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{array}{*{20}{c}} {{u_t} + {\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} + {\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}} \right. $
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.
Citation: Tingfu Feng, Yan Dong, Kelei Zhang, Yan Zhu. Global existence and blow-up to coupled fourth-order parabolic systems arising from modeling epitaxial thin film growth[J]. Communications in Analysis and Mechanics, 2025, 17(1): 263-289. doi: 10.3934/cam.2025011
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{array}{*{20}{c}} {{u_t} + {\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} + {\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}} \right. $
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.
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Tingfu Feng, Yan Dong, Kelei Zhang, Yan Zhu. Global existence and blow-up to coupled fourth-order parabolic systems arising from modeling epitaxial thin film growth[J]. Communications in Analysis and Mechanics, 2025, 17(1): 263-289. doi: 10.3934/cam.2025011
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