Converting LΡ Estimates Into L for Nonlinear Diffusion Models with Memory at the Boundary

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2016 Lloyd Roeling University of Louisiana Lafayette Mathematics Conference: Applied Mathematics

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Lafayette, Louisiana

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Drawing motivation from models of tumor-induced capillary growth, we initiated the study of diffusion models with boundary flux governed by memory around 5 years ago. Considering instances of power laws where supersolution comparison methods are not generally applicable, it turns out to be possible to estimate $L^p$ norms of the solution for any large value of $p$. Although constants in the estimates do not permit passage to the limit for obtaining an $L^\infty$ estimate, one may instead apply an integral form of the maximum principle that has frequently been applied in the case of porous medium equations. We provide an adaptation of the method to also apply equally well in so-called fast diffusion cases. Combined with results on blow-up in finite time, this allows a complete analysis of global solvability for such memory flux models which exactly parallels known results for corresponding localized nonlinear flux models.



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