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A nonlinear kinetic chemotaxis model with internal dynamics incorporating signal transduction and adaptation is considered. This paper is concerned with: (i) the global solution for this model, and, (ii) its fast adaptation limit to Othmer-Dunbar-Alt type model. This limit gives some insight to the molecular origin of the chemotaxis behaviour.First, by using the Schauder fixed point theorem, the global existence of weak solution is proved based on detailed a priori estimates, under quite general assumptions. However, the Schauder theorem does not provide uniqueness, so additional analysis is required to be developed for uniqueness. Next, the fast adaptation limit of this model is derived by extracting a weak convergence subsequence in measure space. For this limit, the first difficulty is to show the concentration effect on the internal state. Another difficulty is the strong compactness argument on the chemical potential, which is essential for passing the nonlinear kinetic equation to the weak limit.

In this paper, the L p convergence rates of planar diffusion waves for multi-dimensional Euler equations with damping are considered. The analysis relies on a newly introduced frequency decomposition and Green function based energy method. It is a combination of the L p estimate on the low frequency component by using an approximate Green function and L 2 estimate on the high frequency component through the energy method. By noticing that the low frequency component in the approximate Green function has the algebraic decay which governs the large time behavior, while the high frequency component has the exponential decay but with singularity, their combination leads to a global algebraic decay estimate. To use the decay property only of the low frequency component in the approximate Green function avoids the singularity in the high frequency component so that it simplifies and improves the previous works on this system. This new approach of the combination of the Green function and energy method through the frequency decomposition can also be applied to the hyperbolicparabolic systems satisfying the Kawashima condition, and also the systems whose derivatives of the coefficients have suitable time decay properties.

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