A bifurcation for a generalized Burgers' equation in dimension one

Abstract

We consider the generalized Burgers' equation \begin{eqnarray*} \left\{ \begin{array}{ll} \partial_t u = \partial_x^2u - u \partial_x u + u^p - \lambda u &\textrm{ in } \overline{\Omega} \textrm{ for } t>0, \\ \mathcal{B}(u)=0 & \textrm{ on } \partial \Omega \textrm{ for } t>0, \\ u(\cdot,0) = \varphi \geq 0 & \textrm{ in } \overline{\Omega}, \end{array} \right. \end{eqnarray*} with $p>1$, $\lambda \in \mathbb{R}$, $\Omega$ a subdomain of $\mathbb{R}$, and where $\mathcal{B}(u)=0$ 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 $\mathcal{B}(u)=0$ stands for the Dirichlet, the Neumann or the dissipative dynamical boundary conditions $\sigma \partial_t u + \partial _\nu u=0$. Finally, for many boundary conditions, global existence and blow up phenomena for solutions of the nonlinear parabolic problem in an unbounded domain $\Omega$ are investigated by using some standard super-solutions and some weighted $L^1-$norms.