Rigorous numerics for nonlinear heat equations in the complex plane of time

Abstract

In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. Using a solution map operator, we construct a simplified Newton operator and show that it has a unique fixed point. The fixed point together with its rigorous bounds provides the local inclusion of the solution of the Cauchy problem. The local inclusion technique is then applied iteratively to compute solutions over long time intervals. This technique is used to prove the existence of a branching singularity in the nonlinear heat equation. Finally, we introduce an approach based on the Lyapunov-Perron method to calculate part of a center-stable manifold and prove that an open set of solutions of the Cauchy problem converge to zero, hence yielding the global existence of the solutions in the complex plane of time.

Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearity

“…Moreover, the present authors with H. Okamoto have studied the Cauchy problem of (2) under the same setting as Cho et al [COS16] in [TLJO19] and have proved two results with computer-assistance. The first one is that there exists a branching singularity at the blow-up time, which extends the mathematical results by Masuda and mathematically proves numerical observations by Cho et al The second one is global existence of the solution to the Cauchy problem of (1) for some θ.…”

“…Furthermore our current proof of unbounded solutions does not establish finite time blowup. In the case θ = 0 we proved in [TLJO19] that the initial data u 0 (x) = 50(1 − cos(2πx)) blows up at time t * ∈ (0.0116, 0.0145) using rigorous numerics. This argument followed the approach of Masuda, wherein we integrated the solution in the complex plane of time to establish a branching singularity.…”

“…In this section we briefly introduce a method of rigorous numerical integration for (1) in forward time. This is firstly proposed in [TLJO19] and then slightly improved for integrating the solution of the nonlinear Schrödinger (NLS) case (θ = π/2) in [JLT20]. For proving the existence of heteroclinic orbits for (1), this integrator will propagate the rigorous inclusion of 1-complex dimensional unstable manifold W u, λ,θ loc (ã, b), which is given by the parameterization method described in Section 2.…”

“…When θ ∈ (−π/2, π/2) then all of the other eigenvalues in the linearization have negative real part. In [TLJO19], we characterized an explicit trapping region about the zero equilibrium using a Lyapunov-Perron argument to demonstrate a stable foliation of the region B(ρ 0 ). However this analysis depended on the eigenvalues having a negative real part, and could not be extended to the case θ = ±π/2.…”

“…The rest of the paper is organized as follows: In the following three sections, we briefly show three separate fundamental techniques of computer-assisted proofs. These have been introduced in previous studies [TLJO19,JLT20], and some minor modifications are made for the current problem. In Section 2 we introduce the parameterization method for 1-complex dimensional unstable manifolds of equilibrium to (1).…”