Asymptotic behavior for a fourth-order parabolic equation involving the Hessian

作者：Xu, Guangyu^*; Zhou, Jun

来源：Zeitschrift fur Angewandte Mathematik und Physik, 2018, 69(6): 147.

DOI：10.1007/s00033-018-1034-6

• Fourth-order parabolic equation
• Hessian term
• Epitaxial growth
• Energy decay
• Exponential growth (decay)
• Blowup
• Global existence

This paper deals with a fourth-order parabolic PDE arising in the theory of epitaxial growth of crystal. For the stationary problem, we find a ground-state solution on its corresponding Nehari manifold by Lagrange multiplier method. As far as the evolution problem is concerned, we study the dynamics for both the global solution and the blow-up solution. Especially, for the global solution, we prove that the energy functional decays exponentially and we get the concretely decay rate. For the blow-up solution, the exponential growth of the solution is obtained, and we also obtain the growth rate. The behavior of the energy functional as is also discussed, where T is the blow-up time. Moreover, we prove that there exists blow-up solution with arbitrary high initial energy. Finally, for the low initial energy case, we give some equivalent conditions for the solutions blow up in finite time and exist globally, respectively. Our results extend the results got by Escudero et al. (Eur J Appl Math 24(3):437-453, 2013; J Differ Equ 254(6):2515-2531, 2013; Math Method Appl Sci 37(6):793-807, 2014; J Math Pures Appl 103(4):924-957, 2015).

• 出版日期2018-12
• 单位贵州师范大学; 重庆大学