Volume 10, Issue 4
Blowup of Volterra Integro-Differential Equations and Applications to Semi-Linear Volterra Diffusion Equations

Numer. Math. Theor. Meth. Appl., 10 (2017), pp. 737-759.

Published online: 2017-11

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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 $\mathbb{R}^N$ with some local integrable kernels. Moreover, the critical exponents of SLVDEs on both bounded domains and the unbounded domain $\mathbb{R}^N$ are investigated for global integrable kernels.

• Keywords

Volterra integro-differential equations, volterra diffusion equations, blowup, global existence, razumikhin theorem.

35K55, 45K05

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@Article{NMTMA-10-737, author = {}, title = {Blowup of Volterra Integro-Differential Equations and Applications to Semi-Linear Volterra Diffusion Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2017}, volume = {10}, number = {4}, pages = {737--759}, 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 $\mathbb{R}^N$ with some local integrable kernels. Moreover, the critical exponents of SLVDEs on both bounded domains and the unbounded domain $\mathbb{R}^N$ are investigated for global integrable kernels.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2016.0001}, url = {http://global-sci.org/intro/article_detail/nmtma/10454.html} }
TY - JOUR T1 - Blowup of Volterra Integro-Differential Equations and Applications to Semi-Linear Volterra Diffusion Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 737 EP - 759 PY - 2017 DA - 2017/11 SN - 10 DO - http://doi.org/10.4208/nmtma.2016.0001 UR - https://global-sci.org/intro/article_detail/nmtma/10454.html KW - Volterra integro-differential equations, volterra diffusion equations, blowup, global existence, razumikhin theorem. AB -

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 $\mathbb{R}^N$ with some local integrable kernels. Moreover, the critical exponents of SLVDEs on both bounded domains and the unbounded domain $\mathbb{R}^N$ are investigated for global integrable kernels.

Zhanwen Yang, Tao Tang & Jiwei Zhang. (2020). Blowup of Volterra Integro-Differential Equations and Applications to Semi-Linear Volterra Diffusion Equations. Numerical Mathematics: Theory, Methods and Applications. 10 (4). 737-759. doi:10.4208/nmtma.2016.0001
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