LOCAL AND GLOBAL EXISTENCE OF SOLUTIONS TO A FOURTH-ORDER PARABOLIC EQUATION MODELING KINETIC ROUGHENING AND COARSENING IN THIN FILMS

In this paper we study both the Cauchy problem and the initial boundary value problem for the equation partial derivative(t)u+div(Delta del u-g(del u))= 0. This equation has been proposed as a continuum model for kinetic roughening and coarsening in thin films. In the Cauchy problem, we obtain that local existence of a weak solution is guaranteed as long as the vector-valued function g is continuous and the initial datum u(0) lies in C-1(R-N) with del u(0)(x) being uniformly continuous and bounded on R-N, and that the global existence assertion also holds true if we assume that g is locally Lipschitz and satisfies the growth condition vertical bar g(xi)vertical bar <= c vertical bar xi vertical bar(alpha) a for some c > 0, alpha is an element of (2,3), sup(R)N vertical bar del u(0)vertical bar < infinity, and the norm of u(0) in the space L(alpha-1)N/3-alpha (R-N) is sufficiently small. This is done by exploring various properties of the biharmonic heat kernel. In the initial boundary value problem, we assume that g is continuous and satisfies the growth condition vertical bar g(xi)vertical bar <= c vertical bar xi vertical bar(alpha) + c for some c, alpha is an element of (0,infinity). Our investigations reveal that if alpha <= 1 we have global existence of a weak solution, while if 1 < alpha < N-2+2N+4/N-2 only a local existence theorem can be established. Our method here is based upon a new interpolation inequality, which may be of interest in its own right.