B-methods for the numerical solution of evolution problems with blow-up solutions part II: Splitting methods

Article Mathematics

Construction of minimal mass blow-up solutions to rough nonlinear Schrodinger equations

Yiming Su et al.

Summary: We study the focusing mass-critical rough nonlinear Schrodinger equations with controlled rough path stochastic integration. We construct minimal mass blow-up solutions in dimensions one and two, which behave similarly to pseudo-conformal blow-up solutions near the blow-up time. Furthermore, we establish global well-posedness for initial data with mass below the ground state. These results show that the mass of the ground state is the exact threshold for global well-posedness and blow-up in the stochastic focusing mass-critical case. Similar results are also obtained for a class of nonlinear Schrodinger equations with lower order perturbations.

JOURNAL OF FUNCTIONAL ANALYSIS (2023)

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Article Mathematics, Applied

Blow up in a periodic semilinear heat equation

M. Fasondini et al.

Summary: The blow-up phenomenon in a one-dimensional semilinear heat equation is investigated using numerical and analytical tools. The focus is on periodic problems initialized with a nearly flat, positive profile. Novel results provide asymptotic approximations of the solution on different timescales, which are valid over the entire space and time interval until the blow-up occurs.

PHYSICA D-NONLINEAR PHENOMENA (2023)

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Article Mathematics, Applied

Refined asymptotics for the blow-up solution of the complex Ginzburg-Landau equation in the subcritical case

Giao Ky Duong et al.

Summary: In this paper, we aim to refine the blow-up behavior of the complex Ginzburg-Landau (CGL) equation in a subcritical case. Specifically, we construct blow-up solutions and refine their blow-up profile to the next order.

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE (2022)

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Article Mathematics