caa a22 4500 477091563 CHVBK 20180405111516.0 cr unu---uuuuu 170330e19961201xx s 000 0 eng 10.1007/BF01193831 doi (NATIONALLICENCE)springer-10.1007/BF01193831 The regularity of solutions of reaction-diffusion equations via Lagrangian coordinates [Elektronische Daten] [Sergei Shmarev, Juan Vazquez] In this paper we study the regularity of nonnegative solutions and their interfaces for the nonlinear reaction-diffusion equation $$u_t = \left( {u^m } \right)_{xx} + f\left( u \right),\left( E \right)$$ wherem>1 andf(u) is aC 1 function withf(0)=0 and is subject to some other technical conditions. This equation has the property of finite propagation which gives rise to interfaces separating regions whereu=0 andu>0. The analysis is carried out by means of Lagrangian coordinates, formally viewing the reaction-diffusion equation as the equation governing the evolution of the density of a certain continuum. Lagrangian coordinates have been successfully applied to study nonlinear diffusion equations posed in one space dimension. The usual formulation applies to equations which can be written in the form of a conservation law, which is not the case here because of the reaction term. In problems exhibiting interfaces such technique has the merit of rendering the interfaces straight lines, much simplifying the analysis. In this paper we present anon-standard Lagrangian formulation that works innon-conservation cases. Equation (E) is then translated into this framework and we find in a natural way the necessary estimates to prove theC 1 regularity of moving interfaces and the regularity of the weak solution near such an interface, that allows us to establish the dynamic properties of the interface for the solutions. We end the paper by describing how the method can be applied to similar problems inseveral space dimensions with radial symmetry. Birkhäuser Verlag, 1996 Nonlinear reaction-diffusion equations nationallicence regularity of solutions and interfaces nationallicence Lagrangian coordinates nationallicence Shmarev Sergei Lavrentiev Institute of Hydrodynamics, 630090, Novosibirsk, Russia aut Vazquez Juan División de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain aut Nonlinear Differential Equations and Applications NoDEA Birkhäuser-Verlag 3/4(1996-12-01), 465-497 1021-9722 3:4<465 1996 3 30 https://doi.org/10.1007/BF01193831 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01193831 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Shmarev Sergei Lavrentiev Institute of Hydrodynamics, 630090, Novosibirsk, Russia aut NATIONALLICENCE 700 1- Vazquez Juan División de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain aut NATIONALLICENCE 773 0- Nonlinear Differential Equations and Applications NoDEA Birkhäuser-Verlag 3/4(1996-12-01), 465-497 1021-9722 3:4<465 1996 3 30 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer