**Author details**

In conclusion it seems interesting to present the plots for *evν*ð Þ*<sup>y</sup>* and *<sup>u</sup>ν*ð Þ*<sup>y</sup>* for different values of *ν* (**Figures 23**–**25**). It is interesting to see that there are irregular solutions for *evν*ð Þ*<sup>y</sup>* ; however, the corresponding solutions for *<sup>u</sup>ν*ð Þ*<sup>y</sup>* are regular [see

5 5 10

2

4

6

8

10 5 5

u y , k C^2 10

u y , k C^2 10

*Mathematical Theorems - Boundary Value Problems and Approximations*

nu 7, k 1, c 1 nu 45, k 1, c 1

nu 1 5, k 1, c 1 nu 8 17, k 1, c 1

We investigate three different one-dimensional parabolic-parabolic Patlak-Keller-Segel models. For each of them, we obtain the exact solutions in terms of traveling wave variables. Not all of these solutions are acceptable for biological interpretation, but there are solutions that require detailed analysis. It seems interesting to consider the latter for the experimental values of the parameters and see

their correspondence with experiment. This question requires further

Eqs. (35)–(37)].

*uν*ð Þ*y ; ν* ¼ *1=5*; *8=17.*

**Figure 25.**

Out[36]=

Out[49]=

**Figure 24.** *uν*ð Þ*y ; ν* ¼ *7*; *45.*

**5. Conclusion**

investigations.

**18**

Maria Vladimirovna Shubina Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russian Federation

\*Address all correspondence to: yurova-m@rambler.ru

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