# ProfessorMichaelRuzhansky

Faculty of Natural SciencesDepartment of Mathematics

Visiting Professor

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### Contact

+44 (0)20 7594 8500m.ruzhansky CV

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### Location

615Huxley BuildingSouth Kensington Campus

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# Citation

## BibTex format

@article{Ruzhansky:2018:10.1016/j.jde.2018.06.033,author = {Ruzhansky, M and Tokmagambetov, N},doi = {10.1016/j.jde.2018.06.033},journal = {Journal of Differential Equations},title = {Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg  group and for Rockland operators on graded Lie groups},url = {http://dx.doi.org/10.1016/j.jde.2018.06.033},year = {2018}}

## RIS format (EndNote, RefMan)

TY  - JOURAB  - In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub-Laplacian on the Heisenberg group. In the case of the positive mass, we show the global in time well-posedness for small data for power like nonlinearities. We also obtain similar well-posedness results for the wave equations for Rockland operators on general graded Lie groups. In particular, this includes higher order operators on and on the Heisenberg group, such as powers of the Laplacian or the sub-Laplacian. In addition, we establish a new family of Gagliardo-Nirenberg inequalities on graded Lie groups that play a crucial role in the proof but which are also of interest on their own: if $G$ is a graded Lie group of homogeneous dimension $Q$ and $a>0$, $1<r<\frac{Q}{a},$ and $1\leq p\leq q\leq \frac{rQ}{Q-ar},$ then we have the following Gagliardo-Nirenberg type inequality $$\|u\|_{L^{q}(G)}\lesssim \|u\|_{\dot{L}_{a}^{r}(G)}^{s} \|u\|_{L^{p}(G)}^{1-s}$$ for $s=\left(\frac1p-\frac1q\right)\left(\frac{a}Q+\frac1p-\frac1r\right)^{-1}\in [0,1]$ provided that$\frac{a}Q+\frac1p-\frac1r\not=0$, where $\dot{L}_{a}^{r}$ is the homogeneous Sobolev space of order $a$ over $L^r$. If $\frac{a}Q+\frac1p-\frac1r=0$, we have $p=q=\frac{rQ}{Q-ar}$, and then the above inequality holds for any $0\leqs\leq 1$.AU  - Ruzhansky,MAU  - Tokmagambetov,NDO  - 10.1016/j.jde.2018.06.033PY  - 2018///SN  - 0022-0396TI  - Nonlinear damped wave equations for the sub-Laplacian on the Heisenberg  group and for Rockland operators on graded Lie groupsT2  - Journal of Differential EquationsUR  - http://dx.doi.org/10.1016/j.jde.2018.06.033UR  - http://arxiv.org/abs/1703.07902v1UR  - http://hdl.handle.net/10044/1/61851ER  -