
-
Previous Article
Attractors for a random evolution equation with infinite memory: Theoretical results
- DCDS-B Home
- This Issue
-
Next Article
Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems"
Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions
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 |
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.
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 ut−∆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] |
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 ut−∆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] |


[1] |
Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839 |
[2] |
Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure and Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032 |
[3] |
Chih-Yuan Chen, Shin-Hwa Wang, Kuo-Chih Hung. S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2589-2608. doi: 10.3934/cpaa.2014.13.2589 |
[4] |
James Walsh. Diffusive heat transport in Budyko's energy balance climate model with a dynamic ice line. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2687-2715. doi: 10.3934/dcdsb.2017131 |
[5] |
Gregorio Díaz, Jesús Ildefonso Díaz. Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing. Discrete and Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021165 |
[6] |
Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045 |
[7] |
Xue Dong He, Roy Kouwenberg, Xun Yu Zhou. Inverse S-shaped probability weighting and its impact on investment. Mathematical Control and Related Fields, 2018, 8 (3&4) : 679-706. doi: 10.3934/mcrf.2018029 |
[8] |
James Walsh, Christopher Rackauckas. On the Budyko-Sellers energy balance climate model with ice line coupling. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2187-2216. doi: 10.3934/dcdsb.2015.20.2187 |
[9] |
Nataliia V. Gorban, Olha V. Khomenko, Liliia S. Paliichuk, Alla M. Tkachuk. Long-time behavior of state functions for climate energy balance model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1887-1897. doi: 10.3934/dcdsb.2017112 |
[10] |
Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4397-4410. doi: 10.3934/dcdsb.2020103 |
[11] |
Yuki Kaneko, Hiroshi Matsuzawa, Yoshio Yamada. A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2719-2745. doi: 10.3934/dcds.2021209 |
[12] |
Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 |
[13] |
Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems and Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49 |
[14] |
Yaodan Huang, Zhengce Zhang, Bei Hu. Bifurcation from stability to instability for a free boundary tumor model with angiogenesis. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2473-2510. doi: 10.3934/dcds.2019105 |
[15] |
Junde Wu. Bifurcation for a free boundary problem modeling the growth of necrotic multilayered tumors. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3399-3411. doi: 10.3934/dcds.2019140 |
[16] |
Jean-Michel Rakotoson. Generalized eigenvalue problem for totally discontinuous operators. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 343-373. doi: 10.3934/dcds.2010.28.343 |
[17] |
Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253 |
[18] |
Yizhao Qin, Yuxia Guo, Peng-Fei Yao. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086 |
[19] |
Bruce Geist and Joyce R. McLaughlin. Eigenvalue formulas for the uniform Timoshenko beam: the free-free problem. Electronic Research Announcements, 1998, 4: 12-17. |
[20] |
Julius Fergy T. Rabago, Hideyuki Azegami. A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem. Evolution Equations and Control Theory, 2019, 8 (4) : 785-824. doi: 10.3934/eect.2019038 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]