We prove that a class of A-stable symplectic Runge–Kutta time semidiscretizations (including the Gauss–Legendre methods) applied to a class of semilinear Hamiltonian PDEs which are well-posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. As a consequence, such timesemidiscretizations conserve the m...

This paper presents a method of numerical verification for existence of a global-in-time solution to a class of semilinear parabolic equations. Such a method is based on two main theorems in this paper. One theorem gives a sufficient condition for proving existence of a solution to the semilinear parabolic equations with the initial point t = t′ ≥ 0. If the sufficient condition does not hold, t...

We consider a class of semilinear elliptic problems in twoand three-dimensional domains with conical points. We introduce Sobolev spaces with detached asymptotics generated by the asymptotical behaviour of solutions of corresponding linearized problems near conical boundary points. We show that the corresponding nonlinear operator acting between these spaces is Fréchet differentiable. Applying ...

In this paper we consider optimal control of stochastic semilinear equations with linearly increasing drift and cylindrical noise. We show existence and uniqueness (up to an additive constant) of solutions to the stationary Hamilton-Jacobi equation associated with the cost functional given by the asymptotic average per unit time cost. As a consequence we nd the optimizing controls given in the ...

In this article, we generalize the main result obtained by Bahaj in [1]. Also our proof is shorter than the original proof.