article

Title: Existence and non-existence of global solutions for a higher-order semilinear parabolic system

Authors: Peter Pang, Fuqin Sun and Mingxin Wang

Issue: Volume 55 (2006), Issue 3, 1113-1134

Abstract:

$In this paper, we study the higher-order semilinear parabolic system \[ \begin{cases} u_t + (-\triangle)^mu = |v|^p,\quad (t,x) \in \mathbb{R}^1_{+} \times \mathbb{R}^N,\\ v_t + (-\triangle)^mv = |u|^q, \quad(t,x) \in \mathbb{R}^1_{+} \times \mathbb{R}^N,\\ u(0,x) = u_0(x),\ v(0,x) = v_0(x), \quad x \in \mathbb{R}^N, \end{cases} \] where $m$ greater than $1$, $p$, $q \geq 1$ and $pq$ greater than $1$. We prove that if $\frac{N}{2m}$ greater than $\max\left\{\frac{1+p}{pq-1}, \frac{1+q}{pq-1}\right\}$, then solutions with small initial data exist globally in time. If the exponents $p$, $q$ meet some additional conditions, we can derive decay estimates $\|u(t)\|_{\infty} \leq C(1 + t)^{-sigma'}$, $\|v(t)\|_{\infty} \leq C(1+t)^{-\sigma''}$, where $\sigma'$ and $\sigma''$ are positive constants. On the other hand, if $N/(2m) \leq \min \{(1 + p)/(pq - 1), (1 + q)/(pq - 1)\}$, then every solution with initial data having positive average value does not exist globally in time.$