Finite time blow-up and global existence for the nonlocal complex Ginzburg–Landau equation

Well-posedness of nonlocal Ginzburg-Landau type equations

“…This equation is originally introduced by Ginzburg and Landau [1] in order to give a mathematical model of superconductivities.The GL equation (GLE), which arises in many mechanical, physical and chemical phenomena, is an important nonlinear evolution equation and has many concrete types. The nonlocal complex GLEs have been used to model electrochemical systems near a supercritical Hopf bifurcation, especially, the reaction-di¤usion systems consisted of oscillators with nonlocal coupling related to di¤usive chemical components [2][3][4][5][6]. In this type results physically and mathematically, the key feature aroused there is the phenomenon of blowing up that solutions become unstable to collapse with time.…”

“…We consider here, more general GL type equation with variable coe¢cients in convolution form. Note that such kind of result was investigated in [6] in particular case, when a (x) 1, g (u) = u n and b (x) = . In the present paper, local and global well-psedness the L p -regularity in terms of convolution, and blow up properties to problem (1:1) is obtained under growth assumptions on coe¢cients functions.…”

Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

“…To date, the GLE has been widely used in the physics fields of superconductivity and superfluidity [3][4][5][6][7][8][9][10][11]. The GLE is used also for describing a number of nonlinear phenomena [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31], such as biophysics systems, nonlinear phenomena in liquid crystals, spatio-temporal chaos, oscillations of different nature (see review papers [32][33][34], where associated references can be found). Additionally, the GLE plays an essential role in nonlinear optics.…”