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SUBCRITICAL NONLINEAR DISSIPATIVE EQUATIONS ON A HALF-LINE

机译：半线上的次子非线性耗散方程

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摘要

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) + K_u = 0, (x,t)R~+×R~+, u(x,0) = u_0(x), xR+, (1) e_x~(j-1)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 |N(u,u)|≤C|c|~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 =∑_(j=n)~ma_je_x~ju where the constants a_n,a_m 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.
机译：在本文中，我们对次临界非线性耗散方程u_t + N（u，u_x）+ K_u = 0，（x，t）R〜+的初边值问题的解的整体存在和长时间行为感兴趣。 ×R〜+，u（x，0）= u_0（x），xR +，（1）e_x〜（j-1）u（0，t）= 0对于j = 1，...，m / 2其中非线性项N（u，u_x）取决于未知函数u及其导数u_x，并且满足估计| N（u，u）|≤C| c |〜ρ| v |〜|σ且σ≥0，ρ ≥1，使得（σ+ρ-1）n + 2 / 2n <1线性算子K（u）定义如下Ku = ∑_（j = n）〜ma_je_x〜ju其中常数a_n，a_m R ，n，m是整数，m> n。本文的目的是证明初边值问题（1）的解的全局性。当方程的非线性项的时间衰减率小于线性项的时间衰减率时，我们找到了次临界情况下解的渐近表示的主要项。并给出了一些一般的方法来获得次临界情况下初边值问题解的整体存在性，并阐述了一般的充分条件以获得解的渐近展开。

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