Evolution of interfaces for the nonlinear parabolic p-Laplacian-type reaction-diffusion equations. II. Fast diffusion vs. absorption

Abstract

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian-type reaction-diffusion equation of non-Newtonian elastic filtration$$u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ 1 \lt p \lt 2, \beta \gt 0.$$If the interface is finite, it may expand, shrink or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β, sign b, and asymptotics of the initial function near its support. In some range of parameters, strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are based on nonlinear scaling laws and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves. A full description of small-time behaviour of the interfaces and local solutions near the interfaces for slow diffusion case when p>2 is presented in a recent paper by Abdulla and Jeli [(2017) Europ. J. Appl. Math.28(5), 827–853].

Interface development for the nonlinear degenerate multidimensional reaction–diffusion equations

“…Full classification of evolution of interfaces and local behavior of solutions near the interfaces for the problem (1.1)-(1.3) with space dimension N = 1 was presented in [2] for slow diffusion case (m > 1), and in [4] for the fast diffusion case (m = 1). The results and methods of [2,4] are extended to solve interface problem for p-Laplacian type reaction-diffusion equations in [9,10], and for the reaction-diffusion equations with double degenerate diffusion in [8]. The method of the proof developed in [2,4] is based on rescaling and application of the one-dimensional theory of reaction-diffusion equations in general non-cylindrical domains with non-smooth boundary curves developed in [1,3].…”

Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source

“…with α > 1 was studied in [33] where K is the stretched manifold with respect to the manifold K with corner-edge singularity and x ∈ K. The operator ∆ p,K + εV (x) with p = 2 arises from a diversity of physical phenomena, like in reaction-diffusion problems, in nonlinear elasticity, in non-Newtonian fluids and petroleum extraction. In [34] the relationship with non-Newtonian Mechanics was considered. Authors present a full classification of the short-time behavior of the interfaces and local solutions to the nonlinear parabolic p-Laplacian type reaction-diffusion equation of non-Newtonian elastic filtration…”

A Sub‐Supersolution Method for p‐Laplacian Equation with Non‐Local Term

“…p-Laplacian equation is a very important part of solving certain issues. For instance, it can be used in the famous non-Newtonian uid theory (for more information, see [1,2]). In addition to this, it can also be used to explain other physical phenomena: reaction-di usion problem, ow through porous media, non-linear elasticity, etc.…”