Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions

Sabri Bensid; Jesús Ildefonso Díaz

doi:10.3934/dcdsb.2017105

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July 2017, 22(5): 1757-1778. doi: 10.3934/dcdsb.2017105

Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions

Sabri Bensid ^1, and Jesús Ildefonso Díaz ^2,,

1.	Department of Mathematics, Faculty of Sciences, University of Tlemcen, B.P. 119, Tlemcen 13000, Algeria
2.	Instituto de Matemática Interdisciplinar, Depto. de Matemática Aplicada, Parque de Ciencias 3,28040{Madrid, Spain

* Corresponding author: Jesús Ildefonso Díaz

Received January 2016 Revised June 2016 Published March 2017

Fund Project: The research of J.I. Díaz was partially supported by the project ref. MTM2014-57113 of the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480) of the UCM. The results of this paper were started during the visit of the first author to the UCM, on October 2015, under the support of the University Aboubekr Belkaid of Tlemcen (Algeria). This author wants to thank this support as well as the received hospitality from the Instituto de Matemática Interdisciplar of the UCM.

We study stability of the nonnegative solutions of a discontinuous elliptic eigenvalue problem relevant in several applications as for instance in climate modeling. After giving the explicit expresion of the S-shaped bifurcation diagram $\left( \lambda ,{{\left\| {{\mu }_{\lambda }} \right\|}_{\infty }} \right)$ we show the instability of the decreasing part of the bifurcation curve and the stability of the increasing part. This extends to the case of non-smooth nonlinear terms the well known 1971 result by M.G. Crandall and P.H. Rabinowitz concerning differentiable nonlinear terms. We point out that, in general, there is a lacking of uniquenees of solutions for the associated parabolic problem. Nevertheless, for nondegenerate solutions (crossing the discontinuity value of u in a transversal way) the comparison principle and the uniqueness of solutions hold. The instability is obtained trough a linearization process leading to an eigenvalue problem in which a Dirac delta distribution appears as a coefficient of the differential operator. The stability proof uses a suitable change of variables, the continuuity of the bifurcation branch and the comparison principle for nondegenerate solutions of the parabolic problem.

Keywords: Nonlinear eigenvalue problem, discontinuous nonlinearity, S-shaped bifurcation curve, stability, free boundary, energy balance climate models.

Mathematics Subject Classification: 35B35, 35J61, 35P30, 35K58, 86A1.

Citation: Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105

References:

