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
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.

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Y. Chan and H.
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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