## 8 Conclusions
### Analytical Solution of the Non-Linear Diffusion Model with Absorption and Production  
We follow the solution of [PERSON] [2], with \(k>1\), \(b>0\), \(c>0\), \(p>0\), and \(q>0\). Skipping details, the solution in this case is  
\[v(x,t)=\big{[}\alpha(t)(\cosh(\lambda x_{i}(t))-\cosh(\lambda x))_{+}\big{]}^{ \frac{1}{k-1}} \tag{101}\]  
with \(\lambda=\mp\frac{k-1}{k}\sqrt{b}\) and \(\alpha(t)\) determined as  
\[\alpha(t)=\frac{2\sqrt{\frac{c}{b}}\Big{(}\sqrt{\frac{c}{b}}+\sqrt{1+\frac{c} {b}}\Big{)}\exp\left[-\sqrt{b}c\frac{\big{(}k^{2}-1\big{)}}{k}t\right]}{\Big{(} \sqrt{\frac{c}{b}}+\sqrt{1+\frac{c}{b}}\Big{)}^{2}-\exp\left[-2\frac{\big{(}k^ {2}-1\big{)}}{k}t\right]} \tag{102}\]  
And, the front position (corresponding to \(\delta(t)\) in our solution method) as  
\[x_{i}(t)=\frac{1}{\lambda}\ln\biggl{(}\left(\sqrt{\frac{c}{b}}+\sqrt{1+\frac{ c}{b}}\right)\biggr{)}+\frac{1}{\lambda}\sqrt{b}c\frac{\big{(}k^{2}-1\big{)}}{k}t \tag{103}\]  
In Figures 11 and 12, the solution (101) is presented in \(1D\) perspective, showing the effect of \(k\) at different time moments.  
In the particular case with \(k=\frac{3}{2}\) and \(b=c=1\), there is \(\lambda=\mp\frac{1}{3}\), and  
\[v(x,t)=\bigg{[}\alpha_{(3/2)}(t)\biggl{(}\cosh\left(\frac{x_{i}(t)}{3}\right) -\cosh\Bigl{(}\frac{x}{3}\Bigr{)}\biggr{)}_{+}\bigg{]}^{2} \tag{104}\]  
\[v^{0}(x)=2\Bigg{[}1-\frac{\cosh\left(\frac{x}{3}\right)}{\sqrt{2}}\Bigg{]}_{+} ^{2} \tag{105}\]\[\alpha_{(3/2)}(t)=\frac{2\Big{(}1+\sqrt{2}\Big{)}\exp\bigl{(}-\frac{5}{6}t\bigr{)}}{ \Big{(}3+2\sqrt{2}\Big{)}-\exp\bigl{(}-\frac{5}{3}t\bigr{)}} \tag{104}\]  
Then, the front propagation in time is (a linear front in time)  
\[x_{i}(t)=3\ln\Bigl{(}1+\sqrt{2}\Big{)}+\frac{5}{2}t \tag{105}\]  
This result is strange, since it is well known that even the equation is degenerate, the front propagates proportional to \(\sqrt{t}\). The \(3D\) profile of this solution (Equation (103)) is shown in Figure A5.  
**Remark A2** (On the simulations of the solutions from [2]).: _In the end, we like to stress the attention on a surprising fact when the solutions of Maire [2] are simulated. Precisely, the plots corresponding to shorter times are located lower than those of longer times. This is physically strange and just opposite to what is presented in Figure A2, where logically the profiles for longer times are more evolved than those at shorter times, even though our simulations (see Figure A5b) confirm the results of Maire plotted in Figure 3 of [2]. Unfortunately, in [2] a physical analysis or at least a comparative analysis of the solution behaviors is missing. To our point of view, this remark is important regarding the results (solutions) developed by the integral-balance method when their physical meaning should be assessed._  
**Remark A3** (A resemblance in the solution constructions).: _Let us rearrange (A24) as_  
\[v(x,t)=\alpha(t)^{\frac{1}{k-1}}[\cosh(\lambda x_{i}(t))]^{\frac{1}{k-1}}\left( 1-\frac{\cosh(\lambda x)}{\cosh(\lambda x_{i}(t))}\right)^{\frac{1}{k-1}} \equiv\left(1-\frac{\cosh(\lambda x)}{\cosh(\lambda x_{i}(t))}\right)^{\frac{1} {k-1}}\] (A31)  
_And, we can see the role of the front \(x_{i}(t)\) in the formulation of the sharp front profile, i.e., when \(x=x_{i}(t)\), the profile goes to zero, the same as within the parabolic Barenblatt profile. However, from this solution we cannot extract a similarity variable either in the manner we did that or as a Boltzmann version; thus, a direct comparison of results of these solutions is hard to carry out._  
_Hence, we use different solution approaches, but the final solution constructions are too close, and all of them use the concept of a final front depth evolving in time. Unfortunately, due to different concepts in the solution developments, direct comparisons to a greater extent are impossible._