Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 35B40, 35B44; Secondary: 35K20.

 Citation:

•  [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] V. Komornik, Exact Controllability and Stabilization, The Multiplier Method, Mason-John Wiley, Paris, 1994. [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. [14] R. Mazzeo, Elliptic theory of differential edge operators, I, Comm. Partial Differential Equations, 16 (1991), 1615-1664.doi: 10.1080/03605309108820815. [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. [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. [17] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, 1980. [18] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal., 30 (1968), 148-172. [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. [20] B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, J. Wiley, Chichester, 1998. [21] J. A. Wheeler and W. H. Zurek, Quantum Theory and Measurements, Princeton Univ. Press, Princetion, 1983.doi: 10.1515/9781400854554.