Blow-up in a slow diffusive \(p\)-Laplace equation with the Neumann boundary conditions. (English) Zbl 1470.35221

Qu, Chengyuan; Liang, Bo

Abstr. Appl. Anal. 2013, Article ID 643819, 5 p. (2013).

Summary: We study a slow diffusive \(p\)-Laplace equation in a bounded domain with the Neumann boundary conditions. A natural energy is associated to the equation. It is shown that the solution blows up in finite time with the nonpositive initial energy, based on an energy technique. Furthermore, under some assumptions of initial data, we prove that the solutions with bounded initial energy also blow up.

MSC:

35K92	Quasilinear parabolic equations with \(p\)-Laplacian
35B44	Blow-up in context of PDEs
35K20	Initial-boundary value problems for second-order parabolic equations
35K55	Nonlinear parabolic equations

References:

[1]	Budd, C.; Dold, B.; Stuart, A., Blowup in a partial differential equation with conserved first integral, SIAM Journal on Applied Mathematics, 53, 3, 718-742 (1993) · Zbl 0784.35009
[2]	Hu, B.; Yin, H.-M., Semilinear parabolic equations with prescribed energy, Rendiconti del Circolo Matematico di Palermo. Serie II, 44, 3, 479-505 (1995) · Zbl 0856.35063
[3]	Alikakos, N. D.; Evans, L. C., Continuity of the gradient for weak solutions of a degenerate parabolic equation, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 62, 3, 253-268 (1983) · Zbl 0529.35039
[4]	Jazar, M.; Kiwan, R., Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 25, 2, 215-218 (2008) · Zbl 1148.35040
[5]	El Soufi, A.; Jazar, M.; Monneau, R., A gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire, 24, 1, 17-39 (2007) · Zbl 1112.35108
[6]	Qu, C. Y.; Bai, X. L.; Zheng, S. N., Blow-up and extinction in a nonlocal \(p\)-Laplace equation with Neumann boundary conditions, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 25, 2, 215-218 (2008) · Zbl 1148.35040
[7]	Fujita, H., On the blowing up of solutions of the Cauchy problem for \(u_t = \Delta u + u^{1 + \alpha}\), Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 13, 109-124 (1966) · Zbl 0163.34002
[8]	Deng, K.; Levine, H. A., The role of critical exponents in blow-up theorems: the sequel, Journal of Mathematical Analysis and Applications, 243, 1, 85-126 (2000) · Zbl 0942.35025
[9]	Li, Y. X.; Xie, C. H., Blow-up for \(p\)-Laplace parabolic equations, Electronic Journal of Differential Equations, 2003, 1-12 (2005)
[10]	Zhao, J. N., Existence and nonexistence of solutions for \(u_t = \operatorname{ div }(| \nabla u |^{p - 2} \nabla u) + f(\nabla u, u, x, t)\), Journal of Mathematical Analysis and Applications, 172, 1, 130-146 (1993) · Zbl 0799.35130
[11]	Zheng, S.; Song, X.; Jiang, Z., Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, Journal of Mathematical Analysis and Applications, 298, 1, 308-324 (2004) · Zbl 1078.35046
[12]	Gao, W.; Han, Y., Blow-up of a nonlocal semilinear parabolic equation with positive initial energy, Applied Mathematics Letters, 24, 5, 784-788 (2011) · Zbl 1213.35131
[13]	Vitillaro, E., Global nonexistence theorems for a class of evolution equations with dissipation, Archive for Rational Mechanics and Analysis, 149, 2, 155-182 (1999) · Zbl 0934.35101