ETERNAL SOLUTIONS FOR A REACTION-DIFFUSION EQUATION WITH WEIGHTED REACTION

We prove existence and uniqueness of eternal solutions in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation partial derivative(t)u = Delta(m)(u) + vertical bar x vertical bar(sigma) u(p), posed in R-N , with m > 1, 0 < p < 1 and the critical value for the weight sigma = 2(1 - p)/m - 1 Existence and uniqueness of some specific solution holds true when m + p >= 2. On the contrary, no eternal solution exists if m + p < 2. We also classify exponential self-similar solutions with a different interface behavior when m + p > 2. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.