October  2018, 38(10): 5297-5337. doi: 10.3934/dcds.2018234

Stability analysis for a family of degenerate semilinear parabolic problems

Département de mathématiques, Station 8, EPFL, Lausanne, CH 1015, Switzerland

Received  April 2018 Published  July 2018

This paper deals with the initial value problem for a class of degenerate nonlinear parabolic equations on a bounded domain in $ \mathbb{R}^N$ for $ N≥2$ with the Dirichlet boundary condition. The assumptions ensure that $ u\equiv0$ is a stationary solution and its stability is analysed. Amongst other things the results show that, in the case of critical degeneracy, the principle of linearized stability fails for some simple smooth nonlinearities. It is also shown that for levels of degeneracy less than the critical one linearized stability is justified for a broad class of nonlinearities including those for which it fails in the critical case.

Citation: Charles A. Stuart. Stability analysis for a family of degenerate semilinear parabolic problems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5297-5337. doi: 10.3934/dcds.2018234
References:
[1]

B. AbdellaouiE. Colorado and I. Peral, Existence and non-existence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities, J. Eur. Math. Soc., 6 (2004), 119-149.   Google Scholar

[2]

B. Abdellaoui and I. Peral, On quasilinear elliptic equation related to some Caffarelli-KohnNirenberg inequalities, Comm. Pure Appl. Anal., 2 (2003), 539-566.  doi: 10.3934/cpaa.2003.2.539.  Google Scholar

[3]

H. W. Alt, Lineare Funktionalanalysis 2nd ed., Springer Lehrbuch, Berlin, 1993. Google Scholar

[4]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Adv. Math. No 34, C. U. P. Cambridge 1993.  Google Scholar

[5]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford, 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.  Google Scholar

[6]

T. BartschS. Peng and Z. Zhang, Existence and non-existence of solutions to elliptic equations related to Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. P.D.E., 30 (2007), 113-136.  doi: 10.1007/s00526-006-0086-1.  Google Scholar

[7]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weight, Compositio Math., 53 (1984), 259-275.   Google Scholar

[8]

P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, NoDEA, 7 (2000), 187-199.  doi: 10.1007/s000300050004.  Google Scholar

[9]

P. Caldiroli and R. Musina, On the existence of extremal functions for a weighted Sobolev embedding with critical exponent, Calc. Var. P.D.E., 8 (1999), 365-387.  doi: 10.1007/s005260050130.  Google Scholar

[10]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence) and symmetry of extremals, Comm. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[11]

F. Chiarenza and R. Serapioni, Degenerate parabolic equations and Harnack inequality, Ann. Mat. Pura Appl., 137 (1984), 139-162.  doi: 10.1007/BF01789392.  Google Scholar

[12]

F. Chiarenza and R. Serapioni, Pointwise estimates for degenerate parabolic equations, Applicable Anal., 23 (1987), 287-299.  doi: 10.1080/00036818708839648.  Google Scholar

[13]

J. W. Cholowa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, LMS Lecture Notes No. 278, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.  Google Scholar

[14]

A. Dall'AglioD. Giachetti and I. Peral, Results on parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities, SIAM J. Math. Anal., 36 (2004), 691-716.  doi: 10.1137/S0036141003432353.  Google Scholar

[15]

D. G. De Figereido, The Ekeland Variational Principle with Applications and Detours, TIFR Lecture Notes, Springer-Verlag, Berlin, 1989.  Google Scholar

[16]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[17]

G. Evéquoz and C. A. Stuart, Bifurcation points of a degenerate elliptic boundary-value problem, Rend. Lincei Mat. Appl., 17 (2006), 309-334.  doi: 10.4171/RLM/471.  Google Scholar

[18]

G. Evéquoz and C. A. Stuart, Bifurcation and concentration of radial solutions of a nonlinear degenerate elliptic eigenvalue problem, Adv. Nonlinear Studies, 6 (2006), 215-232.  doi: 10.1515/ans-2006-0206.  Google Scholar

[19]

T. M. Flett, Differential Analysis, Cambridge University Press, Cambridge, 1980.  Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[21]

