The Optimal Initial Criterion For Global Existence And Blow-up Of Solutions To A Supercritical Thin-film Equation With The Long-wave Unstable Term

In this paper,we mainly study the best initial criterion to distinguish global exis-tence and blow-up of solutions for a supercritical thin-film equation with the long-wave unstable term,including the one-dimensional case and the high-dimensional case(d?2),respectively.For the one-dimensional case,we derive the best initial criterion between global existence and blow-up of solutions for the supercritical thin-film equation with a periodic boundary condition in Section 2.Concretely,by using the Gagliardo-Nirenberg-Sobolev functional inequality in one dimension and decomposing the free energy,it is proved that there is an exact constant such that(i)if ?h0?Lm-1/2 is less than the exact constant,then the L m-1/2 norm of solutions has a uniform upper bound;(ii)if ?h0?Lm-1/2 is lager than the exact constant,then the Lm+1 norm of solutions has a uniform lower bound.The exact constant is from the optimal constant of the Gagliardo-Nirenberg-Sobolev inequality.Furthermore,based on the uniform estimates,global existence and blow-up of solutions are proved.Therefore,we declare that the exact constant is the best initial criterion to distinguish global existence and blow-up of solutions.In the high-dimension,for the two cases that d=2,m>2 and d?3,1+2/d<m<d2+4/d(d-2),Section 3 studies the properties of solutions for the thin-film equation Cauchy problem.Specifically,the Gagliardo-Nirenberg-Sobolev in the high-dimension is used to determine the best initial criterion to distinguish global existence and blow-up of solutions.When the initial free energy is less than a positive constant,if ?h0?Lm+1 is more than the initial criterion,a finite time blow-up occurs for solutions;if ?h0?Lm+1 is less than the initial criterion and a global non-negative entropy weak solution exists,then the second moment goes to infinity as t?? or h(·,tk)(?)0 in L(m-1)d/2(Rd)for some subsequence tk??.This shows that a part of the mass spreads to infinity.