A periodic boundary value problem for a generalized 2D Ginzburg-Landau equation
The authors consider the following generalized Ginzburg-Landau equation (gGL) in 2 spatial dimensions \[ u_t=\alpha_0u+\alpha_1\triangle u+\alpha_2| u|^2u+\alpha_3| u|^2u_x+\alpha_4| u|^2\overline{u}_x+\alpha_5| u|^2u_y+\alpha_6| u|^2\overline{u}_y-\alpha_7| u|^4u\tag{1} \] with periodic boundary condition and initial condition \[ u(x,y+L)=u(x,y-L),\;u(x+L,y)=u(x-L,y),\;u(x,y,0)=f(x)\in H^2(\Omega). \] Here \(\alpha_j=a_j+ib_j,\;j=0,\cdots,7\). For the sake of simplicity, we set \(b_0=0\). Also, both \(Re\alpha_1\) and \(Re\alpha_7\) must be positive, otherwise the solutions either blow up at a finite time or the equation is ill-posed. The authors impose the following two conditions \[ (| b_4-b_3| +| b_6-b_5|)^2<4a_1a_7,\quad a_1>0,a_7>0, \tag{2} \] \[ \sum_{j=3}^6 |\alpha_j|^6 < a_1 \left(\frac{3}{2}-\sqrt{1+(\frac{b_7}{a_7})^2}\right). \tag{3} \] They prove the following theorem: Assume both (2) and (3) hold and \(u(x,y)\in H^3\). Then the gGL equation (1) has a unique classical solution \(u\in L^0(H^2)\cap C^1(L^0)\). Also a blow-up of solutions is found via numerical simulation.
scientific article; zbMATH DE number 1130727