A BIFURCATION FOR A GENERALIZED BURGERS' EQUATION IN DIMENSION ONE

We consider the generalized Burgers' equation GRAPHICS with p > 1, lambda is an element of R, Omega a subdomain of R, and where B(u) = 0 denotes some boundary conditions. First, using some phase plane arguments, we study the existence of stationary solutions under the Dirichlet or the Neumann boundary conditions and prove a bifurcation depending on the parameter lambda. Then, we compare positive solutions of the parabolic equation with appropriate stationary solutions to prove that global existence can occur when B(u) = 0 stands for the Dirichlet, the Neumann or the dissipative dynamical boundary conditions sigma partial derivative(t)u + partial derivative(V)u = 0. Finally, for many boundary conditions, global existence and blow up phenomena for solutions of the nonlinear parabolic problem in an unbounded domain Omega are investigated by using some standard super-solutions and some weighted L-1 norms.