Z. M. GuoF. S. Wan and X. H. Guan, Embeddings of weighted Sobolev spaces and degenerate elliptic problems, Science China Math., 60 (2017), 1399-1418.  doi: 10.1007/s11425-016-0403-6.  Google Scholar

[22]

C. E. Gutierrez, Pointwise estimates for solutions of degenerate parabolic equations, Rev. U. Mat. Argentina, 37 (1991), 261-270.   Google Scholar

[23]

C. E. Gutierrez and R. Wheeden, Harnack's inequality for degenerate parabolic equations, Comm. P.D.E., 16 (1991), 745-770.  doi: 10.1080/03605309108820776.  Google Scholar

[24]

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

[25]

V. A. Ivanov, Boundary value problems for degenerate second order linear parabolic equations, Sem. in Math. V. A. Steklov Inst., Ed. O. A. Ladyzhenskaya, 14 (1971), 22–43. Google Scholar

[26]

S. N. Kruzkov, Boundary value problems for degenerate second order elliptic equations, Mat. Sb. (N.S.), 77 (1968), 299-334.   Google Scholar

[27]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci. No 74, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[28]

D. Motreanu and V. Radulescu, Eigenvalue problems for degenerate nonlinear elliptic equations in anisotropic media, Boundary Value Problems, 2 (2005), 107-127.   Google Scholar

[29]

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1–122 and Errata Corrige, Ibidem, 90 (1971), 413–414. doi: 10.1007/BF02415055.  Google Scholar

[30]

F. Nicolosi, Sulla limitatezza delle soluzioni deaboli dei problemi al contorno per operatori parabolici degeneri, Rendiconti Circ. Mat. Palermo, XXX1 (1982), 23-40.   Google Scholar

[31]

P. J. Rabier, Embeddings of weighted Sobolev spaces and generalized Caffarelli-Kohn-Nirenberg inequalities, J. Anal. Mat., 118 (2012), 251-296.  doi: 10.1007/s11854-012-0035-1.  Google Scholar

[32]

C. A. Stuart, Bifurcation at isolated singular points for a degenerate elliptic eigenvalue problem, Nonlinear Anal. TMA, 119 (2015), 209-221.  doi: 10.1016/j.na.2014.09.022.  Google Scholar

[33]

C. A. Stuart, Bifurcation points of a critically degenerate elliptic Dirichlet problem, in preparation. Google Scholar

[34]

C. A. Stuart, Criteria for the existence of a potential well, Nonlinear Anal.TMA, 158 (2017), 83-108.  doi: 10.1016/j.na.2017.04.001.  Google Scholar

[35]

C. A. Stuart, Bifurcation points of a singular boundary-value problem on (0, 1), J. Diff. Equat., 260 (2016), 6267-6321.  doi: 10.1016/j.jde.2015.12.040.  Google Scholar

[36]

N. S. Trudinger, Linear elliptic operators with measurable coeffients, Ann. Scuola Norm. Super. Pisa, 27 (1973), 265-308.   Google Scholar

[37]

J. Weidmann, Linear Operators in Hilbert Space, Springer, Berlin, 1980.  Google Scholar

show all references

References:
[1]

B. AbdellaouiE. Colorado and I. Peral, Existence and non-existence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities, J. Eur. Math. Soc., 6 (2004), 119-149.   Google Scholar

[2]

B. Abdellaoui and I. Peral, On quasilinear elliptic equation related to some Caffarelli-KohnNirenberg inequalities, Comm. Pure Appl. Anal., 2 (2003), 539-566.  doi: 10.3934/cpaa.2003.2.539.  Google Scholar

[3]

H. W. Alt, Lineare Funktionalanalysis 2nd ed., Springer Lehrbuch, Berlin, 1993. Google Scholar

[4]

A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Adv. Math. No 34, C. U. P. Cambridge 1993.  Google Scholar

[5]

J. M. Ball, Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford, 28 (1977), 473-486.  doi: 10.1093/qmath/28.4.473.  Google Scholar

[6]

T. BartschS. Peng and Z. Zhang, Existence and non-existence of solutions to elliptic equations related to Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. P.D.E., 30 (2007), 113-136.  doi: 10.1007/s00526-006-0086-1.  Google Scholar

[7]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weight, Compositio Math., 53 (1984), 259-275.   Google Scholar

[8]

P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, NoDEA, 7 (2000), 187-199.  doi: 10.1007/s000300050004.  Google Scholar

[9]

P. Caldiroli and R. Musina, On the existence of extremal functions for a weighted Sobolev embedding with critical exponent, Calc. Var. P.D.E., 8 (1999), 365-387.  doi: 10.1007/s005260050130.  Google Scholar

[10]

F. Catrina and Z.-Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence) and symmetry of extremals, Comm. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[11]

F. Chiarenza and R. Serapioni, Degenerate parabolic equations and Harnack inequality, Ann. Mat. Pura Appl., 137 (1984), 139-162.  doi: 10.1007/BF01789392.  Google Scholar

[12]

F. Chiarenza and R. Serapioni, Pointwise estimates for degenerate parabolic equations, Applicable Anal., 23 (1987), 287-299.  doi: 10.1080/00036818708839648.  Google Scholar

[13]

J. W. Cholowa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, LMS Lecture Notes No. 278, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511526404.  Google Scholar

[14]

A. Dall'AglioD. Giachetti and I. Peral, Results on parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities, SIAM J. Math. Anal., 36 (2004), 691-716.  doi: 10.1137/S0036141003432353.  Google Scholar

[15]

D. G. De Figereido, The Ekeland Variational Principle with Applications and Detours, TIFR Lecture Notes, Springer-Verlag, Berlin, 1989.  Google Scholar

[16]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[17]

G. Evéquoz and C. A. Stuart, Bifurcation points of a degenerate elliptic boundary-value problem, Rend. Lincei Mat. Appl., 17 (2006), 309-334.  doi: 10.4171/RLM/471.  Google Scholar

[18]

G. Evéquoz and C. A. Stuart, Bifurcation and concentration of radial solutions of a nonlinear degenerate elliptic eigenvalue problem, Adv. Nonlinear Studies, 6 (2006), 215-232.  doi: 10.1515/ans-2006-0206.  Google Scholar

[19]

T. M. Flett, Differential Analysis, Cambridge University Press, Cambridge, 1980.  Google Scholar

[20]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[21]

Z. M. GuoF. S. Wan and X. H. Guan, Embeddings of weighted Sobolev spaces and degenerate elliptic problems, Science China Math., 60 (2017), 1399-1418.  doi: 10.1007/s11425-016-0403-6.  Google Scholar

[22]

C. E. Gutierrez, Pointwise estimates for solutions of degenerate parabolic equations, Rev. U. Mat. Argentina, 37 (1991), 261-270.   Google Scholar

[23]

C. E. Gutierrez and R. Wheeden, Harnack's inequality for degenerate parabolic equations, Comm. P.D.E., 16 (1991), 745-770.  doi: 10.1080/03605309108820776.  Google Scholar

[24]

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

[25]

V. A. Ivanov, Boundary value problems for degenerate second order linear parabolic equations, Sem. in Math. V. A. Steklov Inst., Ed. O. A. Ladyzhenskaya, 14 (1971), 22–43. Google Scholar

[26]

S. N. Kruzkov, Boundary value problems for degenerate second order elliptic equations, Mat. Sb. (N.S.), 77 (1968), 299-334.   Google Scholar

[27]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Appl. Math. Sci. No 74, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[28]

D. Motreanu and V. Radulescu, Eigenvalue problems for degenerate nonlinear elliptic equations in anisotropic media, Boundary Value Problems, 2 (2005), 107-127.   Google Scholar

[29]

M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1–122 and Errata Corrige, Ibidem, 90 (1971), 413–414. doi: 10.1007/BF02415055.  Google Scholar

[30]

F. Nicolosi, Sulla limitatezza delle soluzioni deaboli dei problemi al contorno per operatori parabolici degeneri, Rendiconti Circ. Mat. Palermo, XXX1 (1982), 23-40.   Google Scholar

