Bounds as \(t\to +\infty\) for solutions of degenerate linear and nonlinear parabolic equations in unbounded domains
The author gives bounds as \(t\to \infty\) of the maximum of the positive solutions of the problem \[ (1)\quad \sum^{n}_{i,k=1}(\partial /\partial x_ i)(a_{ik}(t,x,u,D_ xu)\partial u/\partial x_ k)+f(t,x,u,D_ xu)-\partial u/\partial t=0, \] \[ (2)\quad u|_ E=0,\quad \partial u/\partial \nu |_{S_ 0^{\infty}\setminus E}=0,\quad u|_{t=0}=\phi (x)\in L_ 2(G), \] where \(D_ xu=(u_{x_ 1},...,u_{x_ n})\), G is bounded or unbounded domain in \(R^ n\), \(\partial /\partial \nu\) the conormal derivative, \(S_ 0^{\infty}=\partial G\times [0,\infty)\), E a closed set in \(S_ 0^{\infty}\); \(a_{ik}\) and f satisfy some additional conditions. Together with (1), (2) the problem analogical to (2) in \(G\times [t_ 0,T]\) for the differential inequality \[ \sum^{n}_{i,k=1}(\partial /\partial x_ i)(a_{ik}(t,x)\partial u/\partial x_ k)-\partial u/\partial t\geq 0 \] is considered.
scientific article; zbMATH DE number 3955505