Nonexistence results for the Cauchy problem of some systems of hyperbolic equations

Mokhtar Kirane; Salim Messaoudi

Annales Polonici Mathematici, Tome 79 (2002), p. 39-47 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider the systems of hyperbolic equations ⎧ $u ₜ ₜ = Δ (a (t, x) u) + Δ (b (t, x) v) + {h (t, x) | v |}^{p}$ , t > 0, $x \in ℝ^{N}$ , (S1) ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {k (t, x) | u |}^{q}$ , t > 0, $x \in ℝ^{N}$ ⎧ $u ₜ ₜ = Δ (a (t, x) u) + {h (t, x) | v |}^{p}$ , t > 0, $x \in ℝ^{N}$ , (S2) ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {l (t, x) | v |}^{m} + k (t, x) {| u |}^{q}$ , t > 0, $x \in ℝ^{N}$ , (S3) ⎧ $u ₜ ₜ = Δ (a (t, x) u) + Δ (b (t, x) v) + {h (t, x) | u |}^{p}$ , t > 0, $x \in ℝ^{N}$ , ⎨ ⎩ $v ₜ ₜ = Δ (c (t, x) v) + {k (t, x) | v |}^{q}$ , t > 0, $x \in ℝ^{N}$ , in $(0, \infty) \times ℝ^{N}$ with u(0,x) = u₀(x), v(0,x) = v₀(x), uₜ(0,x) = u₁(x), vₜ(0,x) = v₁(x). We show that, in each case, there exists a bound B on N such that for 1 ≤ N ≤ B solutions to the systems blow up in finite time.

Publié le : 2002-01-01

Zbl 0988.35115

EUDML-ID : urn:eudml:doc:280536

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     author = {Mokhtar Kirane and Salim Messaoudi},
     title = {Nonexistence results for the Cauchy problem of some systems of hyperbolic equations},
     journal = {Annales Polonici Mathematici},
     volume = {79},
     year = {2002},
     pages = {39-47},
     zbl = {0988.35115},
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Mokhtar Kirane; Salim Messaoudi. Nonexistence results for the Cauchy problem of some systems of hyperbolic equations. Annales Polonici Mathematici, Tome 79 (2002) pp. 39-47. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap78-1-5/