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The rescaling method is presented to allow us to establish nonnegative local solutions to the evolution of the Cauchy problem (CP) of the nonlinear degenerate parabolic p-Laplacian process with conservation laws that are posed in one-dimensional space. This equation has specific restrictions in the range of parameters, the non-negative advection coefficient, and a self-similarity representing the main feature. In this work, there are several regions to discuss the qualitative analysis for the local weak solutions and the asymptotic interfaces in irregular domains. The solutions of the CP for degenerate parabolic p-Laplacian type diffusion-advection equations are asymptotically equal to the solutions of p-Laplacian type diffusion or advection equations under some restrictions. Moreover, the blow-up technique, comparison method, and characteristic method are used to estimate the asymptotic local solutions to the CP and the interface functions. The results of this paper can be used to solve problems in the oil and gas industries, such as estimating and controlling the size of oil and gas resources as they evolve through time.
Aal-Rkhais et al. (Sat,) studied this question.