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Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations

100%

Rencławowicz J.

Applicationes Mathematicae

1998-1999

tom 25

nr 3

313-326

We examine the parabolic system of three equations $u_t$ - Δu = $v^p$, $v_t$ - Δv = $w^q$, $w_t$ - Δw = $u^r$, x ∈ $ℝ^N$, t > 0 with p, q, r positive numbers, N ≥ 1, and nonnegative, bounded continuous initial values. We obtain global existence and blow up unconditionally (that is, for any initial data). We prove that if pqr ≤ 1 then any solution is global; when pqr > 1 and max(α,β,γ) ≥ N/2 (α, β, γ are defined in terms of p, q, r) then every nontrivial solution exhibits a finite blow up time.

Blow up, global existence and growth rate estimates in nonlinear parabolic systems

88%

Rencławowicz J.

Colloquium Mathematicum

2000

tom 86

nr 1

43-66

We prove Fujita-type global existence and nonexistence theorems for a system of m equations (m > 1) with different diffusion coefficients, i.e. $u_{it} - d_{i} Δu_{i} = \prod_{k=1}^m u_{k}^{p_k^i}, i=1,...,m, x ∈ ℝ^{N}, t > 0,$ with nonnegative, bounded, continuous initial values and $p_{k}^{i} ≥ 0$, $i,k = 1,...,m$, $d_i > 0$, $i = 1,...,m$. For solutions which blow up at $t = T <≤ ∞$, we derive the following bounds on the blow up rate: $u_i(x,t) ≤ C(T - t)^{-α_{i}}$ with C > 0 and $α_i$ defined in terms of $p_k^i$.

Ograniczanie wyników

1 Applicationes Mathematicae

1 Colloquium Mathematicum

2 Rencławowicz J.

1 2000

1 1998

Wyniki wyszukiwania

Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations

Blow up, global existence and growth rate estimates in nonlinear parabolic systems