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Convergence to steady states of solutions of non-autonomous heat equations in $$\mathbb R^N$$. (English) Zbl 1166.35005
In this paper, the authors consider the following non-autonomous heat equation: $u_{t}-\Delta u+f(u)=g(t,x)\text{ for }(t,x)\in{\mathbb R}_{+}\times \mathbb{R}^{N},\quad u(0,x)=u_{0}(x)\text{ for }x\in\mathbb{R}^{N}\tag{1}$
with $$N\geq3$$ and $$f(u)=u-|u|^{p-1}u$$ $$(u\in\mathbb{R)}$$ for some $$1<p<\frac{N+2}{N-2},g\in L^{2}(\mathbb{R}_{+};L^{2}\cap L^{q})$$ for some $$q>N$$ (where $$L^{p}:=L^{p}(\mathbb{R}^{N}),$$ supp $$g(t,.)\subset K$$ for all $$t\geq0$$ and some $$K\subset\mathbb{R}^{N}$$ compact, $$g(t,x)\geq0$$ for all $$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{N},$$ and $$u_{0}\in H^{1}(\mathbb{R}^{N})\cap L^{\infty}(\mathbb{R}^{N})$$ has compact support.
The main result of the paper states that if $$u\in C(\mathbb{R}_{+};H^{1})$$ is a positive solution of $$(1)$$ such that $$\sup_{t\in\mathbb{R}_{+}}\|u(t)\|_{H^{1}}<+\infty,$$ and if there exists $$\delta>0$$ such that $$\sup_{t\in\mathbb{R}_{+}}t^{1+\delta}\int_{t}^{\infty} \|g(s)\|_{L^{2}(\mathbb{R}^{N})}^{2}ds<+\infty,$$ then $$\lim_{t\rightarrow\infty}u(t):=w$$ exists in $$H^{1},$$ and $$-\Delta w+f(w)=0.$$
This result extends previous results by C. Cortazar, M. del Pino and M. Elguetta [Commun. Partial Differ. Equations 24, No. 11–12, 2147–2172 (1999; Zbl 0940.35107)] and by E. Feireisl and H. Petzeltová [Differ. Integral Equ. 10, No. 1, 181–196 (1997; Zbl 0879.35023)] who considered the autonomous case $$g=0,$$ as well as J. Busca, M. A. Jendoubi and P. Poláčik [Commun. Partial Differ. Equations 27, No. 9–10, 1793–1814 (2002; Zbl 1021.35013)] who considered a larger class of nonlinearities $$f.$$

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations 35K15 Initial value problems for second-order parabolic equations
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