### Impacto

Herrero, Miguel A. and Friedman, Avner
(1990)
*A nonlinear nonlocal wave-equation arising in combustion theory.*
Nonlinear analysis-theory methods & applications, 14
(2).
pp. 93-106.
ISSN 0362-546X

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Official URL: http://www.ima.umn.edu/preprints/Jan88Dec88/462.pdf

## Abstract

The initial value problem for the equation

(∂2 / ∂t2 − ∂2 / ∂x2) ∂T / ∂t = (γ ∂2 / ∂t − ∂2 / ∂x2) eT, γ>1,

is considered. It is proved that under some restrictions on the initial data there is a curve, denoted by t=φγ(x), which is positive, Lipschitz continuous, and satisfies |φ′γ(x)|<1 for all x, such that the above initial value problem admits a unique classical solution for t<φ γ (x). Moreover, the solution blows up on the curve t=φ γ (x), that is, the second derivatives of T are unbounded in {x 0 <x<x 0 +δ, φ γ (x)−δ<t<φ γ (x)} for any x 0 and δ>0. The case of γ=1 is also studied. The solution for γ=1 blows up on t = φ¯¯ (x), and it is proved that under certain conditions the solutions for γ>1 converge to the one for γ=1 as γ→1 and lim inf γ→1 φ γ (x)≥φ¯¯(x).

Item Type: | Article |
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Uncontrolled Keywords: | Nonlocal wave equations; shock, blow-up of solutions; combustion; Cauchy problem; combustible gas; ignition |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 17295 |

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Deposited On: | 03 Dec 2012 10:28 |

Last Modified: | 07 Feb 2014 09:45 |

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