Parabolic Kirchhoff equations with non-homogeneous flux boundary conditions: well-posedness, regularity and asymptotic behavior

We investigate well-posedness, regularity and asymptotic behavior of parabolic Kirchhoff equations partial derivative(t)u - a (integral vertical bar del u vertical bar(2)) Delta u + alpha(x) u = f (x) in Omega x (0, infinity), on bounded domains of R-N, N >= 2, with non-homogeneous flux boundary conditions a (integral vertical bar del u vertical bar(2)) partial derivative u/partial derivative nu + beta(x)u = g(x) on partial derivative Omega x (0, infinity) of Neumann or Robin type. The data in the problem satisfy (f, g, u(0)) is an element of L-2(Omega) x L-2(partial derivative Omega) x H-1(Omega). Approximated solutions are constructed using time rescaling and a complete set in H-1(Omega) relating the equation and the boundary condition. Uniform global estimates are derived and used to prove existence, uniqueness, continuous dependence on data, a priori estimates and higher regularity for the parabolic problem. Existence and uniqueness of stationary solutions are shown, as well as a description about their role on the asymptotic behavior regarding to the evolutionary equation. Furthermore, a sufficient condition for the existence of isolated local energy minimizers is provided. They are shown to be asymptotically stable stationary solutions for the parabolic equation.