Decay rates of solutions to the Cauchy problem for dissipative nonlinear evolution equations (Q866530)

The authors consider the following system of nonlinear evolution equations \[ \begin{aligned} \psi_t&=-(1-\alpha)\psi-\theta_x+ \alpha\psi_{xx},\\ \theta_t&=-(1-\alpha)\theta +\mu \psi_x +(\psi\theta)_x + \alpha \theta_{xx}, \end{aligned} \] with initial data \((\psi,\theta)(x,0)=(\psi_0(x),\theta_0(x))\to (\psi_{+(-)},\theta_{+(-)})\) as \(x\to +(-)\infty\). They study the global existence and the asymptotic behavior of the solutions to the Cauchy problem above in case \(\alpha\) and \(\mu\) are positive constants and \(\alpha <1\), \(\mu<\alpha (1-\alpha)\). Using the energy method, the authors prove that \[ \sup_{x\in \mathbb R}(| (\psi,\theta)(x,t)| +| (\psi_x,\theta_x)(x,t)| )\to 0 \] as \(t\to \infty\) and the solutions decay with exponential rates.