Non-coarsening for solutions to a class of evolution equations (Q2919599)

The authors consider the initial value problem involving an evolution equation in the form \(u_t=A_\sigma-f(u)\), \(t\geq 0\), \(u(0)=u_0\), where \(u:\Omega\times (0,\infty)\to\mathbb R\), \(\Omega\subset\mathbb R^N\) is compact, \(\sigma\) is a positive parameter, the operator \(A_\sigma\) is bounded, linear on \(L^\infty(\Omega)\), and satisfies several other assumptions, and \(f\) is a \(C^1\) function is of bistable type (typically, \(f(u)=u^3-u\)). The result from [\textit{D. B. Duncan}, \textit{M. Grinfeld} and \textit{I. Stoleriu}, Eur. J. Appl. Math. 11, No. 6, 561--572 (2000; Zbl 0973.65131)] concerning non-coarsening of solutions is generalized here. The idea of the proof is to take the initial datum \(u_0\) between a pair of two-phase stationary solutions, and so, by the comparison principle, the solution \(u(t,u_0)\) will remain trapped at each time. Several examples are considered, which show that two known results from the literature are special cases of the main theorem of the paper.