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 filtrationThe interface 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 all cases, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. 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. InrtroductionWe consider the Cauchy problem(CP) for the nonlinear degenerate parabolic equationwhere p > 2, b ∈ R, β > 0, 0 < T ≤ +∞, and u 0 is nonnegative and continuous. We assume that b > 0 if β < 1, and b is arbitrary if β ≥ 1 (see Remark 1.1). Equation (1.1) arises in many applications, such as the filtration of non-Newtonian fluids in porous media ([8] or nonlinear heat conduction ([9]) in the presence of the reaction term expressing additional release (b > 0) or absorption (b < 0) of energy. The goal of this paper is to analyze the behavior of interfaces separating the regions where u = 0 and where u > 0. We present full classification of the short-time evolution of interfaces and local structure of solutions near the interface. Due to invariance of (1.1) with respect to translation, without loss of generality, we will investigate the case when η(0) = 0, where η(t) = sup {x : u(x, t) > 0}. and precisely, we are interested in the short-time behavior of the interface function η(t) and local solution near the interface. We shall assume thatThe direction of the movement of the interface and its asymptotics is an outcome of the competition between the diffusion and reaction terms and depends on the parameters p, b, β, C, and α. Since the main results are local in nature, without loss of generality we may suppose that u 0 either is bounded or satisfies some 1 arXiv:1605.07279v1 [math.AP]

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].