This paper deals with a two-species chemotaxis-competition system involving singular sensitivity and indirect signal production: $ \begin{equation*} \begin{cases} u_{t} = \nabla\cdot(D(u)\nabla u)-\chi_1\nabla\cdot(\frac{u}{z^{k}}\nabla z)+\mu_1 u(1-u-a_1v), &x\in\Omega,\ t>0,\\ v_{t} = \nabla\cdot(D(v)\nabla v)-\chi_2\nabla\cdot(\frac{v}{z^{k}}\nabla z)+\mu_2 v(1-v-a_2 u), &x\in\Omega,\ t>0,\\ w_{t} = \Delta w-w+u+v,&x\in\Omega,\ t>0,\\ z_{t} = \Delta z-z+w,&x\in\Omega,\ t>0,\\ \end{cases} \end{equation*} $ where $ \Omega\subset R^{n} $ is a convex smooth bounded domain with homogeneous Neumann boundary conditions. The diffusion functions $ D(u), D(v) $ are assumed to fulfill $ D(u)\geq(u+1)^{\theta_1} $ and $ D(v)\geq(v+1)^{\theta_2} $ with $ \theta_1, \theta_2 > 0 $, respectively. The parameters are $ k\in (0, \frac{1}{2})\cup (\frac{1}{2}, 1] $, $ \chi_ {i} > 0, (i = 1, 2) $.
View Article and Find Full Text PDFAccording 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.
View Article and Find Full Text PDFIn this paper, we discuss global existence, boundness, blow-up and extinction properties of solutions for the Dirichlet boundary value problem of the $ p $-Laplacian equations with logarithmic nonlinearity $ u_{t}-{\rm{div}}(|\nabla u|^{p-2}\nabla u)+\beta|u|^{q-2}u = \lambda |u|^{r-2}u\ln{|u|} $, where $ 1 < p < 2 $, $ 1 < q\leq2 $, $ r > 1 $, $ \beta, \lambda > 0 $. Under some appropriate conditions, we obtain the global existence of solutions by means of the Galerkin approximations, then we prove that weak solution is globally bounded and blows up at positive infinity by virtue of potential well theory and the Nehari manifold. Moreover, we obtain the decay estimate and the extinction of solutions.
View Article and Find Full Text PDFIn this paper, we study the initial boundary value problem for a class of fractional p-Laplacian Kirchhoff type diffusion equations with logarithmic nonlinearity. Under suitable assumptions, we obtain the extinction property and accurate decay estimates of solutions by virtue of the logarithmic Sobolev inequality. Moreover, we discuss the blow-up property and global boundedness of solutions.
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