LAUSR

where ν is the unit outward normal vector to the boundary. Equation (1.1) can be used to describe the evolution of a crystal surface [6]. In this case, u is the surface height. The fourth order term in the equation represents the diffusion effect, while the lower order terms describe evaporation and deposition. Detailed information can be found in [6]. Epitaxial growth is an important process in forming solid films and other nano-structures. Mathematical modeling of the process has attracted wide attentions [6]. Continuum models involving exponential nonlinearity were first derived in [9] and more recently in [13, 6]. Mathematical analysis of such models in high space dimensions (N ≥ 2) is very challenging due to the lack of estimates for the exponent term. It was first observed in [11] that one had to allow the possibility that the exponent be a measure-valued function. Later, the idea of “exponential singularity” was employed in [2, 4, 5, 14, 18]. However, measure exponents do not arise in the one-dimensional case. See [3, 6]. To remove the singularity in the exponent, the authors in [8, 10] introduced a rather sophisticated critical Wiener algebra space and showed that there existed a strong (no measure) solution as long as the norm of u0 in the Wiener algebra space was suitably small. The proof in [10] employed the Fourier transform of the power series expansion of the exponential term. A similar approach was also adopted in [8]. Here we offer a totally different perspective from which to view the problem. Our method is based upon Lemma 2.7 below, a simple result first introduced in [17]. Denote by ‖ · ‖p,Ω the norm in the space Lp(Ω). Our investigations reveal that we can obtain global existence of a strong solution by requiring the W 2,2(Ω) norm of e−∆u0 and ‖e0‖∞,Ω to be suitably small. Before we state our main theorem, we give our definition of a strong solution.