Blow-up criteria for a parabolic problem due to a concentrated nonlinear source on a semi-infinite interval

Authors:C. Y. Chan and T. Treeyaprasert Journal:
Quart. Appl. Math. 70 (2012), 159-169
MSC (2010):
Primary 35K60, 35K57
Published electronically:
August 30, 2011
MathSciNet review:2920621 Full-text PDF

Abstract: Let , and be positive numbers, , , and . This article studies the first initial-boundary value problem,

where is the Dirac delta function, and and are given functions. We assume that , and its derivatives and are positive for , and is nontrivial, nonnegative and continuous such that , and

in .

It is shown that if blows up, then it blows up in a finite time at the single point only. A criterion for to blow up in a finite time and a criterion for to exist globally are given. It is also shown that there exists a critical position for the nonlinear source to be placed such that no blowup occurs for , and blows up in a finite time for . This also implies that does not blow up in infinite time. The formula for computing is also derived. For illustrations, two examples are given.

2.C.
Y. Chan and H.
Y. Tian, Single-point blow-up for a degenerate parabolic problem
due to a concentrated nonlinear source, Quart. Appl. Math.
61 (2003), no. 2, 363–385. MR 1976376
(2004c:35173)

3.C.
Y. Chan and T.
Treeyaprasert, Blow-up due to a concentrated nonlinear source on a
semi-infinite interval, Dynamic systems and applications. Vol. 5,
Dynamic, Atlanta, GA, 2008, pp. 101–108. MR
2468124

C. Y. Chan and R. Boonklurb, A blow-up criterion for a degenerate parabolic problem due to a concentrated nonlinear source, Quart. Appl. Math. 65 (2007), 781-787. MR 2370360 (2008k:35231)

C. Y. Chan and H. Y. Tian, Single-point blow-up for a degenerate parabolic problem due to a concentrated nonlinear source, Quart. Appl. Math. 61 (2003), 363-385. MR 1976376 (2004c:35173)

C. Y. Chan and T. Treeyaprasert, Blow-up due to a concentrated nonlinear source on a semi-infinite interval, Proceedings of Dynamic Systems and Applications 5 (2008), 101-108. MR 2468124

W. E. Olmstead and C. A. Roberts, Explosion in a diffusive strip due to a concentrated nonlinear source, Methods Appl. Anal. 1 (1994), 435-445. MR 1317023 (95k:35117)

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967, pp. 183-185. MR 0219861 (36:2935)

C. Y. Chan Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
Email:
chan@louisiana.edu

T. Treeyaprasert Affiliation:
Department of Mathematics and Statistics, Thammasat University Rangsit Center, Pathumthani, 12121 Thailand
Email:
tawikan@tu.ac.th