Averaging of a boundary-value problem, periodic with respect to time, for a singularly perturbed weakly nonlinear parabolic equation (Q761624)

The authors consider a boundary value problem with discontinuous coefficients \[ \partial u^{\epsilon}/\partial t-({\bar \partial}/\partial x_ i)(a_{ij}(x,t,x/\epsilon)du^{\epsilon}/dx_ j)+a_ 0(x,t,x/\epsilon)u^{\epsilon}=f(x,t,x/\epsilon,u^{\epsilon}), \] \(x\in \Omega\), \(t\in R\), \(u^{\epsilon}(x,t+T)=u^{\epsilon}(x,t),\quad u^{\epsilon}(x,t)|_{\Gamma}=0\) and investigate the behavior of its solution for \(\epsilon \to +0\). It is supposed that such inequalities \(a_{ij}(x,t,y)\xi_ i\xi_ j\geq \alpha_ 0| \xi |^ 2,\quad a_ 0(x,t,y)\geq 0\) \(\forall (\xi,y)\in R^ n,\) \(x\in \Omega\), \(t\in R\), \(\alpha_ 0=const>0\), are fulfilled and \(a_{ij}(x,t,y)\), \(a_ 0(x,t,y)\), f(x,t,y,u) periodic with respect to \(y\in R^ n\) with the periodicity parallelepiped Y and with respect to t with period T.