Global existence and blow-up to coupled fourth-order parabolic systems arising from modeling epitaxial thin film growth

This paper focuses on a class of fourth-order parabolic systems involving logarithmic and Rellich nonlinearities arising from modeling epitaxial thin film growth:

$$\{\begin{array}{c}{u}_t+{\mathrm{\Delta }}^{2}u={\left|v\right|}^p{\left|u\right|}^{p-2}u\ln \left|uv\right|-\mu \frac{u}{{\left|x\right|}^4},\\ v_t+{\mathrm{\Delta }}^{2}v={\left|u\right|}^p{\left|v\right|}^{p-2}v\ln \left|uv\right|-\gamma \frac{v}{{\left|x\right|}^4}.\end{array}$$

By using some new techniques to deal with the Rellich nonlinearities $\mu \frac{u}{{\left|x\right|}^4}$ and $\gamma \frac{v}{{\left|x\right|}^4}$, as well as the coupled logarithmic nonlinearities ${\left|v\right|}^p{\left|u\right|}^{p-2}u\ln \left|uv\right|$ and ${\left|u\right|}^p{\left|v\right|}^{p-2}v\ln \left|uv\right|$, we prove the global existence and finite time blow-up of weak solutions. Furthermore, we not only obtain a new algebraic decay estimate and study the behavior of global weak solutions, but we also derive a new upper bound estimate for the blow-up time in case of the occurrence of blow-up.