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# Communications in Mathematical Sciences

## Volume 15 (2017)

### Number 8

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

Pages: 2195 – 2218

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n8.a5

#### Author

#### Abstract

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.

Received 7 April 2017

Accepted 23 July 2017

Published 20 December 2017