This paper examines a wide class of ill-posed initial value problems for partial differential equations, and surveys logarithmic convexity results leading to Hoelder-continuous dependence on data for solutions satisfying prescribed bounds. The discussion includes analytic continuation in the unit disc, time-reversed parabolic equations in Lp spaces, the time-reversed Navier-Stokes equations, as well as a large class of non-local evolution equations that can be obtained by randomizing the time variable in well-posed Cauchy problems. It is shown that in many cases, the resulting Hoelder-continuity is much too weak to permit useful continuation from imperfect data. However, considerable reduction in the growth of errors occurs, and continuation becomes feasible, for solutions satisfying the slow evolution from the continuation boundary (SECB) constraint, previously introduced by the author.
Citation: Siam Journal on Mathematical Analysis
Pub Type: Journals
growing diffusion coefficients, ill-posed problems, parabolic equations, SECB constraint