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Blowup of Volterra Integro-Differential Equations and Applications to Semi-Linear Volterra Diffusion Equations

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  12 September 2017

Zhanwen Yang ,
Tao Tang  and
Jiwei Zhang
Zhanwen Yang*
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Tao Tang*
Affiliation:
Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China
Jiwei Zhang*
Affiliation:
Beijing Computational Science Research Center, Zhongguancun Software Park II, Haidian District, Beijing 100094, China
*
*Corresponding author. Email addresses:yangzhan_wen@126.com (Z. Yang), tangt@sustc.edu.cn (T. Tang), jwzhang@csrc.ac.cn (J. Zhang)
*Corresponding author. Email addresses:yangzhan_wen@126.com (Z. Yang), tangt@sustc.edu.cn (T. Tang), jwzhang@csrc.ac.cn (J. Zhang)
*Corresponding author. Email addresses:yangzhan_wen@126.com (Z. Yang), tangt@sustc.edu.cn (T. Tang), jwzhang@csrc.ac.cn (J. Zhang)
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Abstract

In this paper, we discuss the blowup of Volterra integro-differential equations (VIDEs) with a dissipative linear term. To overcome the fluctuation of solutions, we establish a Razumikhin-type theorem to verify the unboundedness of solutions. We also introduce leaving-times and arriving-times for the estimation of the spending-times of solutions to ∞. Based on these two typical techniques, the blowup and global existence of solutions to VIDEs with local and global integrable kernels are presented. As applications, the critical exponents of semi-linear Volterra diffusion equations (SLVDEs) on bounded domains with constant kernel are generalized to SLVDEs on bounded domains and ℝN with some local integrable kernels. Moreover, the critical exponents of SLVDEs on both bounded domains and the unbounded domain ℝN are investigated for global integrable kernels.

Keywords

Volterra integro-differential equationsvolterra diffusion equationsblowupglobal existencerazumikhin theorem

MSC classification

Secondary: 35K55: Nonlinear parabolic equations 45K05: Integro-partial differential equations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 737 - 759
Copyright
Copyright © Global-Science Press 2017 

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