# Decay rates of solutions to dissipative nonlinear evolution equations with ellipticity

• Changjiang Zhu
• Zhian Wang
Original Paper

DOI: 10.1007/s00033-004-3117-9

Zhu, C. & Wang, Z. Z. angew. Math. Phys. (2004) 55: 994. doi:10.1007/s00033-004-3117-9

## Abstract.

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
\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.
(E)
with initial data
$$(\psi ,\theta )(x,0) = (\psi _0 (x),\theta _0 (x)) \to (\psi _ \pm ,\theta _ \pm )\quad {\text{as}}\quad x \to \pm \infty ,$$
(I)
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).

35B4035F2535K45

### Keywords.

Decay ratesenergy methodcorrect functiona priori estimates