Regularity and asymptotic behavior for the second order parabolic equation with nonlinear boundary conditions in Lp

AbstractThe equation Lu = ƒ;(x, u) on B × (0, ∞), B bounded, smooth domain in Rn with nonlinear boundary conditions ∂u∂v = g(x, u) on ∂B × (0, ∞) is studied, L being the uniformly parabolic operator with time independent coefficients. Under suitable conditions on the nonlinearities (that do not involve monotonicity) global existence, uniqueness, compactness of the orbits and certain regularizing effects of the semigroup are established. In the case that L is in divergence form it is shown that under generic conditions orbits tend, as t → + ∞, to some equilibrium and that the stable equilibria attract essentially (Baire category) the whole space L2(B)