KAM techniques in PDE
The long-time asymptotics of certain nonlinear , nonlocal, diffusive equations with a gradient flow structure are analyzed. In particular, a result of Benedetto, Caglioti, Carrillo and Pulvirenti [4] guaranteeing eventual relaxation to equilibrium velocities in a spatially homogencous model of granular flow is extended and quantified by computing explicit relaxation rates. Our arguments rely on establishing generalizations of logarithmic Sobolev inequalities and mass transportation inequalities,...
This work is concerned with the proof of decay estimates for solutions of the Cauchy problem for the Klein-Gordon type equation . The coefficient consists of an increasing smooth function and an oscillating smooth and bounded function b which are uniformly separated from zero. Moreover, is a positive constant. We study under which assumptions for λ and b one can expect as an essential part of the decay rate the classical Klein-Gordon decay rate n/2(1/p-1/q).
We establish an upper bound on the Kolmogorov’s entropy of the locally compact attractor for strongly damped wave equation posed in locally uniform spaces in subcritical case using the method of trajectories.
If is a bounded domain with Lipschitz boundary and is an open subset of , we prove that the following inequality holds for all and , where defines an elliptic differential operator of first order with continuous coefficients on . As a special case we obtain for all vanishing on , where is a continuous mapping with . Next we show that is not valid if , and , but does hold if , and is symmetric and positive definite in .