Type: Article

Subcritical nonlinear dissipative equations on a half-line

Felipe Benitez; Elena I. Kaikina; Hector F. Ruiz-Paredes;

Abstract:

In this paper we are interested in the global existence and large time behavior of solutions to the initial-boundary value problem for sub critical nonlinear dissipative equations {u(t) + N(u, u(x)) + Ku = 0, (x, t) is an element of R(+) x R(+), u(x, 0) = u(0)(x), x is an element of R(+), partial derivative(j-1)(x)u(0, t) = 0 for j = 1, ... , m/2 where the nonlinear term N(u, u(x)) depends on the unknown function u and its derivative u(x) and satisfy the estimate vertical bar N(u, v)vertical bar <= C vertical bar u vertical bar(rho) vertical bar v vertical bar(sigma) with sigma >= 0, rho >= 1, such that (sigma + rho - 1)n+2/2n < 1 The linear operator K(u) is defined as follows: Ku =Sigma(m)(j=n)a(j)partial derivative(j)(x)u where the constants a(n), a(m) is an element of R, n, m are integers, m > n. The aim of this paper is to prove the global existence of solutions to the initial-boundary value Problem (1). We find the main term of the asymptotic representation of solutions in sub critical case, when the nonlinear term of equation has the time decay rate less then that of the linear terms. Also we give some general approach to obtain global existence of solution of initial-boundary value problem in sub critical case and elaborate general sufficient conditions to obtain asymptotic expansion of solution.