## Self-similar radial solutions to a class of strongly coupled reaction-diffusion systems with cross-diffusion.(English)Zbl 1300.34027

Given $$\varepsilon>0$$, $$N\geq 1$$, $$z_1\in \{1, \dots, N\}$$, and non-negative real numbers $$(\mu_l)_{1\leq l \leq z_1}$$, $$(\chi_l)_{1\leq l \leq z_1}$$, $$(\nu_h)_{1 \leq h \leq N-z_1}$$, and $$(\lambda_h)_{1\leq h \leq N-z_1}$$. Consider the Cauchy problem $\psi'' + \left( \frac{1}{r} + \frac{\varepsilon r}{2} \right) \psi' + \sum_{l=1}^{z_1} \mu_l e^{-r^2/4} e^{\chi_l \psi} + \sum_{h=1}^{N-z_1} \lambda_h e^{-r^2/4} e^{-\nu_h \psi} = 0\;, \quad r>0\;,$ with initial conditions $$\psi'(0)=0$$ and $$\psi(0)=a>0$$. It is shown that, if $0 < N \max\{ \mu_l , \lambda_h \}\;\max\{\chi_l\} \frac{\ln{\varepsilon}}{\varepsilon-1} < \frac{1}{e} \;\;\text{ and }\;\; \max\{ \mu_l \}>0\;,$ then there are $$a^*>1/(\max\{\chi_l\})>a_*$$ such that the corresponding solution $$\psi$$ to the above Cauchy problem is positive for all $$r\geq 0$$ and $$r\mapsto r\psi(r)$$ belongs to $$L^1(0,\infty)$$. Such solutions do not exist if $$\left( \sum \mu_l \right) \ln{\varepsilon}/(\varepsilon-1)$$ is sufficiently large. The proof combines the mountain pass principle and ODE arguments. This problem stems from the study of the existence of self-similar radially symmetric solutions to some multi-species chemotaxis systems.

### MSC:

 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 35C06 Self-similar solutions to PDEs 35K40 Second-order parabolic systems
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### References:

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