zbMATH — the first resource for mathematics

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 \(t\to\infty\).

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
Full Text: DOI