×

Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions. (English) Zbl 1227.35157

Summary: We consider nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions. We establish respectively the conditions on nonlinearities to guarantee that \(u(\mathbf x,t)\) exists globally or blows up at some finite time. If blow-up occurs, we obtain upper and lower bounds of the blow-up time.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35B44 Blow-up in context of PDEs
35K58 Semilinear parabolic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Straughan, B., Explosive instabilities in mechanics, (1998), Springer Berlin · Zbl 0911.35002
[2] Quittner, R.; Souplet, P., Superlinear parabolic problems, () · Zbl 1128.35003
[3] Bandle, C.; Brunner, H., Blow-up in diffusion equations. A survey, J. comput. appl. math., 97, 3-22, (1998) · Zbl 0932.65098
[4] Weissler, F.B., Local existence and nonexistence for semilinear parabolic equations in \(L^p\), Indiana univ. math. J., 29, 79-102, (1980) · Zbl 0443.35034
[5] Weissler, F.B., Existence and nonexistence of global solutions for a heat equation, Israel J. math., 38, 1-2, 29-40, (1981) · Zbl 0476.35043
[6] Payne, L.E.; Schaefer, P.W., Lower bound for blow-up time in parabolic problems under Neumann conditions, Appl. anal., 85, 1301-1311, (2006) · Zbl 1110.35032
[7] Payne, L.E.; Schaefer, P.W., Lower bounds for blow-up time in parabolic problems under Dirichlet conditions, J. math. anal. appl., 28, 1196-1205, (2007) · Zbl 1110.35031
[8] Payne, L.E.; Song, J.C., Lower bounds for the blow-up time in a temperature dependent Navier-Stokes flow, J. math. anal. appl., 335, 371-376, (2007) · Zbl 1124.35057
[9] Payne, L.E.; Philippin, G.A.; Schaefer, P.W., Bounds for blow-up time in nonlinear parabolic problems, J. math. anal. appl., 338, 438-447, (2008) · Zbl 1139.35055
[10] Payne, L.E.; Philippin, G.A.; Schaefer, P.W., Blow-up phenomena for some nonlinear parabolic problems, Nonlinear anal., 69, 3495-3502, (2008) · Zbl 1159.35382
[11] Payne, L.E.; Song, J.C., Lower bounds for blow-up time in a nonlinear parabolic problem, J. math. anal. appl., 354, 394-396, (2009) · Zbl 1177.35043
[12] Fujishima, Y.; Ishige, K., Blow-up set for a semilinear heat equation with small diffusion, J. differential equations, 249, 1056-1077, (2010) · Zbl 1204.35054
[13] Payne, L.E.; Schaefer, P.W., Bounds for the blow-up time for heat equation under nonlinear boundary condition, Proc. roy. soc. Edinburgh sect. A, 139, 1289-1296, (2009) · Zbl 1184.35077
[14] Ding, Juntang; Guo, Bao-Zhu, Blow-up and global existence for nonlinear parabolic equations with Neumann boundary conditions, Comput. math. appl., 60, 670-679, (2010) · Zbl 1201.35056
[15] Mizoguchi, N., Blow-up rate of solutions for a semilinear heat equation with Neumann boundary condition, J. differential equations, 193, 212-238, (2003) · Zbl 1029.35128
[16] Ishige, K.; Yagisita, H., Blow-up problems for a semilinear heat equations with large diffusion, J. differential equations, 212, 114-128, (2005) · Zbl 1072.35096
[17] Payne, L.E.; Philippin, G.A.; Vernier Piro, S., Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I, Z. angew. math. phys., 61, 999-1007, (2010) · Zbl 1227.35173
[18] Payne, L.E.; Philippin, G.A.; Vernier Piro, S., Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, II, Nonlinear anal., 73, 971-978, (2010) · Zbl 1198.35131
[19] Levine, H.A.; Smith, R.A., A potential well theory for the heat equation with a nonlinear boundary condition, Math. methods appl. sci., 9, 2, 127-136, (1987) · Zbl 0646.35049
[20] Li, Fushan, Backward solutions to Neumann and Dirichlet problems of heat conduction equation, Appl. math. comput., 210, 211-214, (2009) · Zbl 1167.65442
[21] Li, Fushan; Bai, Yuzhen, Uniform rates of decay for nonlinear viscoelastic Marguerre-von karman shallow shell system, J. math. anal. appl., 351, 522-535, (2009) · Zbl 1155.35058
[22] Li, Fushan, Limit behavior of the solution to nonlinear viscoelastic Marguerre-von karman shallow shells system, J. differential equations, 249, 1241-1257, (2010) · Zbl 1425.74299
[23] Li, Fushan; Zhao, Zengqin; Chen, Yanfu, Global existence uniqueness and decay estimates for nonlinear viscoelastic wave equation with boundary dissipation, Nonlinear anal. real world appl., 12, 1770-1784, (2011) · Zbl 1218.35040
[24] Li, Fushan; Zhao, Cuiling, Uniform energy decay rates for nonlinear viscoelastic wave equation with nonlocal boundary damping, Nonlinear anal., 74, 3468-3477, (2011) · Zbl 1218.35039
[25] Gao, Qingyong; Li, Fushan; Wang, Yanguo, Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation, Cent. eur. J. math., 9, 3, 686-698, (2011) · Zbl 1233.35145
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.