[1]	D. Arcoya, J. 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. Arrieta, A. 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]	M. Belloni and R. W. Robinett, The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics, Physics Reports, 540 (2014), 25-122. doi: 10.1016/j.physrep.2014.02.005. Google Scholar
[4]	S. Bensid and S. M. Bouguima, On a free boundary problem, Nonlinear Anal. T.M.A, 68 (2008), 2328-2348. doi: 10.1016/j.na.2007.01.047. Google Scholar
[5]	S. Bensid and S. M. Bouguima, Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions, Electronic Journal of Differential Equations, 2010 (2010), 1-16. Google Scholar
[6]	M. Bertsch and M. H. A. Klaver, On positive solutions of −∆u+f(u) = 0 with f discontinuous, J. Math. Anal. Appl., 157 (1991), 417-446. doi: 10.1016/0022-247X(91)90099-L. Google Scholar
[7]	H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for u_t−∆u = g(u) revisited, Adv. Differential Equations, 1 (1996), 73-90. Google Scholar
[8]	M. Coti Zelati, A. Huang, I. Kukavica, R. Temam and M. Ziane, The primitive equations of the atmosphere in presence of vapor saturation, Nonlinearity (2015), http://dx.doi.org/10.1088/0951-7715/28/3/625, in press. Google Scholar
[9]	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
[10]	J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London, 1985. Google Scholar
[11]	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
[12]	J. I. Díaz, A. C. Fowler, A. I. Muñoz and E. Schiavi, Mathematical analysis of a model of river channel formation, Pure appl. geophys., 165 (2008), 1663-1682. Google Scholar
[13]	J. I. Diaz and J. Hernández, Global bifurcation and continua of nonegative solutions for a quasilinear elliptic problem, Comptes Rendus Acad. Sci. Paris, 329 (1999), 587-592. doi: 10.1016/S0764-4442(00)80006-3. Google Scholar
[14]	J. I. Díaz, J. Hernández and Y. Ilyasov, Stability criteria on flat and compactly supported ground states of some non-Lipschitz autonomous semilinear equations, To appear in Chinese Annals of Mathematics. Google Scholar
[15]	J. I. Díaz, J. Hernandez and L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in Climatology, Jour. Mathematical Analysis and Applications, 216 (1997), 593-613. doi: 10.1006/jmaa.1997.5691. Google Scholar
[16]	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
[17]	J. I. Díaz, J. 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
[18]	J. I. Díaz, J. F. Padial and J. M. Rakotoson, On some Bernouilli free boundary type problems for general elliptic operators, Proceedings of the Royal Society of Edimburgh, 137 (2007), 895-911. doi: 10.1017/S0308210506000370. Google Scholar
[19]	J. I. Díaz and J. M. Rakotoson, On a nonlocal stationary free boundary problem arising in the confinement of a plasma in a Stellarator geometry, Archive for Rational Mechanics and Analysis, 134 (1996), 53-95. doi: 10.1007/BF00376255. Google Scholar
[20]	J. I. Díaz and S. Shmarev, Langragian approach to level sets: Application to a free boundary problem arising in climatology, Archive for Rational Mechanics and Analysis, 194 (2009), 75-103. doi: 10.1007/s00205-008-0164-y. Google Scholar
[21]	J. I. Diaz 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
[22]	I. H. Farina and R. Aris, Transients in distributed chemical reactors, Part 2: Influence of diffusion in the simplified model, Chem. Engng J., 4 (1972), 149-170. Google Scholar
[23]	E. Fereisel, A note on uniqueness for parabolic problems with discontinuous nonlinearities, Nonlinear Analysis, 16 (1991), 1053-1056. doi: 10.1016/0362-546X(91)90106-B. Google Scholar
[24]	E. Feireisl and J. Norbury, Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh Sect., 119 (1991), 1-17. doi: 10.1017/S0308210500028262. Google Scholar
[25]	B. A. Fleishman and T. J. Mahar, Analytic methods for approximate solution of elliptic free boundary problems, Nonlinear Anal., 1 (1977), 561-569. doi: 10.1016/0362-546X(77)90017-7. Google Scholar
[26]	B. A. Fleishman and T. J. Mahar, A step function model in chemical reactor theory: Multiplicity and stability of solutions, Nonl. Anal., 5 (1981), 645-654. doi: 10.1016/0362-546X(81)90080-8. Google Scholar
[27]	L. E. Fraenkel and M. S. Berger, A global theory of steady vortex rings in an ideal fluid, Acfa Math., 132 (1974), 13-51. doi: 10.1007/BF02392107. Google Scholar
[28]	A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1964. Google Scholar
[29]	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
[30]	A. A. Guetter, A free boundary problem in plasma containment, SIAM J. Appl. Math., 49 (1989), 99-115. doi: 10.1137/0149006. Google Scholar
[31]	A. A. Guetter, On solutions of an elliptic boundary value problem with a discontinuous nonlinearity, Nonl. Anal. T.M.A,, 23 (1994), 1413-1425. doi: 10.1016/0362-546X(94)90136-8. Google Scholar
[32]	D. Henry, Geometric theory of semilinear parabolic equations in Lecture Notes in Mathematics No. 840, Springer-Verlag, New York, 1981. Google Scholar
[33]	J. Hernandez, F. J. Mancebo and J. M. Vega, On the linearization ofsome singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal., 19 (2002), 777-813. doi: 10.1016/S0294-1449(02)00102-6. Google Scholar
[34]	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
[35]	X. Liang, X. 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
[36]	H. P. McKean, Nagumo's equation, Advances in Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X. Google Scholar
[37]	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. doi: 10.1175/1520-0469(1979)036<1189:DBSAMA>2.0.CO;2. Google Scholar
[38]	J. Rauch, Discontinuous semilinear differential equations and multiple valued maps, Proceedings of the American Mathematical Society, 64 (1977), 277-282. doi: 10.1090/S0002-9939-1977-0442453-6. Google Scholar
[39]	I. Stakgold, Free boundary problems in climate modeling. In, Mathematics, Climate and Environment, J. I. Díaz, and J. -L. Lions (Edits. ), Research Notes in Applied Mathematics no 27, Masson, Paris, 1993,179-88. Google Scholar
[40]	C. A. Stuart, The number of solutions of boundary value problems with discontinuous non-linearities, Archive for Rational Mechanics and Analysis, 66 (1977), 225-235. doi: 10.1007/BF00250672. Google Scholar
[41]	R. Temam, A nonlinear eigenvalue problem: The shape at equilibrium of a confined plasma, Arch. Rational. Mech. Anal, 60 (1975), 51-73. doi: 10.1007/BF00281469. Google Scholar
[42]	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
[43]	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
[44]	X. Xu, Existence and regularity theorems for a free boundary problem governing a simple climate model, Aplicable Anal., 42 (1991), 33-57. doi: 10.1080/00036819108840032. Google Scholar
[45]	K. Yosida, Functional Analysis Springer, 1965. Google Scholar

