THE INITIAL-DIRICHLET PROBLEM FOR A 4TH-ORDER PARABOLIC EQUATION IN LIPSCHITZ CYLINDERS

We study the initial-Dirichlet problem for the fourth-order parabolic equation partial t + DELTA-2 on a cylinder (0,T) x D where D subset-of R(n) is a bounded domain with Lipschitz boundary. On the lateral boundary (0,T) x partial D we specify the boundary value of the function to lie in a Sobolev space of functions having one spatial derivative and one quarter of a time derivative in L2. The normal derivative is taken to lie in L2. We only consider the case of homogeneous initial data. With these assumptions, we show that there exists a unique solution whose spatial gradient has parabolic maximal function in L2. The main interest of these results lies in the weak smoothness assumptions on the lateral boundary.