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

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Abstract

This paper focuses on a class of fourth-order parabolic systems involving logarithmic and Rellich nonlinearities arising from modeling epitaxial thin film growth: $\left\{\begin{aligned} u_{t}+\Delta^{2} u & =|v|^{p}|u|^{p-2} u \ln |u v|-\mu \frac{u}{|x|^{4}}, \\ v_{t}+\Delta^{2} v & =|u|^{p}|v|^{p-2} v \ln |u v|-\gamma \frac{v}{|x|^{4}} . \end{aligned}\right.$ By using some new techniques to deal with the Rellich nonlinearities µ $\mu \frac{u}{|x|^{4}}$ and $\gamma \frac{v}{|x|^{4}}$, as well as the coupled logarithmic nonlinearities |v|^p|u|^p−2u ln |uv| and |u|^p|v|^p−2v ln |uv|, 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.

Keywords

epitaxial thin film growth / coupled fourth-order parabolic systems / global existence / blow-up / asymptotic behavior / 35A01 / 35B44 / 35K52

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