Solutions of fourth-order parabolic equation modeling thin film growth

Null controllability of semi-linear fourth order parabolic equations

“…From Lemma 2.1 in [29] we deduce that ∆ 2 u, u | ∂Ω , ∂ u ∂ n | ∂Ω is normally elliptic. We obtain that A is the infinitesimal generator of an analytic semigroup T (t) = e −t∆ 2 and we have the following estimate (see Lemma 2.1 in [29]):…”

Space-Time Estimates of Mild Solutions of a Class of Higher-Order Semilinear Parabolic Equations in Lp

“…(19). Finally, the same procedure can be used to estimate M j and we obtain The proof of Proposition 3.1 is similar to that performed in the papers [24,25]. Equipped with Proposition 3.1, we are now able to sketch the proof of the main theorem of this paper.…”

“…The continuity in time for t ∈ [ , T] follows from standard results on Lebesgue's integral [24] whereas the estimate (8) follows from (16).…”

“…However, with the exception of King et al [14], they do not allow nonlinearities depending on the divergence of the solution. As demonstrated in the papers [24,25], when m = , the well-posedness relies upon scaling invariance of the problem: Suppose that u(x, t) is a smooth solution of Eq. (2) in R n , then for every λ > ,…”

“…This means that well-posedness can only be expected for k = or k = . The case k = was studied in [24]. Note that Melcher [18] studied in the whole space Ω = R n the case k = which works for α > n .…”