Local Well-posedness and Blow-up for the Half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential

TL;DR

This paper investigates the well-posedness and blow-up phenomena for the half Ginzburg-Landau-Kuramoto equation with rough coefficients, establishing conditions for solution blow-up and employing advanced operator estimates.

Contribution

It introduces new techniques for analyzing the hGLK equation with rough coefficients, including commutator estimates and self-adjointness results, to understand solution behavior.

Findings

Solutions blow up under certain conditions with non-positive nonlinearity

Established essential self-adjointness of the elliptic operator with rough metrics

Developed commutator estimates for rough coefficient operators

Abstract

We study the Cauchy problem for the half Ginzburg-Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coefficients and potential type perturbation. The blow-up of solutions for hGLK equation with non-positive nonlinearity is shown by an ODE argument. The key tools in the proof are appropriate commutator estimates and the essential self-adjointness of the symmetric uniformly elliptic operator with rough metric and potential type perturbation.