RT Journal Article
T1 A nonlinear nonlocal wave-equation arising in combustion theory
A1 Herrero, Miguel A.
A1 Friedman, Avner
AB 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).
PB Pergamon-Elsevier Science
SN 0362-546X
YR 1990
FD 1990-01
LK https://hdl.handle.net/20.500.14352/57745
UL https://hdl.handle.net/20.500.14352/57745
LA eng
NO National Science Foundation
NO CICYT
DS Docta Complutense
RD 7 jul 2025