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A nonlinear nonlocal wave-equation arising in combustion theory

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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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
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
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