show all references

References:

[1]	D. Arcoya, J. 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. Arrieta, A. 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]	M. Belloni and R. W. Robinett, The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics, Physics Reports, 540 (2014), 25-122. doi: 10.1016/j.physrep.2014.02.005. Google Scholar
[4]	S. Bensid and S. M. Bouguima, On a free boundary problem, Nonlinear Anal. T.M.A, 68 (2008), 2328-2348. doi: 10.1016/j.na.2007.01.047. Google Scholar
[5]	S. Bensid and S. M. Bouguima, Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions, Electronic Journal of Differential Equations, 2010 (2010), 1-16. Google Scholar
[6]	M. Bertsch and M. H. A. Klaver, On positive solutions of −∆u+f(u) = 0 with f discontinuous, J. Math. Anal. Appl., 157 (1991), 417-446. doi: 10.1016/0022-247X(91)90099-L. Google Scholar
[7]	H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for u_t−∆u = g(u) revisited, Adv. Differential Equations, 1 (1996), 73-90. Google Scholar
[8]	M. Coti Zelati, A. Huang, I. Kukavica, R. Temam and M. Ziane, The primitive equations of the atmosphere in presence of vapor saturation, Nonlinearity (2015), http://dx.doi.org/10.1088/0951-7715/28/3/625, in press. Google Scholar
[9]	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
[10]	J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London, 1985. Google Scholar
[11]	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
[12]	J. I. Díaz, A. C. Fowler, A. I. Muñoz and E. Schiavi, Mathematical analysis of a model of river channel formation, Pure appl. geophys., 165 (2008), 1663-1682. Google Scholar
[13]	J. I. Diaz and J. Hernández, Global bifurcation and continua of nonegative solutions for a quasilinear elliptic problem, Comptes Rendus Acad. Sci. Paris, 329 (1999), 587-592. doi: 10.1016/S0764-4442(00)80006-3. Google Scholar
[14]	J. I. Díaz, J. Hernández and Y. Ilyasov, Stability criteria on flat and compactly supported ground states of some non-Lipschitz autonomous semilinear equations, To appear in Chinese Annals of Mathematics. Google Scholar
[15]	J. I. Díaz, J. Hernandez and L. Tello, On the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in Climatology, Jour. Mathematical Analysis and Applications, 216 (1997), 593-613. doi: 10.1006/jmaa.1997.5691. Google Scholar
[16]	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
[17]	J. I. Díaz, J. 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
[18]	J. I. Díaz, J. F. Padial and J. M. Rakotoson, On some Bernouilli free boundary type problems for general elliptic operators, Proceedings of the Royal Society of Edimburgh, 137 (2007), 895-911. doi: 10.1017/S0308210506000370. Google Scholar
[19]	J. I. Díaz and J. M. Rakotoson, On a nonlocal stationary free boundary problem arising in the confinement of a plasma in a Stellarator geometry, Archive for Rational Mechanics and Analysis, 134 (1996), 53-95. doi: 10.1007/BF00376255. Google Scholar
[20]	J. I. Díaz and S. Shmarev, Langragian approach to level sets: Application to a free boundary problem arising in climatology, Archive for Rational Mechanics and Analysis, 194 (2009), 75-103. doi: 10.1007/s00205-008-0164-y. Google Scholar
[21]	J. I. Diaz 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
[22]	I. H. Farina and R. Aris, Transients in distributed chemical reactors, Part 2: Influence of diffusion in the simplified model, Chem. Engng J., 4 (1972), 149-170. Google Scholar
[23]	E. Fereisel, A note on uniqueness for parabolic problems with discontinuous nonlinearities, Nonlinear Analysis, 16 (1991), 1053-1056. doi: 10.1016/0362-546X(91)90106-B. Google Scholar
