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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.

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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