Communications of the
Korean Mathematical Society CKMS

In this paper, we are concerned with the blow-up phenomena for a class of pseudo-parabolic equations with weighted source $u_{t}-\triangle u-\triangle u_{t}=a(x)f(u)$ subject to Dirichlet (or Neumann) boundary conditions in any smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$. Firstly, we obtain the upper and lower bounds for blow-up time of solutions to these problems. Moreover, we also give the estimates of blow-up rate of solutions under some suitable conditions. Finally, three models are presented to illustrate our main results. In some special cases, we can even get some exact values of blow-up time and blow-up rate.

Keywords: Pseudo-parabolic equation, upper and lower bounds, blow-up rate, weighted source

MSC numbers: Primary 35K61, 35K70, 35A23, 35B44

Supported by: This work was financially supported by the NSF of China (11801108, 11701116), the Scientific Program (2016A030310262) of Guangdong Province, and the College Scientific Research Project (YG2020005) of Guangzhou University.