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In this paper, we investigate the positive solution of nonlinear degenerate equation u t = f (u)( u + a(x) Ω u dx) with Dirichlet boundary condition. The blow-up criteria is obtained. Furthermore, we prove that under certain conditions, the solutions have global blow-up. When f (u) = u p , 0 < p 1, we gained blow-up rate estimate.

We establish the existence, uniqueness and blow-up rate near the boundary of boundary blow-up solutions to p-Laplacian elliptic equations of logistic typeis non-increasing and f (u) is a function whose variation at infinity may be regular or rapid. In particular, our result regarding the blow-up rate reveals the main difference between regular variation function f and rapid variation function f .

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