Solvability of nonlinear diffusion models with flux at the boundary driven by nonmonotone local and memory interactions

Abstract

We investigate local and global solvability for nonlinear diffusion models with boundary flux driven by competing local and memory interactions. New local existence and comparison theories are developed for such models which are not amenable to a-priori arguments for nonnegative solutions, as well as not permitting standard monotonicity techniques of analysis. A transformation used to address nonnegativity subsequently results in classical solutions for strictly positive initial states. An alternating construction introduced to obtain solutions via monotone limits from above and below is somewhat reminiscent of the method of upper and lower solutions. Subsolution/supersolution comparison results that follow are applied to analyze boundedness, positivity, and blow up in finite time of solutions for power law cases of localized absorption and memory driven dissipation at the boundary. This connects the present work with our previous analyses of models having memory driven absorption at the boundary, shown to exactly parallel known results in cases of localized absorption, and begins to reveal comparative strengths of each.