Global existence of the solutions of nonlinear parabolic equations in exterior domains
This paper deals with the following initial boundary value problem of the nonlinear parabolic equation: \[ u_ t=\Delta u+F(u,D_ xu,D^ 2_ xu),\quad (t,x)\in R^+\times \Omega,\quad u(0,x)=\phi (x),\quad x\in \Omega,\quad u|_{\partial \Omega}=0, \] where \(\Omega\) is the exterior domain of a compact set in \(R^ n\) with smooth boundary and F satisfies \(| F(\lambda)| =o(| \lambda |^ 2)\), near \(\lambda =0\). It is proved that when \(n\geq 3\), under the suitable smoothness and compatibility conditions, the above problem has a unique global smooth solution for small initial data. Moreover, it is also proved that the solution has the decay property \(\| u(t)\|_{L^{\infty}(\Omega)}=o(t^{-n/2}),\) as \(t\to +\infty\).
scientific article; zbMATH DE number 4084166