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oai:docta.ucm.es:20.500.14352/577452023-08-10T21:58:14Zcom_20.500.14352_14col_20.500.14352_15

Herrero, Miguel A. Friedman, Avner 2023-06-20T17:05:09Z 2023-06-20T17:05:09Z 1990-01 0362-546X 10.1016/0362-546X(90)90017-B https://hdl.handle.net/20.500.14352/57745 http://www.ima.umn.edu/preprints/Jan88Dec88/462.pdf http://www.ima.umn.edu 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). eng restricted access A nonlinear nonlocal wave-equation arising in combustion theory journal article