On the existence and uniqueness for an age-dependent population model with nonlinear growth. Bouzinab, A. & Arino, O. Facta Univ. Ser. Math. Inform., 1993. abstract bibtex The authors study an age-dependent population model having the form $$∂u\over ∂a (a, t)+ ∂u\over ∂t (a, t)= -\lambda(a, t, u) u,\quad a∈[0, A),\quad t\ge 0,$$ $$u(0, t)= \int\sp A\sb 0 f(a, t, u(a, t))da,\ t > 0,\ u(a, 0)= p(a),\ a∈[0, A).$$ For the function $\lambda(a, t, u)$ they assumed that it is non-decreasing, while the function $f(a, t, u)$ satisfies the estimate $f(a, t, u)\le Cu$, $a∈[0, A)$, $t> 0$. The main result gives existence and uniqueness of the solution. The proof of the existence is based on a suitable compactness argument.
@Article{BouzinabArino1993,
author = {Bouzinab, A. and Arino, Ovide},
title = {On the existence and uniqueness for an age-dependent population model with nonlinear growth},
journal = {Facta Univ. Ser. Math. Inform.},
year = {1993},
number = {8},
pages = {55--68},
issn = {0352-9665},
abstract = {The authors study an age-dependent population model
having the form $${\partial u\over \partial a} (a,
t)+ {\partial u\over \partial t} (a, t)= -\lambda(a,
t, u) u,\quad a\in [0, A),\quad t\ge 0,$$ $$u(0, t)=
\int\sp A\sb 0 f(a, t, u(a, t))da,\ t > 0,\ u(a, 0)=
p(a),\ a\in [0, A).$$ For the function $\lambda(a,
t, u)$ they assumed that it is non-decreasing, while
the function $f(a, t, u)$ satisfies the estimate
$f(a, t, u)\le Cu$, $a\in [0, A)$, $t> 0$. The main
result gives existence and uniqueness of the
solution. The proof of the existence is based on a
suitable compactness argument.},
classmath = {*35A05 General existence and uniqueness theorems (PDE) 35F30 Boundary value problems for first order nonlinear PDE },
fjournal = {Facta Universitatis. Series: Mathematics and Informatics},
keywords = {age-dependent population model},
mrclass = {35K55 (92D25)},
mrnumber = {96f:35070},
mrreviewer = {R. C. MacCamy},
reviewer = {V.Georgiev (Sofia)},
}
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