On the equation <i>u</i>_t = ∆<i>u</i>^α + <i>u</i>^β

Abstract

Synopsis The Cauchy problem of u t , = ∆ u α + uβ , where 0 &lt; α &lt; l and α&gt;1, is studied. It is proved that if 1&lt; β&lt;α + 2/n then every nontrivial non-negative solution is not global in time. But if β&gt;α+ 2/n there exist both blow-up solutions and global positive solutions which decay to zero as t –1/(β–1) when t →∞. Thus the famous Fujita result on u t = ∆u + u p is generalised to the present fast diffusion equation. Furthermore, regarding the equation as an infinite dimensional dynamical system on Sobolev space W 1,s (W 2.s ) with S &gt; 1, a non-uniqueness result is established which shows that there exists a positive solution u(x, t) with u(., t) → 0 in W 1.s ( W 2.s ) as t → 0.