Theses and Dissertations

Advisor

Xu, Xiangsheng

Committee Member

Qian, Chuanxi

Committee Member

Dang, Hai

Committee Member

McBride, Matt

Date of Degree

8-7-2025

Original embargo terms

Immediate Worldwide Access

Document Type

Dissertation - Open Access

Major

Mathematical Sciences

Degree Name

Doctor of Philosophy (Ph.D.)

College

College of Arts and Sciences

Department

Department of Mathematics and Statistics

Abstract

In this dissertation we investigate several PDE models of Epitaxial growth. These models are fourth-order PDE’s featuring exponential nonlinearities as well the p-Laplacian and 1-Laplacian. The presence of the exponential nonlinearity is what provides the main mathematical difficulty. Theexponent in particular does not have enough estimates to guarantee any compactness. Because of this, one has to allow the inclusion of a singular portion to the exponent in the sense of the Lebesgue decomposition theorem. In thefirst chapter weinvestigate a related epitaxial growth, with transition rates of the Metropo lis variety and a linear exponent. The metropolis rates induce an extra term which includes the reciprocal of the exponential part. Weareabletotakeadvantageofthistoprovetheglobalexistence of solutions without any singular portions. In the second chapter we consider the original model with a linear exponent. Building off of the idea in chapter 1, we impose a smallness condition on the initial data to prove the global existence of a solution without a singular portion. The strategy involves deriving a quadratic inequality which depends on the initial data. We are then able to leverage this inequality to show that the exponent remains bounded for all time. In the third chapter we study the model with a nonlinear, p-Laplacian in the exponent. The nonlinear exponent renders many of our earlier estimates untenable. We develop some new estimates for �� in between one and two. With this we are then able to prove enough compactness to justify passing to the limit. With no size restriction on the initial data, we have a singular portion to the exponent. In the fourth chapter, we consider the model with gradient dependent mobility coefficients. Due to the coefficients, the previous tools are no longer enough to handle the model. For this reason we first linearize the exponential term. We also have included the 1-Laplacian in our analysis, which in our case is dominated by the p-Laplacian for all �� greater than one. We prove the global existence of solutions, without any singular portions.

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