Local structure of solutions of the reaction-diffusion equations Original Research Article

We consider the initial-value problem for the nonlinear parabolic equation ul−a(um)xx+buβ=0,−∞0ul−a(um)xx+buβ=0,−∞0 Turn MathJax on with u(x,0)=u0(x),−∞ 0, b ∈ R1,m ≥ 1,β > 0. The inital function has finite support and is supposed to be nonnegative, and continuous. Locating the right-hand edge of the support of uσ(x) at the point x = l, we assume also the initial function to be smooth in (l − δ,l), for some δ > 0. We show that the small-time behaviour of the interface, which emerges from the point (x,t) = (l,0), as well as the local structure of solution near the interface depend crucially on the number View the MathML sourceγ=lim⁡x→l−0(a(u0m)′′/bu0β) Turn MathJax on In all possible cases, when interface either shrinks or remains stationary, the small-time behaviour of the interface is found, together with the local solution.