The paper deals with positive solutions of a nonlocal and degenerate quasilinear parabolic system not in divergence form $$u_t=v^p\left(\Delta u+a{\int }_{\Omega }u\mathrm{d}x\right),v_t=u^q\left(\Delta v+b{\int }_{\Omega }v\mathrm{d}x\right)$$ 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...

We establish the Strichartz estimates for the linear fractional beam equations in Besov spaces. Using these estimates, we obtain global well-posedness for the subcritical and critical defocusing fractional beam equations. Of course, we need to assume small initial data for the critical case. In addition, by the convexity method, we show that blow up occurs for the focusing fractional beam equations with negative energy.