Homogenization of linear and nonlinear ordinary differential equations with time depending coefficients (Q1126552)

Summary: The author studies the effective behaviour of solutions to the linear equation \(\partial_tu^\varepsilon+ a^\varepsilon(t, x)u^\varepsilon= 0\) and to the nonlinear nonhomogeneous Riccati equation \(\partial_tu^\varepsilon+ a^\varepsilon(t, x)(u^\varepsilon)^2= f(t, x)\), \(f>0\), when \(a^\varepsilon\rightharpoonup a\) in \(L^\infty\) weak\(^*\) and data at time \(t= 0\) are nonoscillating. In the first case, the limit is an integro-differential equation and the memory term is described by a resolvent Volterra equation. Existence and uniqueness for the solution to the effective equations are proved by passing to a kinetic formulation. On the other side the Riccati equation reveals the phenomenon of instantaneous memory, described through an asymptotic approach that consists in looking for an expansion by introducing a suitable parameter.