The regularity of solutions of reaction-diffusion equations via Lagrangian coordinates
This article is concerned with the application of the method of Lagrangian coordinates to the reaction-diffusion equation \(u_t=(u^m)_{xx}+f(u)\), where \(m>1\) and \(f\) is a \(C^1\) function with \(f(0)=0\). The authors present a non-standard Lagrangian formulation that works in non-conservation cases and find the necessary estimates to prove the \(C^1\) regularity of moving interfaces and the regularity of the weak solution near such interfaces. The program succeeds for a quite general class of terms \(f(u)\), including all absorption terms of the form \(f(u)=-u^q\) with \(q\geq 1\) and reaction terms of the form \(f(u)=u^q\) if \(q\geq m\). The method can be applied to similar problems in several space dimensions with radial symmetry.
\(C^ 1\) regularity of moving interfaces
scientific article; zbMATH DE number 952687
scientific article