Global solvability for the heat equation with boundary flux governed by nonlinear memory
Authors:
Jeffrey R. Anderson, Keng Deng and Zhihua Dong
Journal:
Quart. Appl. Math. 69 (2011), 759-770
MSC (2010):
Primary 35B44, 35K05, 35K20
DOI:
https://doi.org/10.1090/S0033-569X-2011-01238-X
Published electronically:
July 1, 2011
MathSciNet review:
2893999
Full-text PDF Free Access
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Abstract: We introduce the study of global existence and blowup in finite time for the heat equation with flux at the boundary governed by a nonlinear memory term. Via a simple transformation, the model may be written in a form which has been introduced in previous studies of tumor-induced angiogenesis. The present study is also in the spirit of extending work on models of the heat equation with local, nonlocal, and delay nonlinearities present in the boundary flux. Additionally, we provide a brief summary of related studies regarding heat equation models where memory terms are incorporated within reaction or diffusion.
References
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- P. Souplet, Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys. 55 (2004), 28-31. MR 2033858 (2004k:35199)
- P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal. 29 (1998), 1301-1334. MR 1638054 (99h:35104)
- P. Souplet, Nonexistence of global solutions to some differential inequalities of the second order and applications, Portugal. Math. 52 (1995), 289-299. MR 1355469 (96g:35220)
- P. Vernole, A time dependent parabolic initial boundary value delay problem, J. Integral Equations Appl. 6 (1994), 427-444. MR 1312525 (95m:34147)
- Y. Yamada, Asymptotic stability for some systems of semilinear Volterra diffusion equations, J. Differential Equations 52 (1984), 295-326. MR 744299 (85h:45005)
- Y. Yamada, On a certain class of semilinear Volterra diffusion equations, J. Math. Anal. Appl. 88 (1982), 433-451. MR 667070 (84b:92062)
- J. Yong and X. Zhang, Heat equations with memory, Nonlinear Anal. 63 (2005), e99-e108.
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Additional Information
Jeffrey R. Anderson
Affiliation:
College of Science, Technology, Engineering and Mathematics, University of Wisconsin-Stout, Menomonie, Wisconsin 54751
Email:
andersonjeff@uwstout.edu
Keng Deng
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504
MR Author ID:
225222
Email:
deng@louisiana.edu
Zhihua Dong
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504
Email:
zxd5200@louisiana.edu
Keywords:
Global existence,
finite time blowup,
memory boundary condition
Received by editor(s):
April 28, 2010
Published electronically:
July 1, 2011
Article copyright:
© Copyright 2011
Brown University
The copyright for this article reverts to public domain 28 years after publication.