Abstract as per original application
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The Cahn-Hilliard (CH) equation is a fourth order nonlinear partial differential equation (PDE) originally used to model the separation of a mixture containing two phases of matter, such as water and oil, and is now an ubiquitous component of many mathematical models in applied sciences. It is formulated with a nonconvex potential function, usually taken as a smooth polynomial, that possesses two equal minima, say -1 and 1. Through the addition of appropriate source terms, the CH equation has seen recent applications in modelling the growth of tumours and in repairing damaged black-and-white images (also known as inpainting). However, an unfortunate feature with a polynomial potential is that the solution may not stay bounded in between -1 and 1, and undesirable effects, such as negative mass densities in tumour models or new shades of colour in black-and-white images, can occur.
A remedy is to employ singular potentials, such as a logarithmic-type function, to ensure that the solution to the model stays in between -1 and 1. In exchange, the mathematical analysis of these models with singular potentials become more involved compared to the polynomial potentials. For recent applications in tumour growth and inpainting, the new combination of singular potentials and source terms has not received much attention in the literature, and current analytical studies are mostly confined to establishing the existence of weak solutions.
In this project, we plan to expand the scope of the analysis for these new Cahn-Hilliard models with singular potentials and source terms by investigating the issues of uniqueness and regularity of solutions, as well as addressing the existence of stationary solutions analytically and also numerically. We believe that the proposed methodologies can yield both theoretical and practical contributions towards the active research areas of tumour growth and inpainting, in the form of a novel and alternative strategy to perform inpainting with the CH equation that is built upon a rigorous analytical basis, and a mathematical foundation for practitioners to make meaningful comparisons between tumour model simulations and experimental data.
Cahn-Hilliard（CH）方程是一個四階非線性偏微分方程，最初用於模擬分離含有兩相物質（例如水和油）的混合物，現在是在應用科學許多數學模型中普遍存在的組成部分。它通常包含一個非凸勢函數，具有兩個相等的最小值，比如 -1 和1。通過添加適當的源項，最近CH方程已經應用於模擬腫瘤的生長和修復受損的黑白圖像。然而，具有源項的不幸特徵是該解決方案可能不會保持在-1和1之間，並且具有不期望的效果，例如在腫瘤模型中出現負質量密度或在修復受損的黑白圖像出現新的灰色地帶。
一種補救措施是採用奇異勢函數，以確保模型的解決方案保持在-1和1之間。作為交換，這些具有奇異勢函數的模型的數學分析變得更加複雜。對於腫瘤生長和修復受損圖像這兩種最新應用，這個奇異勢函數和源項的新組合在文獻中並未引起太多關注，目前的分析研究主要局限於建立弱解方案的存在性。
在這個項目中，我們計劃通過研究解決方案獨特性和規律性的問題，以及通過分析和數字方式解決固定解決方案的存在，擴展這些具有奇異勢函數和源項的新型CH模型的分析範圍 。我們相信，所提出的方法可以為腫瘤生長和修復受損圖像的活躍研究領域提供理論和實踐的貢獻，例如新的圖像修復策略建立在嚴格的分析基礎上，以及為從業者提供數學基礎，以便在腫瘤模型模擬和實驗數據之間進行有意義的比較。
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