, Volume 55, Issue 6, pp 994-1014

Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity

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In this paper, we study the global existence and the asymptotic behavior of the solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects (E) $$ \left\{ \begin{aligned} & \psi _t = - (1 - \alpha )\psi - \theta _x + \alpha \psi _{xx} , \\ & \theta _t = - (1 - \alpha )\theta + \nu \psi _x + 2\psi \theta _x + \alpha \theta _{xx} , \\ \end{aligned} \right. $$ with initial data (I) $$ (\psi ,\theta )(x,0) = (\psi _0 (x),\theta _0 (x)) \to (\psi _ \pm ,\theta _ \pm )\quad {\text{as}}\quad x \to \pm \infty , $$ where α and ν are positive constants such that α < 1,ν < α (1−α). Through constructing a correct function \(\hat \theta (x,t)\) defined by (2.13) and using the energy method, we show \(\mathop {\sup }\limits_{x \in \mathbb{R}} (\left| {(\psi ,\theta )(x,t)\left| + \right.\left| {(\psi _x } \right.,\theta _x )(x,t)\left| {) \to 0} \right.} \right.\) as \(t \to \infty \) and the solutions decay with exponential rates. The same problem is studied by Tang and Zhao [10] for the case of (ψ±, θ±)  =  (0,0).

Received: November 18, 2003