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February  2016, 36(2): 661-682. doi: 10.3934/dcds.2016.36.661

Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials

Hua Chen 1, and  Nian Liu 1,

1. 

School of Mathematics and Statistics, Wuhan University, Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China

Received  June 2014 Revised  January 2015 Published  August 2015

This article studies the initial boundary value problem for a class of semilinear edge-degenerate parabolic equations with singular potential term. By introducing a family of potential wells, we derive a threshold of the existence of global solutions with exponential decay, and the blow-up in finite time in both cases with low initial energy and critical initial energy.
Keywords: potential well method, edge type degeneracy, asymptotic stability, blow-up in finite time., Semilinear parabolic equations, singular potential.
Mathematics Subject Classification: Primary: 35B40, 35B44; Secondary: 35K2.
Citation: 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, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
References:
[1]

M. Alimohammady and M. K. Kalleji, Existence result for a class of semilinear totally characteristic hypoelliptic equations with conical degeneration, J. Funct. Anal., 265 (2013), 2331-2356. doi: 10.1016/j.jfa.2013.07.013.  Google Scholar

[2]

H. Chen and G. Liu, Global existence and nonexistence for semilinear parabolic equations with conical degeneration, J. Pseudo-Differ. Oper. Appl., 3 (2012), 329-349. doi: 10.1007/s11868-012-0046-9.  Google Scholar

[3]

H. Chen, X. Liu and Y. Wei, Existence theorem for a class of semilinear totally characteristic elliptic equations with critical cone Sobolev exponents, Ann. Global Anal. Geom., 39 (2011), 27-43. doi: 10.1007/s10455-010-9226-0.  Google Scholar

[4]

H. Chen, X. Liu and Y. Wei, Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on manifold with concial singularities, Calc. Var. Partial Differential Equations, 43 (2012), 463-484. doi: 10.1007/s00526-011-0418-7.  Google Scholar

[5]

H. Chen, X. Liu and Y. Wei, Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents, J. Differential Equations, 252 (2012), 4200-4228. doi: 10.1016/j.jde.2011.12.009.  Google Scholar

[6]

H. Chen, X. Liu and Y. Wei, Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term, J. Differential Equations, 252 (2012), 4289-4314. doi: 10.1016/j.jde.2012.01.011.  Google Scholar

[7]

H. Chen, X. Liu and Y. Wei, Multiple solutions for semi-linear corner degenerate elliptic equations, Journal of Functional Analysis, 266 (2014), 3815-3839. doi: 10.1016/j.jfa.2013.12.012.  Google Scholar

[8]

H. Chen, Y. Wei and B. Zhou, Existence of solutions for degenerate elliptic equations with singular potential on conical singular manifolds, Math. Nachr., 285 (2012), 1370-1384.  Google Scholar

[9]

Ju. V. Egorov and B.-W. Schulze, Pseudo-Differential Operators, Singularities, Appliciations, Oper. Theory Adv. Appl., 93, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8900-1.  Google Scholar

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S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, 33 (2008), 1996-2019. doi: 10.1080/03605300802402633.  Google Scholar

[11]

V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal., 250 (2007), 265-316. doi: 10.1016/j.jfa.2006.10.019.  Google Scholar

[12]

V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Mason-John Wiley, Paris, 1994.  Google Scholar

[13]

Y. Liu and J. Zhao, On potential wells and applications to semiliear hyperbolic and parabolic equations, Nonliear Anal., 64 (2006), 2665-2687. doi: 10.1016/j.na.2005.09.011.  Google Scholar

[14]

R. Mazzeo, Elliptic theory of differential edge operators, I, Comm. Partial Differential Equations, 16 (1991), 1615-1664. doi: 10.1080/03605309108820815.  Google Scholar

[15]

L. E. Payne, G. A. Philippin and P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Anal. TMA, 69 (2008), 3495-3502. doi: 10.1016/j.na.2007.09.035.  Google Scholar

[16]

L. E. Payne and P. W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl., 328 (2007), 1196-1205. doi: 10.1016/j.jmaa.2006.06.015.  Google Scholar

[17]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, 1980.  Google Scholar

[18]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.  Google Scholar

[19]

E. Schrohe and J. Seiler, Ellipticity and invertibility in the cone algebra on $L_p$-Sobolev spaces, Integral Equations Operator Theory, 41 (2001), 93-114. doi: 10.1007/BF01202533.  Google Scholar

[20]

