Applied Partial Differential Equations
Our group's efforts are focused on the study of partial differential equations with particular emphasis on traveling wave solutions, their stability and nonlinear dynamics, asymptotic decay of stable solutions, and existence of attractors. Our primary interest is in hyperbolic PDE and shock waves.
Stability and nonlinear dynamics of shock waves
We investigate the stability of shock-wave solutions in hyperbolic balance laws such as those arising in detonation theory, shallow-water flows, and continuum models of traffic (with L. Faria, R. Rosales, B. Seibold, R. Semenko, Y. Trakhinin.
We investigate the existence of steady solutions of the Gross-Pitaevskii equation in the presence of sources and sinks, such as those present in modeling exciton-polariton condensates (with J. Sierra, P. Markowich, R. Weishaupl)
Decay of solutions of hyperbolic and hyperbolic/parabolic systems
We analyze the rates of decays of dissipative hyperbolic and hyperbolic/parabolic systems arizing in elastic wave propagation using energy methods (with B. Said-Houari, R. Racke)
Blow-up of solutions of nonlinear wave equations
We analyze the blow-up of solutions in nonlinear wave equations (with B. Said-Houari)
We investigate the existence of attractors for systems of reaction-diffusion equations arising in population biology and investigate stability of stationary solutions (with R. Parshad)