We study the following initial value problem:

, (1)

 

where , is a bounded domain with smooth boundary, is in , with either Neumann or Dirichlet boundary conditions. This problem describes a reaction-diffusion system with equal diffusion coefficients defined inside an n-dimensional cylinder .

We assume that , so is a trivial solution of above equations.

If there exists a function such that

then we called it reaction-diffusion system of gradient type.

Traveling solutions play an important role in such systems and a lot of researchers study related topics, including existence and stability of traveling wave solutions in vary domains and corresponding wave speed.

Furthermore, following from ideas in Muratov's study \cite{muratov2004global}, we introduce a wide class of solutions, called "wave-like" solutions, and  show that well-defined instantaneous propagation speed for such solutions monotone converges to a limit at long time. Furthermore, with more assumptions, we could show it converges to traveling wave solution with such speed.