Scholar

[25]

I. Stakgold, Free boundary problems in climate modeling, In Mathematics, Climate and Environment, J. I. Díaz and J.-L.Lions (eds.), Research Notes in Applied Mathematics, Masson, Paris, 27 (1993), 179–188. Google Scholar

[26]

D. Terman, A free boundary problem arising from a bistable reaction–diffusion equation, SIAM J. Math. Anal, 14 (1983), 1107-1129. doi: 10.1137/0514086. Google Scholar

[27]

J. Valero, Attractors of parabolic equations without uniqueness, Journal of Dynamics and Differential Equations, 13 (2001), 711-744. doi: 10.1023/A:1016642525800. Google Scholar

[28]

X. Xu, Existence and regularity theorems for a free boundary problem governing a simple climate model, Applicable Anal., 42 (1991), 33-57. doi: 10.1080/00036819108840032. Google Scholar

[29]

D. G. Zill, A First Course in Differential Equations, 10th edition, Brooks/Cole Publishing Co, Cengage Learning (USA), 2013. doi: 10.1142/8580. Google Scholar

show all references

References:
[1]

D. ArcoyaJ. I. Díaz and L. Tello, S-Shaped bifurcation branch in a quasilinear multivalued model arising in Climatology, Journal of Differential Equations, 150 (1998), 215-225. doi: 10.1006/jdeq.1998.3502. Google Scholar

[2]

J. ArrietaA. Rodríguez-Bernal and J. Valero, Dynamics of a reaction–diffusion equation with a discontinuous nonlinearity, International Journal of Bifurcation and Chaos, 16 (2006), 2965-2984. doi: 10.1142/S0218127406016586. Google Scholar

[3]

S. Bensid and J. I. Diaz, Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions, Discrete and Continuous Dynamical Systems, Series B, 22 (2017), 1757-1778. doi: 10.3934/dcdsb.2017105. Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. Google Scholar

[5]

H. Deguchi, Existence, uniqueness and non-uniqueness of weak solutions of parabolic initial-value problems with discontinuous nonlinearities, Proceedings of the Royal Society of Edinburgh, 135A (2005), 1139-1167. doi: 10.1017/S0308210500004315. Google Scholar

[6]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London, 1985. Google Scholar

[7]

J. I. Díaz, Mathematical analysis of some diffusive energy balance climate models, In Mathematics, Climate and Environment (J. Díaz and J.-L. Lions, eds.) Masson, Paris, 27 (1993), 28–56. Google Scholar

[8]

J. I. DíazA. C. FowlerA. I. Muñoz and E. Schiavi, Mathematical analysis of a model of river channel formation, Pure Appl. Geophys., 165 (2008), 1663-1682. Google Scholar

[9]

J. I. DíazJ. Hernandez and L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in Climatology, Mathematical Analysis and Applications, 216 (1997), 593-613. doi: 10.1006/jmaa.1997.5691. Google Scholar

[10]

J. I. Díaz and G. Hetzer, A Functional Quasilinear Reaction-Diffusion Equation Arising in Climatology, É Quations Aux Dérivées Partielles et Applications. Articles Dédi és à J.-L. Lions, Elsevier, Paris, (1998), 461–480. Google Scholar

[11]

J. I. DíazJ. A. Langa and J. Valero, On the asymptotic behaviour of solutions of a stochastic energy balance climate model, Physica D, 238 (2009), 880-887. doi: 10.1016/j.physd.2009.02.010. Google Scholar

[12]

J. I. Díaz and L. Tello, On a nonlinear parabolic problem on a Riemannian manifold without boundary arising in Climatology, Collectanea Mathematica, 50 (1999), 19-51. Google Scholar

[13]

J. I. Díaz and L. Tello, Infinitely many stationary solutions for a simple climate model via a shooting method, Mathematical Methods in the Applied Sciences, 25 (2002), 327-334. doi: 10.1002/mma.289. Google Scholar

[14]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. Google Scholar

[15]

R. Gianni and J. Hulshof, The semilinear heat equation with a Heaviside source term, Euro. J. of Applied Mathematics, 3 (1992), 367-379. doi: 10.1017/S0956792500000917. Google Scholar

[16]

N. V. GorbanO. V. KhomenkoL. S. Paliichuk and A. M. Tkachuk, Long-time behavior of state functions for climate energy balance model, Discrete and Continuous Dynamical Systems, Series B, 22 (2017), 1887-1897. doi: 10.3934/dcdsb.2017112. Google Scholar

[17]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics No. 840, Springer-Verlag, New York, 1981. Google Scholar

[18]

G. Hetzer, The structure of the principal component for semilinear diffusion equations from energy balance climate models, Houston Journal of Mathematics, 16 (1990), 203-216. Google Scholar

[19]

G. Hetzer, The number of stationary solntions for one-dimensional Budyko-type climate models, Nonlinear Analysis: Real World Applications, 2 (2001), 259-272. doi: 10.1016/S0362-546X(00)00103-6. Google Scholar

[20]

F. Issard-Roch, Subsolutions, supersolutions, and asymptotic behavior of an evolution problem with unilateral constraint, J. Math. Anal. Appl, 140 (1989), 50-66. doi: 10.1016/0022-247X(89)90093-0. Google Scholar

[21]

X. LiangX. Lin and H. Matano, A variational problem associated with the minimal speed of travelling waves for spatially periodic reaction-diffusion equations, Transactions of the American Mathematical Society, 362 (2010), 5605-5633. doi: 10.1090/S0002-9947-2010-04931-1. Google Scholar

[22]

H. P. McKean, Nagumo's equation, Advances in Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X. Google Scholar

[23]

G. R. North and J. A. Coakley, Differences between seasonal and mean annual energy balance model calculations of climate and climate sensitivity, Journal of the Atmospheric Sciences, 36 (1979), 1189-1203. Google Scholar

[24]

B. Schmidt, Bifurcation of Stationary Solutions for Legendre-Type Boundary Value Problems Arising from Climate Modeling, PhD Thesis, Auburn Univ., 1994.Google Scholar

[25]

I. Stakgold, Free boundary problems in climate modeling, In Mathematics, Climate and Environment, J. I. Díaz and J.-L.Lions (eds.), Research Notes in Applied Mathematics, Masson, Paris, 27 (1993), 179–188. Google Scholar

[26]

D. Terman, A free boundary problem arising from a bistable reaction–diffusion equation, SIAM J. Math. Anal, 14 (1983), 1107-1129. doi: 10.1137/0514086. Google Scholar

[27]

J. Valero, Attractors of parabolic equations without uniqueness, Journal of Dynamics and Differential Equations, 13 (2001), 711-744. doi: 10.1023/A:1016642525800. Google Scholar

[28]

X. Xu, Existence and regularity theorems for a free boundary problem governing a simple climate model, Applicable Anal., 42 (1991), 33-57. doi: 10.1080/00036819108840032. Google Scholar

[29]

D. G. Zill, A First Course in Differential Equations, 10th edition, Brooks/Cole Publishing Co, Cengage Learning (USA), 2013. doi: 10.1142/8580. Google Scholar

Figure 1.  Bifucation S-shaped curve
Figure 2.  Qualitative representation of the three surface atmosphere equilibria temperature depending of the equilatitude parallel circles $x\in [-1, 1]$
Figure 3.  Auxiliary barrier functions
Figure 4.  Dynamics of solutions corresponding to suitable initial data closed to the unstable equilibrium $\underline{u}_{\lambda , \mu }$
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