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 - <p style="text-align: justify;">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.</p>