B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, J. Wiley, Chichester, 1998.  Google Scholar

[21]

J. A. Wheeler and W. H. Zurek, Quantum Theory and Measurements, Princeton Univ. Press, Princetion, 1983. doi: 10.1515/9781400854554.  Google Scholar

show all references

References:
[1]

M. Alimohammady and M. K. Kalleji, Existence result for a class of semilinear totally characteristic hypoelliptic equations with conical degeneration, J. Funct. Anal., 265 (2013), 2331-2356. doi: 10.1016/j.jfa.2013.07.013.  Google Scholar

[2]

H. Chen and G. Liu, Global existence and nonexistence for semilinear parabolic equations with conical degeneration, J. Pseudo-Differ. Oper. Appl., 3 (2012), 329-349. doi: 10.1007/s11868-012-0046-9.  Google Scholar

[3]

H. Chen, X. Liu and Y. Wei, Existence theorem for a class of semilinear totally characteristic elliptic equations with critical cone Sobolev exponents, Ann. Global Anal. Geom., 39 (2011), 27-43. doi: 10.1007/s10455-010-9226-0.  Google Scholar

[4]

H. Chen, X. Liu and Y. Wei, Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on manifold with concial singularities, Calc. Var. Partial Differential Equations, 43 (2012), 463-484. doi: 10.1007/s00526-011-0418-7.  Google Scholar

[5]

H. Chen, X. Liu and Y. Wei, Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents, J. Differential Equations, 252 (2012), 4200-4228. doi: 10.1016/j.jde.2011.12.009.  Google Scholar

[6]

H. Chen, X. Liu and Y. Wei, Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term, J. Differential Equations, 252 (2012), 4289-4314. doi: 10.1016/j.jde.2012.01.011.  Google Scholar

[7]

H. Chen, X. Liu and Y. Wei, Multiple solutions for semi-linear corner degenerate elliptic equations, Journal of Functional Analysis, 266 (2014), 3815-3839. doi: 10.1016/j.jfa.2013.12.012.  Google Scholar

[8]

H. Chen, Y. Wei and B. Zhou, Existence of solutions for degenerate elliptic equations with singular potential on conical singular manifolds, Math. Nachr., 285 (2012), 1370-1384.  Google Scholar

[9]

Ju. V. Egorov and B.-W. Schulze, Pseudo-Differential Operators, Singularities, Appliciations, Oper. Theory Adv. Appl., 93, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8900-1.  Google Scholar

[10]

S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, 33 (2008), 1996-2019. doi: 10.1080/03605300802402633.  Google Scholar

[11]

V. Felli, E. M. Marchini and S. Terracini, On Schrödinger operators with multipolar inverse-square potentials, J. Funct. Anal., 250 (2007), 265-316. doi: 10.1016/j.jfa.2006.10.019.  Google Scholar

[12]

V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Mason-John Wiley, Paris, 1994.  Google Scholar

[13]

Y. Liu and J. Zhao, On potential wells and applications to semiliear hyperbolic and parabolic equations, Nonliear Anal., 64 (2006), 2665-2687. doi: 10.1016/j.na.2005.09.011.  Google Scholar

[14]

R. Mazzeo, Elliptic theory of differential edge operators, I, Comm. Partial Differential Equations, 16 (1991), 1615-1664. doi: 10.1080/03605309108820815.  Google Scholar

[15]

L. E. Payne, G. A. Philippin and P. W. Schaefer, Blow-up phenomena for some nonlinear parabolic problems, Nonlinear Anal. TMA, 69 (2008), 3495-3502. doi: 10.1016/j.na.2007.09.035.  Google Scholar

[16]

L. E. Payne and P. W. Schaefer, Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. Math. Anal. Appl., 328 (2007), 1196-1205. doi: 10.1016/j.jmaa.2006.06.015.  Google Scholar

[17]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, 1980.  Google Scholar

[18]

D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172.  Google Scholar

[19]

E. Schrohe and J. Seiler, Ellipticity and invertibility in the cone algebra on $L_p$-Sobolev spaces, Integral Equations Operator Theory, 41 (2001), 93-114. doi: 10.1007/BF01202533.  Google Scholar

[20]

B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, J. Wiley, Chichester, 1998.  Google Scholar

[21]

J. A. Wheeler and W. H. Zurek, Quantum Theory and Measurements, Princeton Univ. Press, Princetion, 1983. doi: 10.1515/9781400854554.  Google Scholar

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