Reaction-diffusion equations are used to describe dynamic evolution of densities or populations for certain forms of initial conditions in space. New results from extensive investigations on such equations are reported. These allow for arbitrary powers of the dependence on density for the diffusion and reaction processes. Primarily, creative reactions are considered but, for comparison, some cases of annihilation reactions are also described. Convenient methods of nonlinear analysis are developed for the purpose of describing in particular the evolution in time of characteristic widths and amplitudes of profiles. The theory is presented in a unified form including one, two and three space dimensions. A rich variety of possible solutions are obtained, including, beside some exact localized solutions, also "collapse" and "anti-collapse" solutions as well as other generalized forms of solutions. The analytical results are supported by computer experiments.

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