Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary

Brändle, Cristina; Quirós, Fernando; Rossi, Julio D.

doi:10.3934/cpaa.2005.4.523

Cited by 35 publications

(26 citation statements)

References 16 publications

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“…This phenomenon is commonly denoted as non-simultaneous blow-up. The possibility of non-simultaneous blow-up in nonlinear parabolic systems was first mentioned in [19], and has been studied more thoroughly later in [3], [15], [16] and [20]. For problem (1.1)-(1.3) this possibility was analyzed in [14], [16]: there exist solutions such that u blows up at time T while v remains bounded up to this time if and only if…”

Section: { U(· T) ∞ + V(· T) ∞ } = ∞mentioning

confidence: 99%

Complete Blow-up and Avalanche Formation for a Parabolic System with Non-Simultaneous Blow-up

Brändle

Quirós

Rossi

2010

Advanced Nonlinear Studies

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We study the possibility of defining a nontrivial continuation after the blow-up time for a system of two heat equations with a nonlinear coupling at the boundary. It turns out that any possible continuation that verify a maximum principle is identically infinity after the blow-up time, that is, both components blow up completely. We also analyze the propagation of the singularity to the whole space, the avalanche, when blow-up is non-simultaneous.

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Section: { U(· T) ∞ + V(· T) ∞ } = ∞mentioning

confidence: 99%

Complete Blow-up and Avalanche Formation for a Parabolic System with Non-Simultaneous Blow-up

Brändle

Quirós

Rossi

2010

Advanced Nonlinear Studies

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“…Inspired by [1], we have a more interesting theorem on the coexistence of simultaneous and non-simultaneous blow-up: Theorem 3.4 (regions C 1 ). Assume m > q + 1 and n > p + 1.…”

Section: Soz Satisfiesmentioning

confidence: 99%

“…See also [8,10,17,19,21,25,29,30,31]. Phenomena of non-simultaneous blow-up for coupled nonlinear parabolic systems were observed and studied by many authors (see, e.g., [1,2,22,23,24,26,27,28,32]). …”

Section: Introductionmentioning

confidence: 99%

Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary fluxes

Liu

Zheng

2007

Z. angew. Math. Phys.

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This paper deals with simultaneous and non-simultaneous blow-up for heat equations coupled via nonlinear boundary fluxes ∂u ∂η = u m + v p , ∂v ∂η = u q + v n . It is proved that, if m < q + 1 and n < p + 1, then blow-up must be simultaneous, and that, for radially symmetric and nondecreasing in time solutions, non-simultaneous blow-up occurs for some initial data if and only if m > q + 1 or n > p + 1. We find three regions: (i) q + 1 < m < p/(p + 1 − n) and n < p + 1, (ii) p + 1 < n < q/(q + 1 − m) and m < q + 1, (iii) m > q + 1 and n > p + 1, where both simultaneous and non-simultaneous blow-up are possible. Four different simultaneous blow-up rates are obtained under different conditions. It is interesting that different initial data may lead to different simultaneous blow-up rates even for the same values of the exponent parameters.

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“…China. 2 Department of Information Engineering, Jiangxi Vocational College of Finance and Economics, Jiujiang, Jiangxi 332000, P.R. China.…”

Section: Competing Interestsmentioning

confidence: 99%

Non-simultaneous blow-up for a parabolic system with nonlinear boundary flux which obey different laws

Zeng

2012

Bound Value Probl

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In this paper, we consider a system of two heat equations with nonlinear boundary flux which obey different laws, one is exponential nonlinearity and another is power nonlinearity. Under certain hypotheses on the initial data, we get the sufficient and necessary conditions, on which there exist initial data such that non-simultaneous blow-up occurs. Moreover, we get some conditions on which simultaneous blow-up must occur. Furthermore, we also get a result on the coexistence of both simultaneous and non-simultaneous blow-ups. MSC: 35B33; 35K65; 35K55

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Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary

Cited by 35 publications

References 16 publications

Complete Blow-up and Avalanche Formation for a Parabolic System with Non-Simultaneous Blow-up

Complete Blow-up and Avalanche Formation for a Parabolic System with Non-Simultaneous Blow-up

Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary fluxes

Non-simultaneous blow-up for a parabolic system with nonlinear boundary flux which obey different laws

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