Search

The asymptotic analysis of a reaction-diffusion system with nonlinear boundary conditions in a thick fractal junctions is presented. In particular, the corresponding homogenized problem is found, the existence and uniqueness of its solution in an anisotropic Sobolev space of multi-sheeted functions is proved, and the approximation for the solution is constructed and justified.

Consider the heat equation with a nonlinear boundary condition $$\displaystyle \mathrm {(P)}\qquad \left \{ \begin {array}{ll} \partial _t u=\Delta u,\qquad & x\in {\mathbf {R}}^N_+,\,\,\,t>0,\\ \displaystyle {-\frac {\partial u}{\partial x_N} u}=u^p, & x\in \partial {\mathbf {R}}^N_+,\,\,\,t>0,\\ u(x,0)=\kappa \psi (x),\qquad & x\in \overline {{\mathbf {R}}^N_+}, \end {array} \right . \qquad \qquad $$ where N ≥ 1, p > 1, κ > 0 and ψ is a nonnegative measurable function in \({\mathbf {R}}^N_+ :=\{y\in {\mathbf {R}}^N:y_N>0 \}\). Let us denote by T(κψ) the life span of solutions to problem (P). We investigate the relationship between the singularity of ψ at the origin and T(κψ) for sufficiently large κ > 0 and the relationship between the behavior of ψ at the space infinity and T(κψ) for sufficiently small κ > 0. Moreover, we obtain sharp estimates of T(κψ), as κ →∞ or κ → +0.