Comparison theorems for infinite systems of parabolic functional-differential equations

Tom 77 / 2001

Danuta Jaruszewska-Walczak Annales Polonici Mathematici 77 (2001), 261-270 MSC: 35R45, 35K50. DOI: 10.4064/ap77-3-5


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.


  • Danuta Jaruszewska-WalczakInstitute of Mathematics
    University of Gda/nsk
    Wita Stwosza 57
    80-952 Gda/nsk, Poland

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