Asymptotic behavior of entire solutions in space and time media

In space and time periodic media, the paper mainly focuses on the existence and the asymptotic behavior of entire solutions for reaction-diffusion equation with advection. We prove that there exists an entire solution tending to a time-periodic traveling wave solution of \(u_t=u_{xx}+f_1(t,u)\) $u_t=u_{\mathit{xx}}+f_1(t,u)$ (as \(t \rightarrow -\infty \) $t\to -\infty$ ), as t increases to \(+\infty \) $+\infty$ , it converges to a space-periodic traveling wave solution of \(u_t=u_{xx}+f_2(x,u)\) $u_t=u_{\mathit{xx}}+f_2(x,u)$ locally uniformly. Moreover, the two types of traveling wave solutions may have different speeds, and the time-periodic traveling wave solution is smaller than the space-periodic traveling wave solution (of course, they can be equal at \(\pm \infty \) $\pm \infty$ ). Finally, we give numerical simulations for the main results.