Global solution to the Cauchy problem of nonlinear thermodiffusion in a solid body

Arkadiusz Szymaniec¹

¹ Institute of Mathematics and Cryptology Faculty of Cybernetics Military University of Technology S. Kaliskiego 2 00-908 Warszawa, Poland

Applicationes Mathematicae, Tome 37 (2010) no. 4, pp. 437-458

Résumé

We consider the initial-value problem for a nonlinear hyperbolic-parabolic system of three coupled partial differential equations of second order describing the process of thermodiffusion in a solid body (in one-dimensional space). We prove global (in time) existence and uniqueness of the solution to the initial-value problem for this nonlinear system. The global existence is proved using time decay estimates for the solution of the associated linearized problem. Next, we prove an energy estimate in Sobolev spaces with constant independent of time. Such an energy estimate allows us to apply the standard continuation argument to continue the local solution to be defined for all times.

Détail

DOI : 10.4064/am37-4-4

Keywords: consider initial value problem nonlinear hyperbolic parabolic system three coupled partial differential equations second order describing process thermodiffusion solid body one dimensional space prove global time existence uniqueness solution initial value problem nonlinear system global existence proved using time decay estimates solution associated linearized problem prove energy estimate sobolev spaces constant independent time energy estimate allows apply standard continuation argument continue local solution defined times