A bifurcation for a generalized Burger's equation in dimension one

Abstract

We consider a generalized Burger's equation (dtu = dxxu - udxu + up - {\lambda}u)in a subdomain of R, under various boundary conditions. First, using some phase plane arguments, we study the existence of stationary solutions under Dirichlet or Neumann boundary conditions and prove a bifurcation depending on the parameters. Then, we compare positive solutions of the parabolic equation with appropriate stationary solutions to prove that global existence can occur for the Dirichlet, the Neumann or the dissipative dynamical boundary conditions {\sigma}dtu+d{\nu}u = 0. Finally, for many boundary conditions, global existence and blow up phenomena for solutions of the nonlinear parabolic problem in an unbounded domain are investigated by using some standard super-solutions and some weighted L1-norms.