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

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Kirane M., Messaoudi S.

Annales Polonici Mathematici

2002

tom 78

nr 1

39-47

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Kirane M.

1 Messaoudi S.

1 2002

Wyniki wyszukiwania

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