[24]	E. Feireisl and J. Norbury, Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, Proc. Roy. Soc. Edinburgh Sect., 119 (1991), 1-17. doi: 10.1017/S0308210500028262. Google Scholar
[25]	B. A. Fleishman and T. J. Mahar, Analytic methods for approximate solution of elliptic free boundary problems, Nonlinear Anal., 1 (1977), 561-569. doi: 10.1016/0362-546X(77)90017-7. Google Scholar
[26]	B. A. Fleishman and T. J. Mahar, A step function model in chemical reactor theory: Multiplicity and stability of solutions, Nonl. Anal., 5 (1981), 645-654. doi: 10.1016/0362-546X(81)90080-8. Google Scholar
[27]	L. E. Fraenkel and M. S. Berger, A global theory of steady vortex rings in an ideal fluid, Acfa Math., 132 (1974), 13-51. doi: 10.1007/BF02392107. Google Scholar
[28]	A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1964. Google Scholar
[29]	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
[30]	A. A. Guetter, A free boundary problem in plasma containment, SIAM J. Appl. Math., 49 (1989), 99-115. doi: 10.1137/0149006. Google Scholar
[31]	A. A. Guetter, On solutions of an elliptic boundary value problem with a discontinuous nonlinearity, Nonl. Anal. T.M.A,, 23 (1994), 1413-1425. doi: 10.1016/0362-546X(94)90136-8. Google Scholar
[32]	D. Henry, Geometric theory of semilinear parabolic equations in Lecture Notes in Mathematics No. 840, Springer-Verlag, New York, 1981. Google Scholar
[33]	J. Hernandez, F. J. Mancebo and J. M. Vega, On the linearization ofsome singular nonlinear elliptic problems and applications, Ann. Inst. H. Poincaré Anal., 19 (2002), 777-813. doi: 10.1016/S0294-1449(02)00102-6. Google Scholar
[34]	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
[35]	X. Liang, X. 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
[36]	H. P. McKean, Nagumo's equation, Advances in Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X. Google Scholar
[37]	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. doi: 10.1175/1520-0469(1979)036<1189:DBSAMA>2.0.CO;2. Google Scholar
[38]	J. Rauch, Discontinuous semilinear differential equations and multiple valued maps, Proceedings of the American Mathematical Society, 64 (1977), 277-282. doi: 10.1090/S0002-9939-1977-0442453-6. Google Scholar
[39]	I. Stakgold, Free boundary problems in climate modeling. In, Mathematics, Climate and Environment, J. I. Díaz, and J. -L. Lions (Edits. ), Research Notes in Applied Mathematics no 27, Masson, Paris, 1993,179-88. Google Scholar
[40]	C. A. Stuart, The number of solutions of boundary value problems with discontinuous non-linearities, Archive for Rational Mechanics and Analysis, 66 (1977), 225-235. doi: 10.1007/BF00250672. Google Scholar
[41]	R. Temam, A nonlinear eigenvalue problem: The shape at equilibrium of a confined plasma, Arch. Rational. Mech. Anal, 60 (1975), 51-73. doi: 10.1007/BF00281469. Google Scholar
[42]	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
[43]	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
[44]	X. Xu, Existence and regularity theorems for a free boundary problem governing a simple climate model, Aplicable Anal., 42 (1991), 33-57. doi: 10.1080/00036819108840032. Google Scholar
[45]	K. Yosida, Functional Analysis Springer, 1965. Google Scholar

Figure 1. A qualitative description of the bifurcation curve

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Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions

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