A bifurcation for a generalized Burgers equation in dimension one (Q428425)

Let \(\Omega\) be an interval of \(\mathbb{R}\) not necessary bounded, \(p>1,\) \(\lambda\in \mathbb{R}\) and \(\varphi\) a non-negative continuous function in \(\overline{\Omega}.\) The nonlinear parabolic problem NEWLINE\[NEWLINE\begin{cases} \partial_{t}u=\partial^{2}_{x}u-u\partial_{x}u+u^{p}-\lambda u,\,\,\,\, \text{in}\,\,\,\overline{\Omega}\,\,\,\text{for}\,\,\,t>0,\\ B(u)=0\,\,\,\, \text{on}\,\,\,\partial\Omega\,\,\,\, \text{for} \,\,\,t>0,\\ u(\cdot,0)=\varphi\,\,\,\, \text{in}\,\,\,\overline{\Omega}\\ \end{cases}\tag{1}NEWLINE\]NEWLINE is considered, where \(B(u)=0\) the Dirichlet or Neumann boundary conditions or the dynamic boundary conditions \(\sigma\partial_{t}u+\partial_{\nu}u=0\) with \(\sigma\geq 0\) a smooth function.NEWLINENEWLINEIn the first part of the article the author proves the bifurcating solutions existence for the stationary problem NEWLINE\[NEWLINEu''-uu'+u|u|^{p-1}-\lambda u=0\tag{2}NEWLINE\]NEWLINE with Dirichlet or Neumann conditions at the usage of phase plane arguments passing to the problem for the system of the first order. Then at the usage of the comparison method [\textit{J. von Below} and \textit{C. De Coster}, J. Inequal. Appl. 5, No. 5, 467--486 (2000; Zbl 0974.35060)] some regular super solutions to (1) are deduced from the solutions to (2) under one of indicated boundary conditions. In conclusion the global existence or blow up phenomena for solutions to (1) in an unbounded domain \(\Omega\) are investigated.