Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term

ABSTRACT

In this paper, we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a nonlinear source term $$\begin{array}{c}\hfill u_t-\mathrm{\Delta }u-\mathrm{\Delta }{u}_t+{\int }_0^{t}g(t-\mathit{\tau })\mathrm{\Delta }u(\mathit{\tau })\mathrm{d}\mathit{\tau }={|u|}^{p-2}u,\text{in}\mathrm{\Omega }\times (0,T),\end{array}$$

where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^n$ ($n\ge 1$) with a Dirichlet boundary condition. Under suitable assumptions on the initial data $u_0$ and the relaxation function g, we obtain the global existence and finite time blow-up of solutions with initial data at low energy level (i.e. $J(u(0))\le d(\infty )$), by using the Galerkin method, the concavity method and an improved potential well method involving time t. We also derive the upper bounds for the blow-up time. Finally, we obtain the existence of solutions which blow up in finite time with initial data at arbitrary energy level.

KEYWORDS:

• Pseudo-parabolic equation
• memory term
• blow-up
• global existence
• potential well method

AMS SUBJECT CLASSIFICATIONS:

• 35K70
• 35A01
• 35B44

Acknowledgements

The authors would like to thank the referee for his/her very important comments that improved the results and the quality of the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors were supported financially by the National Natural Science Foundation of China [grant numer 11371221], [grant numer 11571296].