Continuous dependence of solutions for ill-posed evolution problems

Continuous dependence of solutions for ill-posed evolution problems

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We prove Holder-continuous dependence results for the difference between certain ill-posed and well-posed evolution problems in a Hilbert space.Specifically, given a positive self-adjoint operator D in a Hilbert space, we consider the ill-posed Bottles evolution problem $$displaylines{ frac{du(t)}{dt} = A(t,D)u(t) quad 0leq t<T cr u(0) = chi.}$$ We determine functions $f:[0,T]imes [0,infty)o mathbb{R}$ for which solutions of the well-posed problem $$displaylines{ frac{dv(t)}{dt} = f(t,D)v(t) quad 0leq t<T cr v(0) = LIPSTICK SCARLET SOAKED chi }$$ approximate known solutions of the original ill-posed problem, thereby establishing continuous dependence on modelling for the problems under consideration.