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Volume 33, Issue 3
Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term

QunFei Long

DOI: 10.4208/jpde.v33.n3.3

J. Part. Diff. Eq., 33 (2020), pp. 222-234.

Published online: 2020-06

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  • Abstract

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.

  • Keywords

Pseudo-parabolic equation, Newtonian potential, bounds of lifespan, blow-up, concavity method.

  • AMS Subject Headings

35K70, 35B44

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

201802003@gznu.edu.cn (QunFei Long)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-33-222, author = {Long , QunFei}, title = {Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term}, journal = {Journal of Partial Differential Equations}, year = {2020}, volume = {33}, number = {3}, pages = {222--234}, abstract = {

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.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v33.n3.3}, url = {http://global-sci.org/intro/article_detail/jpde/17071.html} }
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 -

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.

Long , QunFei. (2020). Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term. Journal of Partial Differential Equations. 33 (3). 222-234. doi:10.4208/jpde.v33.n3.3
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