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

TL;DR

This paper analyzes a generalized Burger's equation in one dimension, exploring stationary solutions, bifurcation phenomena, and conditions for global existence or blow-up of solutions under various boundary conditions.

Contribution

It introduces a bifurcation analysis for stationary solutions and compares solution behaviors under different boundary conditions, extending understanding of nonlinear parabolic equations.

Findings

Existence of stationary solutions depends on parameters and boundary conditions.

Global existence can occur under Dirichlet, Neumann, or dissipative boundary conditions.

Conditions for blow-up and global solutions are characterized using super-solutions and weighted norms.

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.