Dirichlet inhomogeneous boundary value problem for the multidimensional fractional Ginzburg-Landau equation

We consider the initial-boundary value problem for the complex Ginzburg-Landau equation with fractional Laplacian on a hyperoctant $$ \{ {array}{c}uₜ(t,x)-∇ βu(t,x)=| u(t,x)| σu(t,x),\, x∈ Dⁿ,\, t>0,\\ u(0,x)=u₀(x),\, x∈ Dⁿ,\\ u _{xⱼ=0} =hⱼ(t,xⱼ),\, j=1,2,...,n,\, t>0, {array} . $$ u t ( t , x ) - ∇ β u ( t , x ) = u ( t , x ) σ u ( t , x ) , x ∈ D n , t > 0 , u ( 0 , x ) = u 0 ( x ) , x ∈ D n , u | x j = 0 = h j ( t , x j ) , j = 1 , 2 , . . . , n , t > 0 , where $$Dⁿ=( x=\{ xⱼ\} ⱼ₌₁ⁿ; xⱼ>0) ,$$ D n = x = x j j = 1 n ; x j > 0 , $$xⱼ=(x₁,x₂,...,xⱼ₋₁,xⱼ₊₁,...,xₙ),$$ x j = ( x 1 , x 2 , . . . , x j - 1 , x j + 1 , . . . , x n ) , $$n∈ N$$ n ∈ N , $$β ∈ ( 3/2,2) , \, σ >0$$ β ∈ 3 2 , 2 , σ > 0 and $$∇ β$$ ∇ β is a fractional Laplacian defined as $${aligned} ∇ βu=∑ ⱼ₌₁ⁿ1/ (2-β )∫ ₀^{xⱼ}{u_{yⱼyⱼ}}{( xⱼ-yⱼ) β -1}dyⱼ. {aligned}$$ ∇ β u = ∑ j = 1 n 1 Γ ( 2 - β ) ∫ 0 x j u y j y j x j - y j β - 1 d y j . We study the main questions of the theory of IBV problems for nonlocal equations: (1) the existence and uniqueness of solutions, (2) the asymptotic behavior of the solution for large time, (3) the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV problems in the L² L 2 based Sobolev spaces to the case of a Dirichlet problem. We give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong in order to minimize the compatibility conditions for initial and boundary data. Our approach is based on the Riemann-Hilbert theory and the multidimensional theory of Laplace transforms.