a b s t r a c tThis paper is concerned with the blow-up of solutions to the following semilinear parabolic equation:under homogeneous Neumann boundary condition in a bounded domain Ω ⊂ R n , n ≥ 1, with smooth boundary.For all p > 1, we prove that the classical solutions to the above equation blow up in finite time when the initial energy is positive and initial data is suitably large. This result improves a recent result by Gao and Han (2011) which asserts the blow-up of classical solutions for n ≥ 3 provided that 1 < p ≤ n+2 n−2 .

This paper deals with the following chemotaxis system: ut=∇·(δ∇u−χu∇v)+f(u),x∈normalΩ,1emt>0,vt=normalΔv−uv,x∈normalΩ,1emt>0, under homogeneous Neumann boundary conditions in a bounded domain normalΩ⊂double-struckRn,n⩾1, with smooth boundary. Here, δ and χ are some positive constants and f is a smooth function that satisfies f(0)⩾0andf(s)⩽as−bsγ,s>0 with some constants a⩾0,b > 0, and γ > 1. We prove that the classical solutions to the preceding system are global and bounded provided that ∥v0∥L∞(normalΩ)<1χδ2(n+1)1.61emπ+2arctan1.61em(1−δ)22(n+1)δ1.61em1.61em,if1em0<δ<1,πχ2(n+1),if1emδ=1,1χδ2(n+1)1.61emπ−2arctan1.61em(δ−1)22(n+1)δ1.61em1.61em,if1emδ>1. Copyright © 2016 John Wiley & Sons, Ltd.

Partial differential equations Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant Les solutions classiques d'un modèle de chimiotaxie avec consommation de chimioattracteurs sont bornées

Communicated by M. GrovesThis paper deals with the blow-up phenomena for a class of fourth-order nonlinear wave equations with a viscous damping term

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