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Self-Similar Intermediate Asymptotics for a Degenerate Parabolic Filtration-Absorption Equation

G. I. Barenblatt·M. Bertsch·A. E. Chertock·V. M. Prostokishin
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Abstract

The equation $$ \partial_tu=u\partial^2_{xx}u-(c-1)(\partial_xu)^2 $$ is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water absorbing fissurized porous rock, therefore we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.

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G. I. Barenblatt, M. Bertsch, A. E. Chertock, V. M. Prostokishin. 2000-06-08. Self-Similar Intermediate Asymptotics for a Degenerate Parabolic Filtration-Absorption Equation. https://doi.org/10.1073/pnas.97.18.9844

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