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The present paper is devoted to the compressible nematic liquid crystal flow in the whole space $ \mathbb{R}^N\,(N≥ 2)$. Here we concentrate on the incompressible limit *in the* $ L^p$ *type critical Besov spaces setting*. We first establish the existence of global solutions in the framework of $ L^p$ type critical spaces provided that the initial data are close to some equilibrium states. Based on the global existence, we then consider the incompressible limit problem in the *ill prepared data case*. We justify the low Mach number convergence to the incompressible flow of liquid crystals in proper function spaces. In addition, the accurate converge rates are obtained.

$u_t=\Delta u^m + au^p\int_\Omega u^q dx,\quad x\in \Omega, t>0$

subject to homogeneous Dirichlet condition. We investigate the influence of the nonlocal source and local term on blow-up properties for this system. It is proved that: (i) when $p\leq 1$, the nonlocal source plays a dominating role, i.e. the system has global blow-up and the blow-up profile which is uniformly away from the boundary either at polynomial scale or logarithmic scale is obtained. (ii) When $p > m$, this system presents single blow-up pattern. In other words, the local term dominates the nonlocal term in the blow-up profile. This extends the work of Li and Xie in Appl. Math. Letter, 16 (2003) 185--192.

$u_t-\Delta_pu=|u|^{q-2}u, \quad (x,t)\in\Omega\times (0,T),$

$u(x,t)=0,\quad (x,t)\in\partial\Omega\times (0,T), $

$ u(x,0)=u_0(x), \quad u_0(x)\geq 0, u_0(x)$ ≠ $0, $

where $\Omega$ is a smooth bounded domain in $R^N(N\geq 3)$, $\Delta_pu=$ div$(|\nabla u|^{p-2}\nabla u )$, $\frac{2N}{N+2}$ < $p$ < $N$, $q=p^\star=\frac{pN}{N-p}$ is the critical Sobolev exponent. In particular, we employ the concentration-compactness principle to prove that the global solutions with the initial data in "stable set" converge strongly to zero in $W_0^{1,p}(\Omega)$.

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