  In this article we present a new p-adic generalization of the Eigen-Schuster
model where the genomes (sequences) are represented by words written in the
alphabet 0,1,,p-1, where p is a prime number, with a time variable length. The
time evolution of the concentration of a sequence is controlled by a p-adic
evolution equation. The long term behavior of the concentration of a sequence
depends on a fitness function f, a mutation measure Q, and an initial
concentration distribution. The new model provides essentially two types of
asymptotic scenarios for evolution. If the complexity of sequences grows at the
right pace, then in the long term the survival is assured. This agrees with the
fact that larger genome size improves the replication fidelity. In other case,
the sequences cannot copy themselves with sufficiently fidelity, and in the
long term they will not survive. Eigen's paradox is one, among the infinitely
many, possible scenarios of the evolution in the long term. The mathematical
formulation of this fact requires solving the Cauchy problem for the p-adic
Eigen-Schuster model in a rigorous mathematical way. This requires imposing
restrictions on the fitness function and on the mutation measure, among other
conditions. The study of the mentioned initial value problem requires
techniques of p-adic wavelets and heat kernels developed in the last 35 years.
