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

Mokhtar Kirane, and Salim Messaoudi. "Nonexistence results for the Cauchy problem of some systems of hyperbolic equations." Annales Polonici Mathematici 78.1 (2002): 39-47. <http://eudml.org/doc/280536>.

@article{MokhtarKirane2002,
abstract = {We consider the systems of hyperbolic equations ⎧$uₜₜ = Δ(a(t,x)u) + Δ(b(t,x)v) + h(t,x)|v|^\{p\}$, t > 0, $x ∈ ℝ^\{N\}$, (S1) ⎨ ⎩$vₜₜ = Δ(c(t,x)v) + k(t,x)|u|^\{q\}$, t > 0, $x ∈ ℝ^\{N\}$ ⎧$uₜₜ = Δ(a(t,x)u) + h(t,x)|v|^\{p\}$, t > 0, $x ∈ ℝ^\{N\}$, (S2) ⎨ ⎩$vₜₜ = Δ(c(t,x)v) + l(t,x)|v|^\{m\} + k(t,x)|u|^\{q\}$, t > 0, $x ∈ ℝ^\{N\}$, (S3) ⎧$uₜₜ = Δ(a(t,x)u) + Δ(b(t,x)v) + h(t,x)|u|^\{p\}$, t > 0, $x ∈ ℝ^\{N\}$, ⎨ ⎩$vₜₜ = Δ(c(t,x)v) + k(t,x)|v|^\{q\}$, t > 0, $x ∈ ℝ^\{N\}$, in $(0,∞) × ℝ^\{N\}$ 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.},
author = {Mokhtar Kirane, Salim Messaoudi},
journal = {Annales Polonici Mathematici},
keywords = {blow up; weak solution},
language = {eng},
number = {1},
pages = {39-47},
title = {Nonexistence results for the Cauchy problem of some systems of hyperbolic equations},
url = {http://eudml.org/doc/280536},
volume = {78},
year = {2002},
}

TY - JOUR
AU - Mokhtar Kirane
AU - Salim Messaoudi
TI - Nonexistence results for the Cauchy problem of some systems of hyperbolic equations
JO - Annales Polonici Mathematici
PY - 2002
VL - 78
IS - 1
SP - 39
EP - 47
AB - We consider the systems of hyperbolic equations ⎧$uₜₜ = Δ(a(t,x)u) + Δ(b(t,x)v) + h(t,x)|v|^{p}$, t > 0, $x ∈ ℝ^{N}$, (S1) ⎨ ⎩$vₜₜ = Δ(c(t,x)v) + k(t,x)|u|^{q}$, t > 0, $x ∈ ℝ^{N}$ ⎧$uₜₜ = Δ(a(t,x)u) + h(t,x)|v|^{p}$, t > 0, $x ∈ ℝ^{N}$, (S2) ⎨ ⎩$vₜₜ = Δ(c(t,x)v) + l(t,x)|v|^{m} + k(t,x)|u|^{q}$, t > 0, $x ∈ ℝ^{N}$, (S3) ⎧$uₜₜ = Δ(a(t,x)u) + Δ(b(t,x)v) + h(t,x)|u|^{p}$, t > 0, $x ∈ ℝ^{N}$, ⎨ ⎩$vₜₜ = Δ(c(t,x)v) + k(t,x)|v|^{q}$, t > 0, $x ∈ ℝ^{N}$, in $(0,∞) × ℝ^{N}$ 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.
LA - eng
KW - blow up; weak solution
UR - http://eudml.org/doc/280536
ER -