Parabolic systems in unbounded domains. I: Existence and dynamics (Q1378615)

The systems considered are of the form \[ u^i_{,t} -L^iu^i =f^i(t,x,u^j),\;t>0,\;x\in \Omega,\;u^i(t,x)= h^i(t,x),\;x\in \partial \Omega,\;u^i(0,x)= u^i_0(x), \tag{S} \] where \(\Omega\) is the exterior of a bounded domain of \(\mathbb{R}^n\), \(\mathbb{R}^n\) or a half space, and \(L^i\) are uniformly elliptic operators whose coefficients may depend on \(t\) and \(x\). The solutions considered are classical ones (under the appropriate smoothness conditions on the data). The author proves existence and uniqueness of the solution of (S) and also asymptotic behavior. The main technique used are upper and lower solutions.