In this paper, we study a class of dispersive wave equations on the Heisenberg group $$H^n$$ . Based Fourier transform , properties Laguerre functions and stationary phase lemma, establish decay estimates for semigroup given by $$e^{\textrm{it}\phi ({\mathscr {L}})}$$ where $$\phi : {\mathbb {R}}^+ \rightarrow {R}}$$ is smooth, $${\mathscr {L}}$$ sub-Laplacian Finally, using duality arguments, ...

Some uniform decay estimates are established for solutions of the following type retarded integral inequalities:y(t)?E(t,?)?y??+??tK1(t,s)?ys?ds+?t?K2(t,s)?ys?ds+?,t???0. As a simple example application, scalar functional differential equation x?=?a(t)x+B(t,xt) is considered, and global asymptotic stability proved under weaker conditions. Another ODE system x?=F0(t,x)+?i=1mFi(t,x(t?ri(t))) on R...

In the present paper we consider Cauchy-type problem associated to space-time fractional differential equation \begin{document}$ \partial_t u+\partial_t^{\beta}(-\Delta)^{1-\beta}u-\Delta u = g(t, x), \quad t>0, \ x\in \mathbb R^n $\end{document} with $ \beta\in (0, 1) $, where derivative \partial_t^{\beta} is in Caputo sense and (-\Delta)^{1-\beta} Laplace operator of order 1-\beta $. We pr...

In this work, we consider the Cauchy problem for a diffusive Oldroyd-B model in three dimensions. Some optimal time-decay rates of solutions are derived via analysis upper and lower estimates provided that initial data small absolute value Fourier transform velocity is bounded below away from zero low-frequency region. It worth noticing independent fluid viscosity or coefficient, which differen...

We prove some weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates. As application we obtain improved spherical average decay rates of the transform fractal measures, therefore improve results for Falconer distance set conjecture in three higher dimensions.