Convergence to Diffusion Waves for Nonlinear Evolution Equations with Ellipticity and Damping, and with Different End States

Abstract

In this paper, we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects:

$$ \left\{ {\begin{array}{*{20}l} {{\psi _{t} } \hfill} & {{ = - {\left( {1 - \alpha } \right)}\psi - \theta _{x} + \alpha \psi _{{xx}} ,} \hfill} \\ {{\theta _{t} } \hfill} & {{ = - {\left( {1 - \alpha } \right)}\theta + \nu \psi _{x} + {\left( {\psi \theta } \right)}_{x} + \alpha \theta _{{xx}} ,} \hfill} \\ \end{array} } \right. $$

((E))

with initial data

$$ \left( \psi,\theta \right)\left( {x,0} \right) = \left( \psi _{0} \left( x \right),\theta _{0} \left( x \right) \right) \to \left( \psi _\pm ,\theta _\pm \right)\;{\rm as}\;x \to \pm \infty $$

((I))

where α and ν are positive constants such that α < 1, ν < 4α(1 − α). Under the assumption that $\left| \psi_{ + } - \psi_{ - } \right| + \left| \theta _{ + } - \theta _{ - } \right|$ is sufficiently small, we show the global existence of the solutions to Cauchy problem (E) and (I) if the initial data is a small perturbation. And the decay rates of the solutions with exponential rates also are obtained. The analysis is based on the energy method.