REACTION DIFFUSION-EQUATIONS IN ONE DIMENSION - PARTICULAR SOLUTIONS AND RELAXATION

Reaction-diffusion equations for the concentration of one species in one spatial dimension are considered, where the diffusion coefficient, as well as creation and annihilation terms are monomials. When the exponent of the annihilation term is smaller than the one of the creation term, unstable equilibrium solutions may exist. In the opposite case, travelling wave solutions are found in phase space, yielding the value of speed, even when the wave form cannot be written in closed form. The stability of some solutions is studied numerically, showing their robustness. In the case of nonlinear diffusion it is found that localized initial conditions evolve to the stable steady state via the travelling waves. The invasion of the stable state into the unstable one is via a travelling front with the largest possible gradient. This front can be interpreted as the final result of the growth of the most unstable mode. The exact solutions allow a modulation approach to study the problem with slowly varying coefficients. It is shown that when the creation term is of compact support in space, equilibria of compact support are also obtained.