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Blow-up of solutions for the non-Newtonian polytropic filtration equation with a generalized source

Jun Zhou — 2016

Annales Polonici Mathematici

This paper deals with the blow-up properties of the non-Newtonian polytropic filtration equation u t - d i v ( | 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.

Rigidity of Einstein manifolds and generalized quasi-Einstein manifolds

Yi Hua DengLi Ping LuoLi Jun Zhou — 2015

Annales Polonici Mathematici

We discuss the rigidity of Einstein manifolds and generalized quasi-Einstein manifolds. We improve a pinching condition used in a theorem on the rigidity of compact Einstein manifolds. Under an additional condition, we confirm a conjecture on the rigidity of compact Einstein manifolds. In addition, we prove that every closed generalized quasi-Einstein manifold is an Einstein manifold provided μ = -1/(n-2), λ ≤ 0 and β ≤ 0.

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