In this article, we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities. Firstly, we introduce the corner type weighted $p$-Sobolev spaces and the weighted corner type Sobolev inequality, the Poincar$\acute{e}$ inequality, and the Hardy inequality. Then, by using the potential well method and the inequality mentioned above, we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy. Significantly, the relation between the above two phenomena is derived as a sharp condition. Moreover, we show that the global existence also holds for the case of a potential well family.
Hua CHEN
,
Nian LIU
. ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS[J]. Acta mathematica scientia, Series B, 2021
, 41(1)
: 257
-282
.
DOI: 10.1007/s10473-021-0115-3
[1] Ambrosetti A, Rabinowitz P H. Dual variational methods in critical point theory and applications. J Funct Anal, 1973, 14:349-381
[2] Alimohammady M, Kalleji M K. Existence result for a class of semilinear totally characteristic hypoelliptic equations with conical degeneration. J Funct Anal, 2013, 265:2331-2356
[3] Chen H, Liu N. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete Contin Dyn Syst, 2016, 36(2):661-682
[4] Chen H, Liu G. Global existence and nonexistence for semilinear parabolic equations with conical degeneration. J Pseudo-Differ Oper Appl, 2012, 3(3):329-349
[5] Chen H, Liu G. Global existence, uniform decay and exponential growth for a class of semi-linear wave equation with strong damping. Acta Math Sci, 2013, 33B(1):41-58
[6] Chen H, Liu X, Wei Y. Existence theorem for a class of semilinear totally characteristic elliptic equations with critical cone Sobolev exponents. Ann Global Anal Geom, 2011, 39:27-43
[7] Chen H, Liu X, Wei Y. Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on manifold with concial singularities. Calc Var Partial Differ Equ, 2012, 43:463-484
[8] Chen H, Liu X, Wei Y. Multiple solutions for semilinear totally characteristic elliptic equations with subcritical or critical cone Sobolev exponents. J Differential Equations, 2012, 252(7):4200-4228
[9] Chen H, Wei Y, Zhou B. Existence of solutions for degenerate elliptic equations with singular potential on concial singular manifolds. Math Nachr, 2012, 285:1370-1384
[10] Chen H, Liu X, Wei Y. Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term. J Differ Equ, 2012, 252:4289-4314
[11] Chen H, Liu X C, Wei Y W. Multiple solutions for semi-linear corner degenerate elliptic equations. J Funct Anal, 2014, 266:3815-3839
[12] Chen H, Tian S Y, Wei Y W. Multiple sign changing solutions for semi-linear corner degenerate elliptic equations with singular potential. J Funct Anal, 2016, 270:1602-1621
[13] Chen H, Tian S Y, Wei Y W. Multiple solutions for semi-linear corner degenerate elliptic equations with singular potential term. Communications in Contemporary Mathematics, DOI:10.1142/S0219199716500437
[14] Egorov Ju V, Schulze B-W. Pseudo-Differential Operators, Singularities, Appliciations. Oper Theory Adv Appl, Vol 93. Basel:Birkhäuser Verlag, 1997
[15] Komornik V. Exact Controllability and Stabilization. The Multiplier Method. Paris:Mason-John Wiley, 1994
[16] Liu Y, Zhao J. On potential wells and applications to semiliear hyperbolic and parabolic equations. Nonliear Anal, 2006, 64:2665-2687
[17] Mazzeo R. Elliptic theory of differential edge operators, I. Comm Partial Differ Equ, 1991, 16(10):1616-1664
[18] Melrose R B, Mendoza G A. Elliptic operators of totally characteristic type, preprint. Math Sci Res Institute, 1983, MSRI047-83
[19] Melrose R B, Piazza P. Analytic K-theory on manifolds with corners. Adv Math, 1992, 92(1):1-26
[20] Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Isr J Math, 1975, 22:273-303
[21] Sattinger D H. On global solution of nonlinear hyperbolic equations. Arch Ration Mech Anal, 1968, 30:148-172
[22] Schrohe E, Seiler J. Ellipticity and invertibility in the cone algebra on Lp-Sobolev spaces. Integral Equations Operator Theory, 2001, 41:93-114
[23] Schulze B-W. Boundary Value Problems and Singular Pseudo-Differential Operators. Chichester:John Wiley, 1998
[24] Schulze B-W. Mellin representations of pseudo-differential operators on manifolds with corners. Ann Global Anal Geom, 1990, 8(3):261-297
[25] Willem M. Minimax Theorems. Birkhäuser, 1996
