Type: Article

Whitham equation with Landau damping on a half-line

Felipe Benitez; Elena I. Kaikina;

Abstract:

We consider the initial-boundary value problem on a half-line for the evolution equation (partial derivative(t) + R(1/2)partial derivative(2)(x) + K)u(x, t) = -u(x)u, where R(alpha)phi = 1/2 Gamma(alpha)sin(pi/2 alpha) integral(+infinity)(0) phi(y)/vertical bar x - y vertical bar(1-alpha) dy is the modified Riesz potential and Ku = 1/2 pi i theta(x) integral(i infinity)(-i infinity) e(px)p root tanh vertical bar p vertical bar/vertical bar p vertical bar((u) over cap (p, t) - u(0, t)/p)dp, where theta(x) is the Heaviside step function. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.