Global Existence Of Solutions For Thin-film Equation With The Long-wave Unstable Term

This paper mainly studies the properties of solutions of thin film equations with long-wave unstable term.It is widely used in physics,for example,it can be used to describe the height change of fluid on solid surface or the thickness change of film in cracks.The study provides important theoretical guidance for people to understand physical phenomena and study physical laws.Therefore,it is of great theoretical significance and application value to analyze the properties of this kind of the solution of the model.In this paper,the global existence of solutions of thin film equations with long-wave unstable term is studied for subcritical,critical and supercritical cases respectively.For the subcritical case,it is given that the solution of the model exists globally for any initial value.Specifically,firstly,the regularization problem is constructed,the local existence of non-negative entropy weak solutions for thin film equations is proved.Secondly,the uniform estimation of the H¹norm of the solution of model is proved by the Gagliardo-Nirenberg-Sobolev functional inequality with the best constant in high dimension.Finally,the global existence of the solution is proved by the uniform estimation of the H¹norm of the solution and iterative technique.In the critical case,it is proved that the solution of the model exists globally when M₀<M_c.Different from the subcritical case,the conclusion depends on the initial mass.Particularly,the H¹norm of the solution is proved by applying the Sz.Nagy inequality.For the supercritical case,when||h₀||_Lm_-n+2_（R）2is less than a certain constant,the solution of the model exists globally.Mainly,by using the techniques of G-N-S inequality with the best constant in high dimensions,the property of decreasing free energy and the decomposition of free energy,the critical index of global existence is given,then the uniform boundedness of the H¹norm of the solution is proved.In this paper,it is found that the existence of solutions of thin film equations with long-wave unstable term depends not only on the relationship between the mobility co-efficient n and the reverse diffusion index m,but also on the initial mass.So this paper discusses the global existence of the solution of the model in detail from these two angles.</sub>