Ginzburg–Landau equation with fractional Laplacian on a upper- right quarter plane

Abstract

We consider the initial-boundary value problem for the Ginzburg–Landau equation with fractional Laplacian on a upper-right quarter plane 0\\\\ u\\left(0,\\mathbf{x}\\right) = u_{0}\\left(\\mathbf{x}\\right),\\mathbf{x\\in D},\\\\ u\\left\\vert _{x_{i} = 0}\\right. = h_{j}\\left(t,x_{j}\\right),\\ x_{j}>0,\\ j = 1,2,t>0, \\end{array} \\right. $?> {ut(t,x)−∇βu(t,x)=|u(t,x)|σu(t,x),x∈D,t>0u(0,x)=u0(x),x∈D,u|xi=0=hj(t,xj), xj>0, j=1,2,t>0, where D={x1>0,x2>0} , β∈(32,2), σ>0 and ∇β is a fractional Laplacian defined as ∇βu=∑j1Γ(2−β)∫0xjuyjyj(xj−yj)β−1dy. We study the main questions of the theory of IBV- problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV- problem in the L2− based Sobolev spaces to the case of a multidimensional domain. We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong. The lower order compatibility conditions between initial and boundary data are also discussed.