## 5 Nonlinear Diffusion Equation  
If we ask ourselves how to describe the evolution of aftershocks not only in time, but also in space-time, then this immediately puts us in a difficult position. On the one hand, a number of methods are known for modeling space-time distributions, but on the other hand, in our case, there is a strong limitation, which consists in the fact that when averaging over the epicentral zone, we want to obtain the Omori law in the form (2), or in the form (6). Fortunately for us, it turns out here that we can use the well-known Kolmogorov-Petrovsky-Piskunov equation (abbreviated KPP), which describes nonlinear diffusion [14]. It is convenient for us to represent it in the following form:  
\[\frac{\partial n}{\partial t}=n(\gamma-\sigma n)+D\frac{\partial^{2}n}{ \partial x^{2}}, \tag{17}\]  
where \(n(x,t)\) is the spatio-temporal distribution of aftershocks, the \(x\) axis is directed along the earth's surface (for simplicity, we limited ourselves to a one-dimensional model), and \(D\) is a new phenomenological parameter (diffusion coefficient). At \(D=0\), (17) turns into the logistic equation of the evolution of aftershocks (6), and under the additional condition \(\gamma=0\) into Omori law (2). It is useful to derive (17) from the integro-differential equation  
\[\frac{\partial n}{\partial t}=\Phi(n)+\int\limits_{-\infty}^{\infty}K(x-y) \cdot n(y,t)\cdot dy. \tag{18}\]  
Here \(\Phi(n)\) is some kind of functional. This will make it possible to express the parameters \(\gamma\) and \(D\) in terms of the kernel \(K(x-y)\). Indeed, suppose that \(K(x-y)=K(y-x)\), i.e., the core is symmetrical. If \(K\to 0\) for \(|x-y|\to\infty\), then expanding \(n(x-z,t)\) into a Taylor series in powers of \(x\), we obtain  
\[\frac{\partial n}{\partial t}=\gamma n+\Phi(n)+D\frac{\partial^{2}n}{\partial x ^{2}}+\ldots \tag{19}\]  
where  
\[\gamma=\int_{-\infty}^{\infty}K(z)dz,\ D=\frac{1}{2}\int_{-\infty}^{\infty}z^ {2}K(z)dz, \tag{20}\]  
and \(z=x-y\) (e.g., see [11; 15; 16]). Let us \(\Phi(n)=-\sigma n^{2}\), and confine the first two terms in the series. In this approximation we obtain Equation (17) from which after phenomenological reduction follows the Omori law in the form (2). When constructing a phenomenological theory of aftershocks, there is one most important condition, only one, but absolutely necessary: the phenomenological coefficients, whatever meaning we put into them, we must be able to measure experimentally, since they cannot be calculated on the basis of a more fundamental theory--because we simply do not have such a theory. In Section 6.1 we show how the deactivation factor we introduced in Section 2 is estimated. In the current section, we have significantly complicated the theory and introduced the diffusion coefficient \(D\). We want to briefly describe the result of observing aftershocks, which actually led us to master equation (17), and then indicate how, at least in principle, the parameter \(D\) can be estimated experimentally. (In this regard, it is appropriate to mention the recent interesting work [17]. It provides additional arguments in favor of the idea of the applicability of the KPP equation for modeling aftershocks). The study of aftershocks in space and time led us to the idea of using the KPP equation as the master equation. The main step forward in the study of the space-time distribution was the discovery that, apparently, at least some of the aftershocks tend to propagate like waves with a speed much lower than the speed of seismic waves [8; 18]. The rate of propagation varies widely from case to case. It is roughly a few kilometers per hour. This value is three orders of magnitude less than the velocities of elastic waves in the crust, which suggests the propagation of a nonlinear diffusion wave excited by the main shock. Equation (17) has self-similar solutions in the form of a traveling wave \(n(x,t)=n(x\pm Ult)\)[14; 19]. It is this circumstance that played a role in our choice of the KPP equation as the master equation.