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

We establish the global existence and decaying results for the Cauchy problem of nonlinear evolution equations, 'mathematical equation present' for initial data with different end states, 'mathematical equation present' which displays the complexity in between ellipticity and dissipation. Due to smoothing effect of the parabolic operator, we detail the regularity property and estimates when t > 0 for the higher order spatial derivatives despite its relatively lower regularity of the initial data. Also we discuss the decay estimates without the restriction of L1 bound as in Tang and Zhao [17], Wang [20]. Related to recent work by [15], our derivation may also establish the same estimates directly if under the same condition.