Slow decay of solutions in a semilinear dissipative parabolic equation
The author of this interesting paper considers a Cauchy problem \(u_t=u_{xx}-|u|^{p-1}u\) in \(\mathbb{R}\times(0,\infty)\) and \(u(x,0)= \varphi (x)\) in \(\mathbb{R}\), where \(p>1\) and \(\varphi\in L^\infty(\mathbb{R})\). The solution \(u(x,t)\) of the problem under consideration is said to decay fast as \(t\to \infty\) if \(\lim t^{1/(p-1)} u(x,t)=0\) uniformly in \(\mathbb{R}\). Otherwise the solution \(u(x,t)\) is said to decay slowly as \(t\to\infty\). It is shown with perfect precision that in case \(\varphi(x)\) does not decays faster than \(|x|^{-q} (q<2/(p-1))\) as \(x\to\infty\) or \(x\to-\infty\), then \(u\) decays slowly as \(t\to\infty\). Also an estimate of solutions near the spatial infinity for general initial data can be seen, which implies that none of zeros of \(u(x,t)\) goes to \(\pm\infty\) at each \(t>0\).
scientific article; zbMATH DE number 1384460