Decay of heat semigroups in \(L^\infty\) and applications to nonlinear parabolic problems in unbounded domains
The author describes an interesting characterization of the Poincaré inequality on the unbounded domain \(\Omega\) in terms of the solutions \(u\) to \[ -\Delta u= f,\quad u\in L^q(\Omega),\quad f\in L^q(\Omega),\quad u=0\quad\text{on }\partial\Omega\tag{1} \] on the one hand, and in terms of the global existence of solutions \(u\) to \[ u_t=\Delta u+\lambda g(u),\quad u\in L^\infty(\Omega),\quad u=0\quad\text{on }\partial\Omega,\quad u(0)= 0,\quad\lambda> 0,\tag{2} \] on the other hand; here \(g:[0, b)\to \mathbb{R}_+\) is locally Lipschitz and satisfies \(\lim_sg(s)= \infty\) as \(s\to b\). We list two representative results; (Pi) is short for Poincaré's inequality. Theorem A: For any domain \(\Omega\subseteq \mathbb{R}^n\), (i)--(iii) below are equivalent: (i) for all \(1\leq q\leq\infty\), \(f\in L^q(\Omega)\) there is \(u\in L^q(\Omega)\) with \(-\Delta u= f\) in \({\mathcal D}'(\Omega)\), i.e. in the distributional sense, (ii) there is \(u\in L^\infty(\Omega)\) with \(-\Delta u=1\) in \({\mathcal D}'(\Omega)\), (iii) (Pi) holds on \(\Omega\). Theorem B: For any \(\Omega\subseteq \mathbb{R}^n\), (i), (ii) below are equivalent: (i) for all \(\lambda> 0\) sufficiently small, the solution \(u(t)\) to (2) exists globally, (ii) (Pi) holds on \(\Omega\). There are further interesting results of a similar kind; we refer to the paper for reasons of length. The proofs are based on subtle estimates and on a series of results on the heat semigroup, at disposal in the literature.
scientific article; zbMATH DE number 1484765
scientific article