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

Joanna Rencławowicz¹

¹

Colloquium Mathematicum, Tome 86 (2000) no. 1, pp. 43-66.

Résumé

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 $p_k^i$.

Détail

DOI : 10.4064/cm-86-1-43-66

Keywords: invariant manifold, reaction-diffusion system, invariant region, global existence, blow up