## 2 A Preliminary Basis the New Solution Developed
### Barenblatt's Parabolic Profile  
For the approximate solution, applying the integral-balance approach with the double-integration technique, the assumed parabolic profile is the so-called Bareblatt's profile [37] (see the comments further in this section):  
\[u_{a}=u_{s}\bigg{(}1-\frac{x}{\delta_{u}}\bigg{)}^{n} \tag{12}\]  
The Barenblatt profile obeys the Goodman conditions for any positive values of the exponent \(n\)[30].  
The profile of Equation (12) originates from the similarity solution of [PERSON] and [PERSON] [38] (see also [10]) and is sometimes called the Barenblatt-Pattle profile as used in the point source solution of [PERSON] [39]. It was extensively explored in solutions of high-order non-linear diffusions [31; 32] with exponent \(n=1/m\) this was confirmed independently in [29] applying optimization of the solution concerning the optimal value of \(n\) because the boundary conditions are not enough to define it at the solution beginning [30]. It is worth noting, for the sake of clarity and easy understanding, the comparative analyses concerning the present solutions and those existing in the literature, taking into account the relationship between the degeneracy exponents \(m\) and \(k\) (see Remark 1), that the common element in the existing solutions summarized in the Appendix A is the exponent \(1/(k-1)\), instead of \(1/m\). The construction of Equation (12), with \(n=1/m\), could also be related to a more general solution approach to degenerate diffusion models; see Remarks A1 and A3 in Appendix A.  
The sharp front solution with the Barenblatt profile could be related, to some extent, to the so-called type \(l\)_sharp front traveling waves_[40]. Moreover, as commented in [26], concerning the left side of the moving front (\(x\to\delta\)), in the form \(u(x,t)\propto(x-l(t))^{\alpha}\) (in terms of [26]), presented as \(u\propto(x-l(t))^{1/m}\), can be paraphrased, in terms of present work, as \(u\propto(1-x/l(t))^{1/m}\). For a detailed analysis with many possible situations concerning model (1) and traveling wave solutions, we refer to [24; 25]. For instance, as a clarifying comment, traveling waves in population dynamics and combustion theory appear as _invasion fronts_ and _combustion fronts_, respectively [41], corresponding to the propagation of the penetration depth \(\delta(t)\) in the integral-balance method.  
As proved in [23], the traveling wave with a sharp front should be unique. This means that at the front, there is a discontinuous derivative (since \(du/dz=du/dx\), we will further use \(du/dx\) in light of the Barenblatt profile): that is, the left derivative \(du/dx(\delta^{-})\
eq 0\), while the right derivative tends to zero, i.e., \(du/dx(\delta^{+})=0\) (see the analysis about the wave shapes concerning Figure 2.1 in [23]). It is easy to check that the Barenblatt profile \((1-x/\delta)^{n}\) obeys these properties at the front; the second condition (about the right derivative) is, in fact, implicitly provided by the Goodman boundary conditions.  
To make the exposition clear, the plots in Figure 1 illustrate what we use as a general property of the Barenblatt profile and understand what its behavior with various exponents is. In general, for \(m>1\), the profiles are convex (panel (**b**) with exponents \(n=1/m\).