Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source

This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. For the considered class of models, we establish existence of a new type of ultra-singular self-similar solutions. These solutions arise as limits of the solutions of the initial value problem with zero initial data and infinitely strong source at the boundary. We prove existence and uniqueness of such solutions in the suitable weighted energy spaces. Moreover, we prove that the obtained self-similar solutions are the long-time limits of the solutions of the initial value problem with zero initial data and a time-independent boundary source.

💡 Research Summary

The paper investigates the long‑time asymptotics of a class of nonlinear reaction‑diffusion equations that model morphogen gradient formation in developing tissues. The governing equation is a one‑dimensional diffusion equation with a power‑law absorption term,
(u_t = D u_{xx} - k u^p) (with (p>1)),
supplemented by a boundary source at (x=0). Two types of boundary conditions are considered: a finite Neumann flux (u_x(0,t) = -\alpha) with (\alpha>0), and the limiting case of an infinitely strong source ((\alpha\to\infty)). The authors focus on the latter case, where the initial data are taken to be zero, and they discover a new family of “ultra‑singular” self‑similar solutions.

By introducing the scaling ansatz
(u(x,t)=t^{-1/(p-1)} f!\big(x/\sqrt{t}\big)),
the problem reduces to a nonlinear ordinary differential equation for the profile (f(\xi)) defined on (\xi>0). The ODE is accompanied by the singular boundary condition (f(0)=\infty) and the decay condition (f(\xi)\to0) as (\xi\to\infty). Because the profile diverges at the origin, classical existence theory does not apply. The authors overcome this difficulty by working in a weighted Sobolev space (H^1_{\omega}(\mathbb{R}_+)) with weight (\omega(\xi)=e^{\xi^2/4}). They formulate a variational problem for the weighted energy functional
(E