Global existence and blow up of solutions for a class of coupled parabolic systems with logarithmic nonlinearity

Qigang Deng; Fugeng Zeng; Dongxiu Wang; Qigang Deng; Fugeng Zeng; Dongxiu Wang

doi:10.3934/mbe.2022398

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 8: 8580-8600. doi: 10.3934/mbe.2022398

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Global existence and blow up of solutions for a class of coupled parabolic systems with logarithmic nonlinearity

School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, China

Academic Editor: Runzhang Xu

Received: 21 April 2022 Revised: 27 May 2022 Accepted: 06 June 2022 Published: 13 June 2022

According to the difference of the initial energy, we consider three cases about the global existence and blow-up of the solutions for a class of coupled parabolic systems with logarithmic nonlinearity. The three cases are the low initial energy, critical initial energy and high initial energy, respectively. For the low initial energy and critical initial energy $ J(u_0, v_0)\leq d $, we prove the existence of global solutions with $ I(u_0, v_0)\geq 0 $ and blow up of solutions at finite time $ T < +\infty $ with $ I(u_0, v_0) < 0 $, where $ I $ is Nehari functional. On the other hand, we give sufficient conditions for global existence and blow up of solutions in the case of high initial energy $ J(u_0, v_0) > d $.
- coupled parabolic systems,
- global existence,
- blow up,
- logarithmic nonlinearity
Citation: Qigang Deng, Fugeng Zeng, Dongxiu Wang. Global existence and blow up of solutions for a class of coupled parabolic systems with logarithmic nonlinearity[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 8580-8600. doi: 10.3934/mbe.2022398

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Abstract

According to the difference of the initial energy, we consider three cases about the global existence and blow-up of the solutions for a class of coupled parabolic systems with logarithmic nonlinearity. The three cases are the low initial energy, critical initial energy and high initial energy, respectively. For the low initial energy and critical initial energy $ J(u_0, v_0)\leq d $, we prove the existence of global solutions with $ I(u_0, v_0)\geq 0 $ and blow up of solutions at finite time $ T < +\infty $ with $ I(u_0, v_0) < 0 $, where $ I $ is Nehari functional. On the other hand, we give sufficient conditions for global existence and blow up of solutions in the case of high initial energy $ J(u_0, v_0) > d $.

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