Leonori, Tommaso; Petitta, Francesco Asymptotic behavior for solutions of parabolic equations with natural growth terms and irregular data. (English) Zbl 1145.35346 Asymptotic Anal. 48, No. 3, 219-233 (2006). The authors deal with the asymptotic behaviour as \(t\to\infty\) of solutions \(u\in L^2(0,T; H^1_0(\Omega))\) of the problem \[ \begin{aligned} u_t- \Delta_x u+ g(u)|\nabla_x u|^2 & = f\quad\text{in }\Omega\times (0,T),\\ u(x,0) & = u_0(x)\quad\text{in }\Omega,\\ u(x,t) & = 0\quad\text{on }\partial\Omega\times (0,T),\end{aligned} \]where \(\Omega\subset\mathbb{R}^N\) is a bounded open set, \(N\geq 3\) \(u_0\in L^1(\Omega)\) nonnegative, while \(g\), \(f\) and \(u_0\) satisfy suitable assumptions. The goal of the authors is to prove that \(u(x,t)\) converges, as \(t\to\infty\), to \(v(x)\) which is the unique solution of the elliptic problem \[ \begin{aligned} -\Delta v+ g(v)|\nabla v|^2& = f\quad\text{in }\Omega,\\ v(x) & = 0\quad\text{on }\partial\Omega.\end{aligned} \] Reviewer: Messoud A. Efendiev (Berlin) Cited in 15 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations Keywords:asymptotic behaviour; irregular data PDFBibTeX XMLCite \textit{T. Leonori} and \textit{F. Petitta}, Asymptotic Anal. 48, No. 3, 219--233 (2006; Zbl 1145.35346)