Cortázar, Carmen; del Pino, Manuel; Elgueta, Manuel The problem of uniqueness of the limit in a semilinear heat equation. (English) Zbl 0940.35107 Commun. Partial Differ. Equations 24, No. 11-12, 2147-2172 (1999). Let \(N\in{\mathbb N}\), \(p\in (1,{{N+2}\over{N-2}})\) for \(N\geq 3\) and \(u\in C([0,\infty),H^1({\mathbb R}^N))\) be a (global) solution of \[ u_t=\Delta u-u+u^p \] subject to the initial condition \(u(0,\cdot)\in H^1({\mathbb R}^N)\cap C({\mathbb R})\). The authors show that \(u(t,\cdot)\) converges to some stationary solution in \(H^1({\mathbb R}^N)\) as \(t\to\infty\). Moreover, they show for each compactly supported nonnegative nontrivial initial condition \(\psi\) that there exists a \(\lambda_0>0\) such that the solution starting at \(\lambda \psi\) converges to 0, if \(\lambda<\lambda_0\), to a nontrivial stationary solution, if \(\lambda=\lambda_0\), and blows up in finite time otherwise. Reviewer: G.Hetzer (Auburn) Cited in 1 ReviewCited in 23 Documents MSC: 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations Keywords:compactly supported nonnegative nontrivial initial condition PDFBibTeX XMLCite \textit{C. Cortázar} et al., Commun. Partial Differ. Equations 24, No. 11--12, 2147--2172 (1999; Zbl 0940.35107) Full Text: DOI References: [1] DOI: 10.1090/S0002-9947-1983-0712265-1 · doi:10.1090/S0002-9947-1983-0712265-1 [2] DOI: 10.1080/03605308408820353 · Zbl 0555.35067 · doi:10.1080/03605308408820353 [3] DOI: 10.1080/03605309108820811 · Zbl 0753.35034 · doi:10.1080/03605309108820811 [4] Gidas B., Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies 7 pp 369– (1981) [5] DOI: 10.1512/iumj.1987.36.36001 · Zbl 0601.35052 · doi:10.1512/iumj.1987.36.36001 [6] DOI: 10.1007/s002220050285 · Zbl 0958.53032 · doi:10.1007/s002220050285 [7] DOI: 10.1007/BF00251502 · Zbl 0676.35032 · doi:10.1007/BF00251502 [8] DOI: 10.1215/S0012-7094-93-07004-4 · Zbl 0796.35056 · doi:10.1215/S0012-7094-93-07004-4 [9] DOI: 10.2307/2006981 · Zbl 0549.35071 · doi:10.2307/2006981 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.