Abstract
The objective of this paper aims to prove positivity of solutions for the following semilinear partial differential equationu\(u_t = - \alpha u_{xxxx} + (u^2 )_{xx} + u(1 - u^2 )\). This equation represents a generalised model of the so-called porous medium equation. It arises in a variety of meaningful physical situations including gas flows, diffusion of an electron-ion plasma and the dynamics of biological populations whose mobility is density dependent. In all these situations the solutions of the equation must be positive functions.
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Bartuccelli, M., Gourley, S. On the Problem of Asymptotic Positivity of Solutions for Dissipative Partial Differential Equations. Journal of Biological Physics 25, 65–71 (1999). https://doi.org/10.1023/A:1005113116218
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DOI: https://doi.org/10.1023/A:1005113116218