Asymptotics of solutions of a time-periodic boundary-value problem for a singularly perturbed nonlinear parabolic equation with rapidly oscillating coefficients (Q761625)

The authors consider the boundary value problem (1) \[ \epsilon \partial u^{\epsilon}/\partial t-(\partial /\partial x_ i)(a_{ij}(x,t,x/\epsilon)\partial u^{\epsilon}/\partial x_ j)+a_ 0(x,t,x/\epsilon)u^{\epsilon}=f(x,t,x/\epsilon,u^{\epsilon}),\quad x\in \Omega,\quad t\in R, \] \[ u^{\epsilon}(x,t+T)=u^{\epsilon}(x,t),\quad u^{\epsilon}(x,t)|_{\Gamma}=0, \] where \(\Omega\) is a bounded domain in \(R^ n\) with boundary \(\Gamma\), \(\epsilon >0\) is a smooth parameter. The coefficients of the equation are supposed to be periodic and smooth. It is assumed that \(| f| \leq f_ 0+c| u|^ r\), where \(f_ 0\in L_{2r}(Q_ T)\), \(c=const\), \(Q_ T=\Omega \times (0,T)\) and \(1<r<(n+1)/(n-1).\) Using the Leray-Schauder principle the existence of a unique solution \(u^{\epsilon}\in H^ 1(Q_ T)\) for (1) is proved. The asymptotic behavior as \(\epsilon\) \(\to 0\) of this solution is investigated. Sufficient conditions for strong convergence of the solution of this problem in the space \(L_ 2(0,T;H^ 1(\Omega))\) are found.