In this paper, we study the initial boundary value problem of the pseudo-parabolic p(x)-Laplacian equation with logarithmic nonlinearity. The existence of the global solution is obtained by using the potential well method and the logarithmic inequality. In addition, the sufficient conditions of the blow-up are obtained by concavity method.

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<abstract><p>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 &lt; +\infty $ with $ I(u_0, v_0) &lt; 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) &gt; d $.</p></abstract>

In this paper, we study the initial-boundary value problem of the singular non-Newton filtration equation with logarithmic nonlinearity. By using the concavity method, we obtain the existence of finite time blow-up solutions at initial energy J (u 0 ) d . Furthermore, we discuss the asymptotic behavior of the weak solution and prove that the weak solution converges to the corresponding stationary solution as t → +∞. Finally, we give sufficient conditions for global existence and blow-up of solutions at initial energy J (u 0 ) > d .

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