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

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.