J.evol.equ. 7 (2007), 347–372 © 2007 Birkh¨ auser Verlag, Basel 1424-3199/06/020347-26, published online February 07, 2007 DOI 10.1007/s00028-006-0294-3 Uniform and optimal estimates for solutions to singularly perturbed parabolic equations D. R. Akhmetov and R. Spigler Abstract. The Cauchy problem for singularly perturbed parabolic equations is considered, and weighted L 2 -estimates as well as certain decay properties of bounded classical solutions to it are established. These do not depend on the value of the small perturbation parameter, and allow to prove global in time existence of strong solutions to certain boundary-value problems for ultraparabolic equations with unbounded coefficients. Optimal decay estimates are proved for such solutions. All results concerning ultraparabolic equations apply, in particular, to the Kolmogorov equation for diffusion with inertia, to the (linear) Fokker-Planck equation, to the linearized Boltzmann equation, and to some nonlinear integro-differential ultraparabolic equations of the Fokker-Planck type, arising from biophysics. Optimal decay estimates are derived for global in time strong solutions to such equations. Introduction In this paper, we study ultraparabolic partial differential equations through their parabolic regularization, with a small parameter multiplying an additional diffusion term. Ultra- parabolic equations have been intensively investigated since the beginning of the 20th century. However, a comprehensive theory for such equations does not exist to date. The interest for ultraparabolic equations is motivated by a number of applications, see the Intro- duction to [6], e.g., for a short review. A few interesting papers have been published in 2005, concerning the qualitative theory of boundary value problems for ultaparabolic equations and systems. In [21], classical solvability has been established for the Cauchy problem for a general system of linear ultraparabolic equations. In [22], existence and uniqueness of the generalized solution have been proved for a boundary value problem for a quasilinear ultraparabolic equation arising in hydrodynamics (unidirectional flow in a tube). The well- posedness of the Cauchy problems for a quasilinear ultraparabolic equation with partial diffusion and discontinuous convection coefficients has been established in [17] for both entropy and kinetic formulations (see also [18]). AMS Mathematics Subject Classifications (2000): 35A05, 35B25, 35K20, 35K70. Key words: Parabolic equations, ultraparabolic equations, Fokker-Planck equation, unbounded coefficients, singular perturbations, boundary-layer.