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The problem of uniqueness of the limit in a semilinear heat equation. (English) Zbl 0940.35107

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)

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
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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
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