Global solution of nonlinear Heat equation with solutions in a Hilbert Manifold

Abstract

The objective of this paper is to deal with the deterministic problem consisting of non-linear heat equation of gradient type. The evolution equation emerges as projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearity of degree 2n−1, onto the tangent space of a sphere M in a Hilbert space ℋ. We are going to deal with questions of the existence and the uniqueness of a global solution, and the invariance of the manifold, M i.e. if the suitable initial data lives on M then all trajectories of solutions also belong to M. Finally, we will also show the global solution generates a gradient flow.