The paper deals with positive solutions of a nonlocal and degenerate quasilinear parabolic system not in divergence form $$ u_t = v^p\biggl (\Delta u + a\int _\Omega u \,{\rm d} x\biggr ),\quad v_t =u^q\biggl (\Delta v + b\int _\Omega v \,{\rm d} x\biggr ) $$ with null Dirichlet boundary conditions. By using the standard approximation method, we first give a series of fine a priori estimates for the solution of the corresponding approximate problem. Then using the diagonal method, we get the local existence and the bounds of the solution $(u,v)$ to this problem. Moreover, a necessary and sufficient condition for the non-global existence of the solution is obtained. Under some further conditions on the initial data, we get criteria for the finite time blow-up of the solution.
By deriving a variant of interpolation inequality, we obtain a sharp criterion for global existence and blow-up of solutions to the inhomogeneous nonlinear Schrödinger equation with harmonic potential $$ {\rm i}\varphi _t=-\triangle \varphi +|x|^2\varphi -|x|^b|\varphi |^{p-2}\varphi . $$ We also prove the existence of unstable standing-wave solutions via blow-up under certain conditions on the unbounded inhomogeneity and the power of nonlinearity, as well as the frequency of the wave.
The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed., Sen Ming, Han Yang, Zili Chen, Ls Yong., and Obsahuje bibliografii