## Section: New Results

### Nonlinear dynamics of travelling waves for reaction-diffusion equations

#### Existence of waves for a nonlocal reaction-diffusion equation

Participant : Vitaly Volpert.

*In collaboration with I. Demin (Novartis Pharma in Basel, Switzerland)*.

We studied in [17] a nonlocal reaction-diffusion equation arising in population dynamics. The integral term in the nonlinearity describes nonlocal stimulation of reproduction. We proved existence of travelling wave solutions by the Leray-Schauder method using topological degree for Fredholm and proper operators and special a priori estimates of solutions in weighted HÃ¶lder spaces.

**Keywords:** *integro-differential equation, travelling waves, Leray-Schauder method*.

#### Linear stability analysis of reaction fronts propagation in liquids with vibrations

Participant : Vitaly Volpert.

*In collaboration with K. Allali, F. Bikany and A. Taik (University of Mohammedia, Morocco)*.

Influence of vibrations on the onset of convective instability of reaction fronts in a liquid medium was studied in [14] . The model consisted of a reaction-diffusion system coupled with the Navier-Stokes equations under the Boussinesq approximation. Linear stability analysis of the problem was fulfilled, and the convective instability boundary was found.

**Keywords:** *convective instability, reaction front propagation, vibrations*.