This paper deals with the blow-up properties of the non-Newtonian polytropic filtration equation $u_t - div(|∇u^m|^{p-2} ∇u^m) = f(u)$ with homogeneous Dirichlet boundary conditions. The blow-up conditions, upper and lower bounds of the blow-up time, and the blow-up rate are established by using the energy method and differential inequality techniques.
Let G be a mixed graph. We discuss the relation between the second largest eigenvalue λ₂(G) and the second largest degree d₂(G), and present a sufficient condition for λ₂(G) ≥ d₂(G).
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