Recent development on multiplicity result in semilinear parabolic equations (Q2766303)

In this lecture the author considers the problem NEWLINE\[NEWLINE \begin{gathered} \frac{\partial u}{\partial t} - \Delta_x u = g(x,t,u)\quad \text{in }\mathbb{R}\times\Omega,\\ u(t,x)=0\text{ on }\mathbb{R}\times\partial\Omega,\quad u(0,x)=u(2\pi,x)\text{ on } \Omega . \end{gathered} NEWLINE\]NEWLINE Under the Ambrosetti-Prodi type conditions, the author investigates some situations of the solution absence, the existence at least one or at least two solutions, or at least one stable and one unstable solution. See also \textit{N. Hirano} and \textit{W. S. Kim} [Discrete Contin. Dyn. Syst. 2, No. 2, 271-280 (1996; Zbl 0948.35068); Nonlinear Anal., Theory Methods Appl. 26, No. 6, 1143-1160 (1996; Zbl 0902.35057)].NEWLINENEWLINEFor the entire collection see [Zbl 0973.00042].