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 $\partial_t u + \mathrm{div}(\nabla \Delta_u - \mathbf{g}(\nabla_u))=0$. 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 $\mathbf{g}$ 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 $\lvert \mathbf{g}(\xi ) \rvert \leq c {\lvert \xi \rvert}^\alpha$ for some $c \gt 0 , \alpha \in (2,3) , \mathrm{sup}_{\mathbb{R}^N} \lvert \nabla u_0 \rvert \lt \infty$, 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 $\lvert \mathbf{g}(\xi) \rvert \leq c {\lvert \xi \rvert}^\alpha +c$ for some $c, \alpha \in (0,\infty)$. Our investigations reveal that if $\alpha \leq 1$ we have global existence of a weak solution, while if $1 \lt \alpha \lt \frac{N^2 + 2N + 4}{N^2}$ 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.

Keywords

biharmonic heat kernel, interpolation inequality, local and global existence of weak solutions, nonlinear fourth order parabolic equations, thin film growth

2010 Mathematics Subject Classification

Primary 35A01, 35A02, 35A35, 35K55. Secondary 35D30, 35Q99.

Full Text (PDF format)

Received 7 April 2017

Accepted 23 July 2017

Published 20 December 2017