  In this paper, we propose a new mathematical model nonlinear
reaction-diffusion PDE's describing the dynamics of propagation of cancer. Here
the mixed problem for the proposed PDE's is investigated and by applying
obtained results conclusions on the dynamics of propagation of cancer are
drawn. These problems have nonlocal nonlinearity with variable exponents and
possess special properties: these can be to remain either dissipative all time
or become non-dissipative after a finite time. Here the solvability and
behavior of solutions both when problems are yet dissipative and when become
nondissipative are proved. It is shown that if the studied process gets become
nondissipative can have various states, e.g. an infinite number of different
unstable solutions with varying speeds, in addition, their propagation can
become chaotic. The behavior of these solutions is analyzed in detail and it is
explained how space-time chaos can arise. Investigation of this mathematics
model allows explaining the dynamics of propagation of cancer, which are
provided here as conclusions for each case.
