Stability and instability for a viscoelastic Kirchhoff heat equation with variable sources

Abstract

This work is devoted to studying the existence and nonexistence of global weak solutions for the following class of diffusion equations of Kirchhoff type with viscoelastic term and variable exponent sources𝑢𝑡−𝑀⁡(‖∇𝑢‖2)⁢Δu+∫𝑡0𝑔⁢(𝑡−𝑠)⁢Δu⁡(𝑠)d𝑠=|𝑢|𝑝⁡(𝑥)−2⁢𝑢.More specifically, we prove some results of the finite time blow-up of weak solutions as well as provide the lower and upper bounds of the blow-up time. In addition, we also establish the sufficient conditions under which the global weak solutions exist and decay generally at infinity. In particular, we show an explicit and optimal decay rate of the potential energy which is driven by the decay rate of relaxation function g. Our results extend and improve some previous papers [Han and Li: Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy. Comput Math App. 2018;75:3283–3297; Han, Gao and Sun, et al.: Upper and lower bounds of blow-up time to a parabolic type Kirchhoff equation with arbitrary initial energy. Comput Math Appl. 2018;76(10):2477–2483; Liu and Chen: Global and blow-up of solutions for a quasilinear parabolic system with viscoelastic and source terms. Math Methods Appl Sci. 2014;37:148–156; Messaoudi: Blow-up of solutions of a semilinear heat equation with a visco-elastic term. Prog Nonlinear Differ Equ Appl. 2005;64:351–356; Truong and Nguyen: On a class of nonlinear heat equations with viscoelastic term. Comput Math Appl. 2016;72(1):216–232; Zheng and Chipot: Asymptotic behavior of solutions to nonlinear parabolic equations with nonlocal terms. Asympt Anal. 2000;45(3):35–40].