Subcritical nonlinear nonlocal equations on a half-line

Abstract

We study nonlinear nonlocal equations on a half-line in the subcritical case (0.1) { ∂ t u + β | u | ρ u + K u = 0 , x > 0 , t > 0 , u ( 0 , x ) = u 0 ( x ) , x > 0 , ∂ x j − 1 u ( 0 , t ) = 0 , j = 1 , … , M , where β ∈ C , ρ ∈ ( 0 , α ) . The linear operator K is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol K ( p ) = E α p α , the number M = [ α 2 ] . The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) and to find the main term of the large time asymptotic representation of solutions in the subcritical case, when the time decay rate of the nonlinearity is less than that of the linear part of the equation.