## Spectral properties and time asymptotic behaviour of linear transport equations in slab geometry.(English)Zbl 1103.82027

The authors consider a one-dimensional transport equation in a slab, whose abstract form reads as follows: $\dot \psi(t) = \big( T_H + K) \psi(t), \qquad \psi^-(t) = H\psi^+(t).$ Here $$K$$ is a bounded perturbation, accounting for scattering phenomena, $$T_H$$ is a unbounded operator, accounting for streaming and absorption, and $$H$$ is a boundary operator relating the inflow $$\psi^-$$ with the outflow $$\psi^+$$. The functional setting of the evolution problem is that of $$L^p$$ functions on the phase-space, with $$p \in [1,\infty)$$.
In the case of perfect reflection or periodic boundary conditions, the authors are able to compute the explicit expression of the group $$e^{tT_H}$$. This allows them to prove the compactness of the second-order remainder term in the Dyson-Phillips expansion of the complete group $$e^{t(T_H+K)}$$. Thus, applying a result due to Mokhtar-Kharroubi [Mathematical topics in neutron transport theory, World Scientific (1997; Zbl 0997.82047)] one can conclude that $$e^{tT_H}$$ and $$e^{t(T_H+K)}$$ have the same essential type, meaning that the time asymptotic behaviour of the solution is determined by the part of $$e^{t(T_H+K)}$$ in a finite-dimensional space.
This result extends to the more difficult case of reflection/periodic boundary conditions analogous results obtained so far in the case of vacuum boundary conditions ($$H = 0$$).

### MSC:

 82C70 Transport processes in time-dependent statistical mechanics 47D06 One-parameter semigroups and linear evolution equations 35B40 Asymptotic behavior of solutions to PDEs 47N55 Applications of operator theory in statistical physics (MSC2000)

Zbl 0997.82047
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