Continuous dependence on modeling for related Cauchy problems of a class of evolution equations
The authors investigate the Cauchy problem \(Pu_t+ Mu= F(t,u)\) on some interval \([0,T)\), under the initial datum \(u(0)= f\in H\), where \(P\) and \(M\) are linear operators with dense domain in the Hilbert space \(H\). \(F\) stands for a (nonlinear) map from \([0,T) \times H\) into \(H\). \(P\) and \(M\) are assumed to be symmetric and commuting, while \(P\) is strictly positive. \(F\) is supposed to satisfy a Lipschitz condition in the second argument. The problem is to justify the validity of the approximation scheme based on the singular perturbation described by the equation \[ Pw_t+ Mw- \varepsilon Pw_{tt} =F(t,w), \] under initial conditions \(w(0)=f\), \(w_t (0) =g\). Under further technical assumptions one obtains in \(L^2\) estimate for the difference \(u-w\). A nonlinear differential inequality is the key for obtaining the estimate which guarantees the validity of the procedure.
scientific article; zbMATH DE number 1115436
scientific article