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Blow-up results for some reaction-diffusion equations with time delay

Hongliang Wang, Yujuan Chen, Haihua Lu (2012)

Annales Polonici Mathematici

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.

Blowup solutions to Keller-Segel system and its simplified systems

Takasi Senba (2006)

Banach Center Publications

In this paper, we will consider blowup solutions to the so called Keller-Segel system and its simplified form. The Keller-Segel system was introduced to describe how cellular slime molds aggregate, owing to the motion of the cells toward a higher concentration of a chemical substance produced by themselves. We will describe a common conjecture in connection with blowup solutions to the Keller-Segel system, and some results for solutions to simplified versions of the Keller-Segel system, giving the...

Blow-up versus global existence of solutions to aggregation equations

Grzegorz Karch, Kanako Suzuki (2011)

Applicationes Mathematicae

A class of nonlinear viscous transport equations describing aggregation phenomena in biology is considered. General conditions on an interaction potential are obtained which lead either to the existence or to the nonexistence of global-in-time solutions.

Boundary blow-up solutions for a cooperative system involving the p-Laplacian

Li Chen, Yujuan Chen, Dang Luo (2013)

Annales Polonici Mathematici

We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system Δ p u = g ( u - α v ) , Δ p v = f ( v - β u ) in a smooth bounded domain of ℝ N , where Δ p is the p-Laplacian operator defined by Δ p u = d i v ( | ∇ u | p - 2 ∇ 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.

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