Asymptotic behavior of periodic solutions of parabolic equations with weakly nonlinear perturbation (Q1907425)

The author considers the periodic problem for the operator differential equation \[ u'+ A(u)+ \varepsilon B(u)= f,\quad u(0)= u(T),\tag{1} \] with a small parameter \(\varepsilon> 0\). The solution of the problem (1) is sought in the space \(X= L_2(0, T; H)\), where \(H\) is a real Hilbert space; \(f\in X^*\). It is assumed that the operator \(A: X\to X^*\) is monotone, bounded and radially continuous and the operator \(B\) is pseudomonotone and weakly nonlinear. It is shown that independent of the solvability of the limiting problem \[ u'+ A(u)= f,\quad u(0)= u(T),\tag{2} \] the problem (1) with \(\varepsilon> 0\) is solvable and a sequence of its solutions corresponding to values \(\varepsilon\to 0\) defines a minimal correction to the right-hand side of the problem (2) turning it into a solvable one.