Recent development on multiplicity result in semilinear parabolic equations (Q2707984)

In the paper the author studies the following boundary-value problem for a nonlinear parabolic equation NEWLINE\[NEWLINE\begin{cases} \frac{\partial u}{\partial t}-\Delta_x u-\lambda_1 u+g(u)=s\phi(x)+h(t,x);\;x\in \Omega,\;t\in (0,2\pi),\\ u(t,x)=0\text{ on }(0,2\pi)\times\partial\Omega,\\ u(0,x)=u(2\pi,x)\text{ on }\Omega,\end{cases}NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain with \(C^2\)-smooth boundary, \(s\in\mathbb{R}\), \(\lambda_1\) is the first eigenvalue of \(-\Delta\) with zero boundary condition and \(\phi\) is the corresponding normalized eigenfunction, \(g\in C^1(\mathbb{R})\), and \(h : [0,2\pi]\times \Omega\to\mathbb{R}\) is orthogonal to \(\phi\). The purpose of the paper is to investigate the multiplicity and stability for weak solutions of the above problem. Precisely: specifying the regularity of \(h\) and assuming the coercive growth condition of \(g\) as well as its sublinear behavior at \(-\infty\), the author gives a result in the spirit of Ambrosetti-Prodi, i.e., he shows the existence of real parameters \(s_0,s_1\) such that the problem has no solutions for \(s<s_0\), has at least one solution for \(s= s_1\), and has at least one stable solution one unstable solution for \(s > s_1\). The paper uses topological methods such as the Mawhin version of the Leray-Schauder degree and the fixed point index on cones together with the method of upper and lower solutions. The provided results are discussed in detail and very well-placed in the recent development of the subject.