The Palais-Smale condition for the energy of some semilinear parabolic equations
The author of this interesting paper considers an initial and boundary value problem for the nonlinear parabolic equation \(u_t(t,x)-\triangle u(t,x)=|u(t,x)|^{p-1}u(t,x)\), \((t,x)\in (0,T)\times\Omega \), \(u(0,x)=u_0(x)\), \(x\in \Omega \), \(u|_{\partial \Omega }=0\), \(t\in (0,T)\), where \(1<p\leq (N+2)/(N-2)\) and \(\Omega \subset \mathbb{R}^N\) \((N\geq 3)\) is a bounded domain with smooth boundary \(\partial \Omega \). Of special interest here are the solutions which neither blowup nor decay. The main result is that all the global solutions for some semilinear parabolic equations naturally contain a Palais-Smale sequence as a subsequence and then one can apply a global compactness result due to M. Struwe. The finite-time blowup problems are discussed as well.
scientific article; zbMATH DE number 1484787