Uniform and optimal estimates for solutions to singularly perturbed parabolic equations |
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Authors: | D R Akhmetov R Spigler |
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Institution: | (1) Sobolev Institute of Mathematics, 4 Acad. Koptyug prosp., 630090 Novosibirsk, Russia;(2) Dipartimento di Matematica, Università “Roma Tre”, 1 Largo San Leonardo Murialdo, 00146 Rome, Italy |
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Abstract: | The Cauchy problem for singularly perturbed parabolic equations is considered, and weighted L2-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. |
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Keywords: | AMS Mathematics Subject Classifications (2000)" target="_blank">AMS Mathematics Subject Classifications (2000) 35A05 35B25 35K20 35K70 |
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