An \(L^p\) theory of invariant manifolds for parabolic partial differential equations on \(\mathbb{R}^d\) (Q1598376)

It is studied the existence of finite-dimensional invariant manifolds for nonlinear heat equations of the form \[ \partial u/\partial\tau=\Delta u+F(u,\nabla u)\quad\text{on }\quad {\mathbb R}^d\times [1,\infty). \] It is shown that in spite of the fact that the linearized equation has continuous spectrum extending from negative infinity to zero, there exist finite-dimensional invariant manifolds which control the long time asymptotics of solutions. It is considered the problem for these equations in the framework of weighted Sobolev spaces of \(L^p\) type. It is obtained an \(L^\infty\) estimate of the long-time asymptotics of solutions under natural assumptions on the nonlinear term \(F\) and their initial data.