Front motion in the reaction–diffusion problem in the case of a balance between reaction and diffusion

We show the existence and uniqueness of the solution with a moving internal transition layer in the initial boundary-value problem for the singularly perturbed parabolic reaction–diffusion equation in the case of a balance between reaction and diffusion. Using the Vasil'eva method of boundary functions, we construct an asymptotic approximation of the solution of the front form. We prove the existence and uniqueness theorem using the asymptotic method of Nefedov's differential inequalities. The obtained results can be used to develop effective numerical algorithms for solving hard problems appearing in the theory of nonlinear heat conduction and in population dynamics.