Global Solvability for Degenerate Diffusion Models with Memory Boundary Conditions Arising in the Study of Tumor Induced Angiogenesis

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IPFW Department of Mathematical Sciences Third Annual Mini-Symposium on Analysis

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IPFW, Fort Wayne, IN

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Drawing motivation from the work of Judah Folkman in the 1970’s on angiogenesis and possible cancer treatments, a system of differential equations has been previously introduced with the goal of building accurate mathematical models for the phenomenon of tumor induced capillary growth. The system features coupled degenerate diffusion, mean curvature flow, and ordinary differential equations both in the domain (here, the extracellular matrix) and on the boundary (which contains the interface with the tumor and a nearby existing capillary). Although numerical studies have produced results comparing well with laboratory observations, qualitative mathematical studies are yet in development.

Toward analyzing a subset of the full problem, we study one component of the model, which reduces to a degenerate diffusion equation with memory type boundary condition not previously studied in the literature. An analysis of power law cases reveals similarities between global solvability of the simplified model and a corresponding one with more standard localized boundary conditions. In this presentation, we give a brief account of the background related to angiogenesis, difficulties inherent in degenerate diffusion equations, and the methods of analysis used for establishing global solvability in the presence of memory boundary conditions.

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