Lower and upper solutions for the heat equation on a polygonal domain of \(\mathbb {R}^2\). (Q2444519)

The authors study the time-periodic Dirichlet boundary value problem for the semilinear heat equation \(u_t-\Delta u=f(x,t,u)\) in a polygonal domain \(\Omega \subset \mathbb {R}^2\), where the boundary is the union of a finite number of line segments. The nonlinearity is assumed to be \(L^p(0,T;L^p_\mu (\Omega ))\)-Carathéodory with a weight \(r^{\mu }\), where \(r^\mu(x)\) expresses the distance of \(x\) from the corners of \(\Omega \). They give assumptions on the nonlinearity which allow to prove the existence of solutions even in the situation that upper and lower solutions are not well-ordered. The crucial role plays the space of functions that are comparable to the first eigenvalue, \(\varphi _1\), of the linear problem. \(C_{\varphi _1}=\{ u\in C(\Omega \times (-\pi ,\pi ): \exists a>0\;\forall (x,t)\;| u(x,t)| \leq a\varphi _1(x,t)\}\). The paper extends the result obtained by the authors for the elliptic equation [J. Differ. Equations 244, No. 3, 599--629 (2008; Zbl 1138.35013)].