  In this paper, we introduce a general constructive method to compute
solutions of initial value problems of semilinear parabolic partial
differential equations via semigroup theory and computer-assisted proofs. Once
a numerical candidate for the solution is obtained via a finite dimensional
projection, Chebyshev series expansions are used to solve the linearized
equations about the approximation from which a solution map operator is
constructed. Using the solution operator (which exists from semigroup theory),
we define an infinite dimensional contraction operator whose unique fixed point
together with its rigorous bounds provide the local inclusion of the solution.
Applying this technique for multiple time steps leads to constructive proofs of
existence of solutions over long time intervals. As applications, we study the
3D/2D Swift-Hohenberg, where we combine our method with explicit constructions
of trapping regions to prove global existence of solutions of initial value
problems converging asymptotically to nontrivial equilibria. A second
application consists of the 2D Ohta-Kawasaki equation, providing a framework
for handling derivatives in nonlinear terms.
