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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London, 1985.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1964.
[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.
[30]	A. A. Guetter, A free boundary problem in plasma containment, SIAM J. Appl. Math., 49 (1989), 99-115. doi: 10.1137/0149006.
[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.
[32]	D. Henry, Geometric theory of semilinear parabolic equations in Lecture Notes in Mathematics No. 840, Springer-Verlag, New York, 1981.
[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.
[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.
[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.
[36]	H. P. McKean, Nagumo's equation, Advances in Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X.
[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.
[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.
[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.
[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.
[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.
[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.
[43]	J. Valero, Attractors of parabolic equations without uniqueness, Journal of Dynamics and Differential Equations, 13 (2001), 711-744. doi: 10.1023/A:1016642525800.
[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.
[45]	K. Yosida, Functional Analysis Springer, 1965.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Pitman, London, 1985.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc. , Englewood Cliffs, N. J. , 1964.
[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.
[30]	A. A. Guetter, A free boundary problem in plasma containment, SIAM J. Appl. Math., 49 (1989), 99-115. doi: 10.1137/0149006.
[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.
[32]	D. Henry, Geometric theory of semilinear parabolic equations in Lecture Notes in Mathematics No. 840, Springer-Verlag, New York, 1981.
[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.
[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.
[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.
[36]	H. P. McKean, Nagumo's equation, Advances in Math., 4 (1970), 209-223. doi: 10.1016/0001-8708(70)90023-X.
[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.
[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.
[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.
[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.
[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.
[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.
[43]	J. Valero, Attractors of parabolic equations without uniqueness, Journal of Dynamics and Differential Equations, 13 (2001), 711-744. doi: 10.1023/A:1016642525800.
[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.
[45]	K. Yosida, Functional Analysis Springer, 1965.

Figure 1. A qualitative description of the bifurcation curve

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Figure 2. Security neighborhood

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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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