The authors discuss the Cauchy problem governed by a transport equation with Maxwell boundary conditions arising in a growing cell population: \[ \frac{\partial \psi }{\partial t}(\mu ,v,t)=-v\frac{\partial \psi }{\partial \mu }(\mu ,v,t)-\sigma (\mu ,v)\psi (\mu ,v,t)+ \]
\[ +\int\limits_0^c {r(\mu ,v,{v}')\psi (\mu ,{v}',t)\,d{v}'} =A_K \psi (\mu ,v,t)= \]
\[ =S_K \psi (\mu ,v,t)+B\psi (\mu ,v,t), \]
\[ \psi (\mu ,v,0)=\psi_0 (\mu ,v), \]
\[ \psi | {_{\Gamma_0 } } =K(\psi | {_{\Gamma_a } } ), \] where \(\Gamma_0 =\{0\}\times [0,c], \quad \Gamma_a =\{a\}\times [0,c]\); \(K\) denotes a linear operator from a suitable space on \(\Gamma_a\) into a similar one on \(\Gamma_0\), \(A_K \) is the transport operator, \(S_K \) is the streaming operator and \(B\), the integral part of \(A_K \), is the collision operator.
The time asymptotic behavior of the solution to the above-mentioned problem on \(L^1([0,a]\times [0,c],d\mu\,dv)\) is discussed. This paper continues the analysis of the work B. Lods and M. Sbihi [Math. Methods Appl. Sci. 29, No. 5, 499–523 (2006; Zbl 1092.35057)], where the time asymptotic behavior of the solution to the considered problem on \(L^p([0,a]\times [0,c],d\mu\,dv)\)-spaces with \(1 < p < \infty \) was discussed.
The paper consists of four sections. Section 1 is an introduction. In Section 2, the functional setting of the problem is introduced, the different notations and facts needed in the sequel are fixed and the main result of the paper is stated. The proof of the main result is the topic of Section 3, while Section 4 is devoted to some technical lemmas.