Local and global existence of solutions to a fourth-order parabolic equation modeling kinetic roughening and coarsening in thin films

Abstract

In this paper we study both the Cauchy problem and the initial boundary value problem for the equation $\partial_tu+\mbox{div}\left(\nablaΔu-{\bf g}(\nabla u)\right)=0$. 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 $\mathbf{g}$ is continuous and the initial datum $u_0$ lies in $C^1({\mathbb{R}}^N)$ with $\nabla u_0(x)$ being uniformly continuous and bounded on ${\mathbb{R}}^N$ and that the global existence assertion also holds true if we assume that $\mathbf{g}$ is locally Lipschitz and satisfies the growth condition $|\mathbf{g}(\xi )|\le c|\xi {|}^{\alpha }$ for some $c>0,\alpha \in (2,3)$, ${\sup }_{{\mathbb{R}}^N}|\nabla u_0|<\infty$, and the norm of $u_0$ in the space $L^{\frac{(\alpha -1)N}{3-\alpha }}({\mathbb{R}}^N)$ is sufficiently small. This is done by exploring various properties of the biharmonic heat kernel. In the initial boundary value problem, we assume that $\mathbf{g}$ is continuous and satisfies the growth condition $|\mathbf{g}(\xi )|\le c|\xi {|}^{\alpha }+c$ for some $c,\alpha \in (0,\infty )$. Our investigations reveal that if $\alpha \le 1$ we have global existence of a weak solution, while if $1<\alpha <\frac{N^2+2N+4}{N^2}$ 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.