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

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].

Persistence Properties of the Two-Component b-Family System

In this paper we study the persistence properties of decay for the solutions to the two component b-family system. Using the method of characteristics, we establish that certain decay properties of the initial data persist as long as the solution exists. We also examine the propagation behavior of compactly supported solutions. We show that solutions have an infinite speed of propagation, that is, a non-trivial strong solution with compact initial value can not be compactly supported at any later time.

A NOTE ON HYPERBOLIC HEAT CONDUCTION IN A RIGID SOLID

Two hyperbolic heat conduction theories which predict a finite speed of propagation of thermal effects are considered. Numerical results obtained from these theories are presented for a one spatial dimension boundary-initial value problem, which involves transient heat conduction in a rigid half space. These results are compared with results from the classical theory, which predicts an infinite speed of propagation of thermal effects. It is shown that, for the problem considered, results from the three theories are very similar at times which are large compared to the relaxation time of the hyperbolic theories.

The Logical Basis of Electrical Circuit Theory

Two assumptions underlie circuit theory: infinite speed of propagation of force and existence at all times of equilibrium in matter. The theory is therefore essentially of static states. Voltages across all circuit components, including inductors, transformers, generators and others, are differences in potential, traceable to Coulomb forces and obeying Poisson's equation.