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

Original Paper

DOI: 10.1007/s00033-004-3117-9

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

- 16 Citations
- 62 Downloads

## 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
with initial data
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

$$
\left\{ \begin{aligned}
& \psi _t = - (1 - \alpha )\psi - \theta _x + \alpha \psi _{xx} , \\
& \theta _t = - (1 - \alpha )\theta + \nu \psi _x + 2\psi \theta _x + \alpha \theta _{xx} , \\
\end{aligned} \right.
$$

(E)

$$
(\psi ,\theta )(x,0) = (\psi _0 (x),\theta _0 (x)) \to (\psi _ \pm ,\theta _ \pm )\quad {\text{as}}\quad x \to \pm \infty ,
$$

(I)

_{±}, θ_{±}) = (0,0).### Mathematics Subject Classification (2000).

35B4035F2535K45### Keywords.

Decay ratesenergy methodcorrect function*a priori*estimates

## Copyright information

© Birkhäuser Verlag, Basel 2004