Families of stable manifolds for one-dimensional parabolic equations (Q1387289)

It is proved the existence of a family of stable manifolds associated with an invariant subset of the dynamical system generated by the one-dimensional parabolic equation \[ u_t=-Lu+B(u,t), \quad u(t_0,x)=\phi(x), \quad x\in (0,1) \] with periodic or homogeneous boundary conditions. The operator \(L\) is elliptic with smooth coefficients and the operator \(B(u,t)\) is given by the expression \(B(u,t)=f(u,u_x,x,t)\) where the function \(f\) is smooth and satisfies some essential conditions. The author considers only autonomous equations, but all results can be generalized to the nonautonomous case.