This paper establishes some existence and nonexistence results of self-similar radial symmetric solutions to some class of strongly coupled reaction-diffusion systems with cross-diffusion. The considered class of systems allows to reduce the problem to a single equation with exponential source terms. Using the famous Mountain Pass Theorem and some smallness conditions on the system parameters it is possible to generalize wellknown results on self-similar radial solutions for a related problem that have been established by Y. Mizutani, N. Muramoto and K. Yoshida in 1999. As an application of the results derived in the present paper it is possible to conclude the existence and nonexistence of self-similar radial solutions for multi-species chemotaxis-model in the conflict-free setting and in the presence of a conflict of interests.
"Self-similar radial solutions to a class of strongly coupled reaction-diffusion systems with cross-diffusion." Hiroshima Math. J. 43 (3) 305 - 333, November 2013. https://doi.org/10.32917/hmj/1389102578