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

Huafei Di; Yadong Shang; Jiali Yu; Huafei Di; Yadong Shang; Jiali Yu

doi:10.3934/math.2020220

AIMS Mathematics

2020, Volume 5, Issue 4: 3408-3422. doi: 10.3934/math.2020220

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Research article

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

Received: 13 December 2019 Accepted: 08 March 2020 Published: 02 April 2020
MSC : 35B44, 35D40, 35K61, 35K70

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

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Abstract

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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