Comparison theorems for infinite systems of parabolic functional-differential equations

Danuta Jaruszewska-Walczak¹

¹ Institute of Mathematics University of Gda/nsk Wita Stwosza 57 80-952 Gda/nsk, Poland

Annales Polonici Mathematici, Tome 77 (2001) no. 3, pp. 261-270

Résumé

The paper deals with a weakly coupled system of functional-differential equations $$ \partial _t u_i(t,x)=f_i(t,x,u(t,x),u,\partial_x u_i(t,x), \partial_{xx}u_i(t,x)),\quad i\in S, $$ where $(t,x)=(t,x_1,\ldots ,x_n)\in (0,a)\times G$, $u=\{u_i\}_{i\in S}$ and $S$ is an arbitrary set of indices. Initial boundary conditions are considered and the following questions are discussed: estimates of solutions, criteria of uniqueness, continuous dependence of solutions on given functions. The right hand sides of the equations satisfy nonlinear estimates of the Perron type with respect to the unknown functions. The results are based on a theorem on extremal solutions of an initial problem for infinite systems of ordinary functional-differential equations.

Détail

DOI : 10.4064/ap77-3-5

Keywords: paper deals weakly coupled system functional differential equations partial t x partial t partial x quad where ldots times arbitrary set indices initial boundary conditions considered following questions discussed estimates solutions criteria uniqueness continuous dependence solutions given functions right sides equations satisfy nonlinear estimates perron type respect unknown functions results based theorem extremal solutions initial problem infinite systems ordinary functional differential equations