Global well-posedness, asymptotic behavior and blow-up of solutions for a class of degenerate parabolic equations. (English) Zbl 1445.35222
Summary: The initial-boundary value problem for a class of degenerate parabolic equations is studied. Some new results on global existence of solutions are established by introducing a family of potential wells. In addition, asymptotic behavior and finite time blow-up of solutions are obtained in the case of subcritical initial energy and critical initial energy, respectively.
MSC:
| 35K59 | Quasilinear parabolic equations |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
Keywords:
degenerate parabolic equations; global well-posedness; asymptotic behavior; blow-up; family of potential wellsReferences:
| [1] | Abdulla, U. G.; Jeli, R., Evolution of interfaces for the non-linear parabolic \(p\)-Laplacian type reaction-diffusion equations, European J. Appl. Math., 28, 5, 827-853 (2017) · Zbl 1386.35196 |
| [2] | Astarita, G.; Marrucci, G., Principles of Non-Newtonian Fluid Mechanics (1974), McGraw-Hill: McGraw-Hill London · Zbl 0316.73001 |
| [3] | Aubin, J. P., Un théorème de compacité, C. R. Acad. Sci., Paris, 256, 5042-5044 (1963) · Zbl 0195.13002 |
| [4] | DiBenedetto, E., Degenerate Parabolic Equations (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0794.35090 |
| [5] | Galaktionov, V. A.; Levine, H. A., On critical fujita exponents for heat equations with nonlinear flux conditions on the boundary, Israel J. Math., 94, 1, 125-146 (1996) · Zbl 0851.35067 |
| [6] | Levine, H. A., Instability and nonexistence of global solutions to nonlinear wave equations of the form \(P u_{t t} = - A u + \mathcal{F} ( u )\), Trans. Amer. Math. Soc., 192, 1-21 (1974) · Zbl 0288.35003 |
| [7] | Levine, H. A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5, 138-146 (1974) · Zbl 0243.35069 |
| [8] | Lian, W.; Radulescu, V. D.; Xu, R.; Yang, Y.; Zhao, N., Global well-posedness for a class of fourth order nonlinear strongly damped wave equations, Adv. Calc. Var. (2019) |
| [9] | Lian, W.; Xu, R., Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9, 1, 613-632 (2019) · Zbl 1421.35222 |
| [10] | Lions, J. L., Quelques Méthodes de Résolution Des Problèmes Aux Limites Nonlinéaires (1969), Dunod: Dunod Paris · Zbl 0189.40603 |
| [11] | Liu, Y.; Zhao, J., Nonlinear parabolic equations with critical initial conditions \(J ( u_0 ) = d\) or \(I ( u_0 ) = 0\), Nonlinear Anal., 58, 7-8, 873-883 (2004) · Zbl 1059.35064 |
| [12] | Liu, Y.; Zhao, J., On potential wells and applications to semilinear hyperbolic equations and parabolic equations, Nonlinear Anal., 64, 12, 2665-2687 (2006) · Zbl 1096.35089 |
| [13] | Nakao, M., Periodic solutions of some nonlinear degenerate parabolic equations, J. Math. Anal. Appl., 104, 2, 554-567 (1984) · Zbl 0565.35057 |
| [14] | Papageorgiou, N. S.; Radulescu, V. D., Semilinear Neumann problems with indefinite and unbounded potential and crossing nonlinearity, Contemp. Math., 595, 293-315 (2013) · Zbl 1301.35052 |
| [15] | Papageorgiou, N. S.; Radulescu, V. D., Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations, 256, 7, 2449-2479 (2014) · Zbl 1287.35010 |
| [16] | Papageorgiou, N. S.; Radulescu, V. D., Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim., 69, 3, 393-430 (2014) · Zbl 1304.35077 |
| [17] | Papageorgiou, N. S.; Radulescu, V. D., Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential, Trans. Amer. Math. Soc., 367, 12, 8723-8756 (2015) · Zbl 1341.35028 |
| [18] | Papageorgiou, N. S.; Radulescu, V. D., Robin problems with indefinite, unbounded potential and reaction of arbitrary growth, Rev. Mat. Complut., 29, 1, 91-126 (2016) · Zbl 1338.35137 |
| [19] | Payne, L. E.; Sattinger, D. H., Sadle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22, 3-4, 273-303 (1975) · Zbl 0317.35059 |
| [20] | Tsutsumi, M., Existence and nonexistence of global solutions of nonlinear parabolic equations, Publ. RIMS Kyoto Univ., 8, 211-229 (1972) · Zbl 0248.35074 |
| [21] | Xu, R., Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level, Math. Comput. Simulation, 80, 4, 808-813 (2009) · Zbl 1180.35137 |
| [22] | Xu, R.; Cao, X.; Yu, T., Finite time blow-up and global solutions for a class of semilinear parabolic equations at high energy level, Nonlinear Anal., 13, 1, 197-202 (2012) · Zbl 1238.35047 |
| [23] | Xu, R.; Lian, W.; Niu, Y., Global well-posedness of coupled parabolic systems, Sci. China Math. (2019) |
| [24] | Xu, R.; Su, J., Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264, 12, 2732-2763 (2013) · Zbl 1279.35065 |
| [25] | Xu, R.; Zhang, M.; Chen, S.; Yang, Y.; Shen, J., The initial-boundary value problems for a class of six order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37, 11, 5631-5649 (2019) · Zbl 1368.35168 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
