Abstract: | We consider the equation (?1)m?m (p?mu) + ?u = ? in ?n × (0, ∞) for arbitrary positive integers m and n and under the assumptions p ? 1, ? ? C(?n) and p > 0. Even if the differential operator (?1)m?m (p?mu) has no eigenvalues, the solution u(x,t) may increase as t → ∞ for 2m ≥ n. For this case, we derive necessary and sufficient conditions for the convergence of u(x,t) as t → ∞. Furthermore, we characterize the functions occurring in these conditions as solutions of the homogeneous static equation (?1)m?m (p?mu) = 0, which satisfy appropriate asymptotic conditions at infinity. We also give an asymptotic characterization of the static limit. |