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

Abstract

Abstract We investigate well-posedness, regularity and asymptotic behavior of parabolic Kirchhoff equations ∂ t u − a ∫ | ∇ u | 2 Δ u + α ( x ) u = f ( x ) in Ω × ( 0 , ∞ ) , on bounded domains of R N , N ⩾ 2, with non-homogeneous flux boundary conditions a ∫ | ∇ u | 2 ∂ u ∂ ν + β ( x ) u = g ( x ) on ∂ Ω × ( 0 , ∞ ) of Neumann or Robin type. The data in the problem satisfy (f, g, u(0)) ∈ L 2(Ω) × L 2(∂Ω) × H 1(Ω). Approximated solutions are constructed using time rescaling and a complete set in H 1(Ω) 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.