AIMS Mathematics, 2020, 5(4): 3408-3422. doi: 10.3934/math.2020220.

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Blow-up analysis of a nonlinear pseudo-parabolic equation with memory term

1 School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, 510006, P. R. China
2 School of Science, Dalian Jiaotong University, Dalian, Liaoning, 116028, P. R. China

This paper deals with the blow-up phenomena for a nonlinear pseudo-parabolic equation with a memory term $u_{t}-\triangle{u}-\triangle{u}_{t}+\int_{0}^{t}g(t-\tau)\triangle{u}(\tau)d\tau=|{u}|^{p}{u}$ in a bounded domain, with the initial and Dirichlet boundary conditions. We first obtain the finite time blow-up results for the solutions with initial data at non-positive energy level as well as arbitrary positive energy level, and give some upper bounds for the blow-up time $T^{*}$ depending on the sign and size of initial energy $E(0)$. In addition, a lower bound for the life span $T^{*}$ is derived by means of a differential inequality technique if blow-up does occur.
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Keywords pseudo-parabolic equation; memory term; blow up; upper bound; lower bound

Citation: Huafei Di, Yadong Shang, Jiali Yu. Blow-up analysis of a nonlinear pseudo-parabolic equation with memory term. AIMS Mathematics, 2020, 5(4): 3408-3422. doi: 10.3934/math.2020220

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