Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity

Wenjun Liu; Jiangyong Yu; Gang Li

doi:10.3934/dcdss.2021121

Article Contents

2021, Volume 14, Issue 12: 4337-4366. Doi: 10.3934/dcdss.2021121 open access

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Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

^* Corresponding author
^* Corresponding author

Received: June 30, 2021

Revised: August 31, 2021

Early access: October 2021

Published: December 2021

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Abstract

In this paper, we study the fractional pseudo-parabolic equations $ u_{t} + \left(-\Delta\right)^{s} u + \left(-\Delta\right)^{s} u_{t} = u\log \left| u \right| $. Firstly, we recall the relationship between the fractional Laplace operator $ \left(-\Delta\right)^{s} $ and the fractional Sobolev space $ H^{s} $ and discuss the invariant sets and the vacuum isolating behavior of solutions with the help of a family of potential wells. Then, we derive a threshold result of existence of weak solution: for the low initial energy case (i.e., $ J(u_{0}) < d $), the solution is global in time with $ I(u_{0}) > 0 $ or $ \Vert u_{0}\Vert_{{X_{0}(\Omega)}} = 0 $ and blows up at $ +\infty $ with $ I(u_{0}) < 0 $; for the critical initial energy case (i.e., $ J(u_{0}) = d $), the solution is global in time with $ I(u_{0}) \geq0 $ and blows up at $ +\infty $ with $ I(u_{0}) < 0 $. The decay estimate of the energy functional for the global solution is also given.

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Mathematics Subject Classification: 35R11, 35A01, 45K05.

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Figure 1. $ J(\lambda u) $ curve about $ \lambda $

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Figure 2. Space partition diagram with $ \gamma(\delta) $ and $ I_{\delta}(u) $

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Figure 3. $ d(\delta) $ curve about $ \delta $

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Figure 4. Space partition diagram with $ \sqrt{\frac{d(\delta)}{a(\delta)}} $ and $ I_{\delta}(u) $

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Figure 5. Vacuum region $ U_{e} $

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