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Asymptotic behavior for solutions of parabolic equations with natural growth terms and irregular data. (English) Zbl 1145.35346

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
\[ \begin{aligned} -\Delta v+ g(v)|\nabla v|^2& = f\quad\text{in }\Omega,\\ v(x) & = 0\quad\text{on }\partial\Omega.\end{aligned} \]

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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