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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:
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