Subcritical nonlinear dissipative equations on a half-line

Abstract EN

-In this paper we are interested in the global existence and large time behavior of solutions to the initialboundary value problem for sub critical nonlinear dissipative equations ⎧⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎩ ut + N(u, ux) + Ku = 0, (x, t) ∈ R+ × R+, u(x, 0) = u0(x), x∈ R+, ∂j−1 x u(0, t) = 0 for j = 1, ..., m 2 (1) where the nonlinear term N(u, ux) depends on the unknown function u and its derivative ux and satisfy the estimate |N(u, v)| ≤ C |u|ρ |v|σ with σ ≥ 0, ρ ≥ 1, such that (σ + ρ − 1)n + 2 2n < 1 The linear operator K(u) is defined as follows Ku = m j=n aj∂jx u where the constants an, am ∈ 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.