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Convergence for semilinear degenerate parabolic equations in several space dimensions. (English) Zbl 0977.35069
The authors study the long-time behavior of solutions to the degenerate reaction-diffusion equation $u_t - \Delta\varphi(u) + f(u) = 0,\qquad u = u(x,t),\quad t > 0,\quad x\in\Omega$ with the homogeneous Dirichlet conditions $$\left.u\right|_{\partial\Omega}$$ where $$\Omega\subset \mathbb{R}^N$$ is a bounded domain with regular boundary. $$f$$ is a restriction of a real analytic function defined on a sector containing the half-line $$[0,\infty)$$ and $$f(u^{1/m})$$, $$m>0$$, is a continuously differentiable function of $$u$$. The main result of the article reads as follows:
Theorem. Let $$\Omega\subset\mathbb{R}^N$$ be a bounded domain of the class $$C^{2, \gamma}$$, $$\gamma > 0$$. Assume that $\varphi(u) = u^m,\qquad f(u) = \sum_{i=1}^na_iu^{p_i} - a_0, \qquad \text{for all } u\in [0,\infty)$ where $$a_0 \geq 0$$ is a nonnegative constant and $0 < \min\{1,m\}\leq \max\{1,m\}\leq p_1\leq p_2\leq\cdots \leq p_n\qquad (m,p_i,a_i\in \mathbb{R})$ for all $$i=1,\dots, n$$. Let $$u$$ be a weak solution of the problem under consideration on $$(0,\infty)$$ such that $0\leq u(t,x) \leq U\qquad \text{for a.e. } (t,x)\in (0,\infty)\times\Omega.$ Then $$u$$ is continuous in $$(0,\infty)\times\overline\Omega$$ and there exists a stationary solution $$w$$ such that $$u(t)\to w$$ in $$C(\overline\Omega)$$ as $$t\to\infty$$.

##### MSC:
 35K65 Degenerate parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs 35B45 A priori estimates in context of PDEs
##### Keywords:
long-time behavior; stabilization
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