Mixed Initial-Boundary Value Problem for the Capillary Wave Equation
Abstract: We study the mixed initial-boundary value problem for the capillary wave equation:iut+u2u=∂x3/2u, t>0, x>0; u(x,0)=u0(x), x>0; u(0,t)+βux(0,t)=h(t), t>0, where∂x3/2u=(1/2π)∫0∞signx-y/x-yuyy(y) dy. We prove the global in-time existence of solutions of IBV problem for nonlinear capillary equation with inhomogeneous Robin boundary conditions. Also we are interested in the study of the asymptotic behavior of solutions.
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References 17 publications
Ginzburg–Landau equation with fractional Laplacian on a upper- right quarter plane
We consider the initial-boundary value problem for the Ginzburg–Landau equation with fractional Laplacian on a upper-right quarter plane { u t ( t , x ) − ∇ β u ( t , x ) = | u ( t , x ) | σ u ( t , x ) , x ∈ D , t > 0 u ( 0 , x ) = u 0 ( x ) , x ∈ D , u | x i = 0 = h j ( t , x j ) , x j > 0 , j = 1 , 2 , t > 0 , where D = { x 1 > 0 , x 2 > 0 } , β ∈ ( 3 2 , 2 ) , σ > 0 and ∇ β is a fractional Laplacian defined as ∇ β u = ∑ j 1 Γ ( 2 − β ) ∫ 0 x j u y j y j ( x j − y j ) β − 1 d y . 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 L 2 − 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.
Ginzburg–Landau equation with fractional Laplacian on a upper- right quarter plane
We consider the initial-boundary value problem for the Ginzburg–Landau equation with fractional Laplacian on a upper-right quarter plane { u t ( t , x ) − ∇ β u ( t , x ) = | u ( t , x ) | σ u ( t , x ) , x ∈ D , t > 0 u ( 0 , x ) = u 0 ( x ) , x ∈ D , u | x i = 0 = h j ( t , x j ) , x j > 0 , j = 1 , 2 , t > 0 , where D = { x 1 > 0 , x 2 > 0 } , β ∈ ( 3 2 , 2 ) , σ > 0 and ∇ β is a fractional Laplacian defined as ∇ β u = ∑ j 1 Γ ( 2 − β ) ∫ 0 x j u y j y j ( x j − y j ) β − 1 d y . 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 L 2 − 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.
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