Properties of global decaying solution to the Cauchy problem of nonlinear evolution equations (Q1001481)

The authors establish global existence and decaying results for the Cauchy problem for the following nonlinear evolution equations \[ \begin{aligned} \psi _t&= -(1-\alpha )\psi-\theta_x+ \psi\psi_x+ \alpha\psi_{xx}, \\ \theta _t&= -(1-\alpha )\theta+\nu\psi_x+ (\psi\theta)_x+ \alpha\theta_{xx}\end{aligned} \] with initial data of different end states \[ (\psi (x,0),\theta (x,0))= (\psi _0 (x),\theta _{0} (x))\to (\psi_\pm,\theta_\pm)\quad {\text{as}}\quad x \to {\pm}\infty. \] which displays the complexity between ellipticity and dissipation. Thanks to the smoothing effect of the parabolic operator, the authors obtain the regularity property and estimates when \(t > 0\) for the higher order spatial derivatives despite its relatively lower regularity of the initial data. Some decay estimates are also discussed.