Shaomei Fang

The global existence and time-decay for the solutions of the fractional pseudo-parabolic equation☆

Abstract We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space R n , n ≥ 1 . Here, the fractional order α is related to the diffusion-type source term behaving as the usual diffusion term on the high frequency part. It has a feature of regularity-gain and regularity-loss for α > 1 and 0 α 1 , respectively. We establish the global existence and time-decay rates for small-amplitude solutions to the Cauchy problem for α > 0 . In the case that 0 α 1 , we introduce the time-weighted energy method to overcome the weakly dissipative property of the equation.

Large time behavior of solutions for a Cauchy problem on a nonlinear conservation law with large initial data in the whole space: Large time behavior of solutions for a Cauchy problem on a nonlinear conservation law with large initial data in the whole space

We consider the Cauchy problem on a nonlinear conversation law with large initial data. By Green’s function methods, energy methods, Fourier analysis, frequency decomposition, pseudo-differential operators, we obtain the global existence and the optimal decay estimate of t.