[31]

P. J. Rabier, Embeddings of weighted Sobolev spaces and generalized Caffarelli-Kohn-Nirenberg inequalities, J. Anal. Mat., 118 (2012), 251-296.  doi: 10.1007/s11854-012-0035-1.  Google Scholar

[32]

C. A. Stuart, Bifurcation at isolated singular points for a degenerate elliptic eigenvalue problem, Nonlinear Anal. TMA, 119 (2015), 209-221.  doi: 10.1016/j.na.2014.09.022.  Google Scholar

[33]

C. A. Stuart, Bifurcation points of a critically degenerate elliptic Dirichlet problem, in preparation. Google Scholar

[34]

C. A. Stuart, Criteria for the existence of a potential well, Nonlinear Anal.TMA, 158 (2017), 83-108.  doi: 10.1016/j.na.2017.04.001.  Google Scholar

[35]

C. A. Stuart, Bifurcation points of a singular boundary-value problem on (0, 1), J. Diff. Equat., 260 (2016), 6267-6321.  doi: 10.1016/j.jde.2015.12.040.  Google Scholar

[36]

N. S. Trudinger, Linear elliptic operators with measurable coeffients, Ann. Scuola Norm. Super. Pisa, 27 (1973), 265-308.   Google Scholar

[37]

J. Weidmann, Linear Operators in Hilbert Space, Springer, Berlin, 1980.  Google Scholar

[1]

Teemu Lukkari, Mikko Parviainen. Stability of degenerate parabolic Cauchy problems. Communications on Pure & Applied Analysis, 2015, 14 (1) : 201-216. doi: 10.3934/cpaa.2015.14.201

[2]

Peter Howard, K. Zumbrun. The Evans function and stability criteria for degenerate viscous shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 837-855. doi: 10.3934/dcds.2004.10.837

[3]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081

[4]

Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661

[5]

Stephen Pankavich, Petronela Radu. Nonlinear instability of solutions in parabolic and hyperbolic diffusion. Evolution Equations & Control Theory, 2013, 2 (2) : 403-422. doi: 10.3934/eect.2013.2.403

[6]

Toshi Ogawa. Degenerate Hopf instability in oscillatory reaction-diffusion equations. Conference Publications, 2007, 2007 (Special) : 784-793. doi: 10.3934/proc.2007.2007.784

[7]

Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671

[8]

Luis Barreira, Claudia Valls. Stability of nonautonomous equations and Lyapunov functions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2631-2650. doi: 10.3934/dcds.2013.33.2631

[9]

Jacopo De Simoi. Stability and instability results in a model of Fermi acceleration. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 719-750. doi: 10.3934/dcds.2009.25.719

[10]

Takayoshi Ogawa, Hiroshi Wakui. Stability and instability of solutions to the drift-diffusion system. Evolution Equations & Control Theory, 2017, 6 (4) : 587-597. doi: 10.3934/eect.2017029

[11]

Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048

[12]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101

[13]

Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57

[14]

Łukasz Struski, Jacek Tabor. Expansivity implies existence of Hölder continuous Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3575-3589. doi: 10.3934/dcdsb.2017180

[15]

Robert Baier, Lars Grüne, Sigurđur Freyr Hafstein. Linear programming based Lyapunov function computation for differential inclusions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 33-56. doi: 10.3934/dcdsb.2012.17.33

[16]

Peter Giesl. Construction of a finite-time Lyapunov function by meshless collocation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2387-2412. doi: 10.3934/dcdsb.2012.17.2387

[17]

Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas, Skuli Gudmundsson. Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4247-4269. doi: 10.3934/dcdsb.2019080

[18]

Julien Grivaux, Pascal Hubert. Loci in strata of meromorphic quadratic differentials with fully degenerate Lyapunov spectrum. Journal of Modern Dynamics, 2014, 8 (1) : 61-73. doi: 10.3934/jmd.2014.8.61

[19]

Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071

[20]

Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (86)
  • HTML views (68)
  • Cited by (0)

Other articles
by authors

[Back to Top]