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On a linear runs and tumbles equation. (English) Zbl 1357.35048
Summary: We consider a linear runs and tumbles equation in dimension $$d \geq 1$$ for which we establish the existence of a unique positive and normalized steady state as well as its asymptotic stability, improving similar results obtained by V. Calvez et al. [Kinet. Relat. Models 8, No. 4, 651–666 (2015; Zbl 1337.35155)] in dimension $$d=1$$. Our analysis is based on the Krein-Rutman theory revisited in [S. Mischler and J. Scher, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 33, No. 3, 849–898 (2016; Zbl 1357.47044)] together with some new moment estimates for proving confinement mechanism as well as dispersion, multiplicator and averaging lemma arguments for proving some regularity property of suitable iterated averaging quantities.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35Q92 PDEs in connection with biology, chemistry and other natural sciences 47D06 One-parameter semigroups and linear evolution equations 92C17 Cell movement (chemotaxis, etc.)
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