TY - JOUR
T1 - Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term
AU - Long , QunFei
JO - Journal of Partial Differential Equations
VL - 3
SP - 222
EP - 234
PY - 2020
DA - 2020/06
SN - 33
DO - http://doi.org/10.4208/jpde.v33.n3.3
UR - https://global-sci.org/intro/article_detail/jpde/17071.html
KW - Pseudo-parabolic equation, Newtonian potential, bounds of lifespan, blow-up, concavity method.
AB - <p style="text-align: justify;">We investigate the initial boundary value problem of the pseudo-parabolic equation $u_{t} - \triangle u_{t} - \triangle u = \phi_{u}u + |u|^{p - 1}u,$ where $\phi_{u}$ is the Newtonian potential, which was studied by Zhu et al. (Appl. Math. Comput., 329 (2018) 38-51), and the global existence and the finite time blow-up of the solutions were studied by the potential well method under the subcritical and critical initial energy levels. We in this note determine the upper and lower bounds for the blow-up time. While estimating the upper bound of blow-up time, we also find a sufficient condition of the solution blowing-up in finite time at arbitrary initial energy level. Moreover, we also refine the upper bounds for the blow-up time under the negative initial energy.</p>