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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)

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
Full Text: DOI
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