Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

On finite-time blow-up for a nonlocal parabolic problem arising from shear bands in metals

Author(s): Gao-Feng Zheng
Journal: Proc. Amer. Math. Soc. 135 (2007), 1487-1494.
MSC (2000): Primary 35K10, 35K57, 35K60.
Posted: November 27, 2006
MathSciNet review: 2276658
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: Results on finite-time blow-up of solutions to the nonlocal parabolic problem

$\begin{displaymath} \left\{ \begin{array}{ll} u_t=\Delta u +\delta \displaystyle... ... u(x,0)=u_0(x)\geqslant 0,& x\in \Omega \end{array}\right. \end{displaymath}$

are established. They extend some known results to higher dimensions.

References:

1.: H. Bellout, A criterion for blow-up of solutions to semilinear heat equations, SIAM J. Math. Anal. 18 (1987), 722-727. MR 0883564 (88m:35074)
2.: J.W. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Applied Mathematical Sciences, Vol 83, Springer-Verlag, New York, 1989. MR 1012946 (91d:35165)
3.: J.W. Bebernes and A.A. Lacey, Global existence and finite-time blow-up for a class of nonlocal parabolic problems, Adv. Differential Equations 2 (1997), 927-953. MR 1606347 (99a:35113)
4.: J.W. Bebernes and P. Talaga, Nonlocal problems modelling shear banding, Comm. Appl. Nonlinear Anal. 3 (1996), 79-103. MR 1379441 (97a:35090)
5.: J.W. Bebernes, C. Li and P. Talaga, Single-point blow-up for nonlocal parabolic problems, Phys. D 134 (1999), 48-60. MR 1711136 (2000e:74043)
6.: H. Brezis, T. Cazenave, Y. Martel, and A. Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations 1 (1996), 73-90. MR 1357955 (96i:35063)
7.: D.G. de Figueiredo, P.L. Lions and R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), 41-63. MR 0664341 (83h:35039)
8.: B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243. MR 0544879 (80h:35043)
9.: D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. MR 1814364 (2001k:35004)
10.: D.D. Joseph and T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/1973), 241-269. MR 0340701 (49:5452)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35K10, 35K57, 35K60.

Retrieve articles in all Journals with MSC (2000): 35K10, 35K57, 35K60.

Additional Information:

Gao-Feng Zheng
Affiliation: Department of Mathematics, Huazhong Normal University, Wuhan, People's Republic of China
Email: gfzheng76@yahoo.com.cn

DOI: 10.1090/S0002-9939-06-08925-8
PII: S 0002-9939(06)08925-8
Keywords: Nonlocal parabolic equations, finite-time blow-up, method of moving planes.
Received by editor(s): October 5, 2005
Received by editor(s) in revised form: December 20, 2005
Posted: November 27, 2006
Communicated by: David S. Tartakoff
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|