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.
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $uₜ - Δ(a_{11}u) = h(t,x)|v|^{p}$, $vₜ -Δ(a_{21}u) - Δ(a_{22}v) = k(t,x)|w|^{q}$, $wₜ - Δ(a_{31}u) - Δ(a_{32}v) - Δ(a_{33}w) = l(t,x)|u|^{r}$, for $x ∈ ℝ^{N}$, t > 0, p > 0, q > 0, r > 0, $a_{ij} = a_{ij}(t,x,u,v)$, under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for $x ∈ ℝ^{N}$, where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the above system, generalizing results of H. Fujita for the scalar Cauchy problem, of M. Escobedo and M. A. Herrero, of Fila, Levine and Uda, and of J. Rencławowicz for systems.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.