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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 discuss the effect of time delay on blow-up of solutions to initial-boundary value problems for nonlinear reaction-diffusion equations. Firstly, two examples are given, which indicate that the delay can both induce and prevent the blow-up of solutions. Then we show that adding a new term with delay may not change the blow-up character of solutions.

We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system ${\Delta }_{p}u=g(u-\alpha v),{\Delta }_{p}v=f(v-\beta u)$ in a smooth bounded domain of ${ℝ}^N$, where ${\Delta }_p$ is the p-Laplacian operator defined by ${\Delta }_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ with p > 1, f and g are nondecreasing, nonnegative C¹ functions, and α and β are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for p = 2.