Local Well-Posedness And Blow-Up For The Half Ginzburg-Landau-Kuramoto Equation With Rough Coefficients And Potential

We study the Cauchy problem for the half Ginzburg- Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coecients 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.