  In this paper, we study the mixed problem for new class of nonlinear
reaction-diffusion PDEs with the nonlocal nonlinearity with variable exponents.
Here we obtain results on solvability and behavior of solutions both when these
are yet dissipative and when these get become non-dissipative problems. These
problems are possess special properties: these can be to remain dissipative all
time or can get become non dissipative after finite time. It is shown that if
the studied problems get become non-dissipative can have an infinite number of
different unstable solutions with varying speeds and also an infinite number of
different states of spatio-temporal (diffusion) chaos that are generated by
cascades of bifurcations governed by the corresponding steady states. The
behavior of these solutions is analyzed in detail and it is explained how
space-time chaos can arise. Since these problems desctibe the dynamics of
cancer, we explain the behavior of the dynamics of cancer using obtained here
results.
