Long-term Behavior And Convergence Of Solutions To Several Types Of Partial Differential Equation

In this paper,we first study the additive eigenvalue problem of Poisson equation with a prescribed contact angle boundary on the bounded region Ω (?) Rn with a smooth boundary.By considering the C0,C1 estimates of its perturbed equation,set (?)→0,we obtain the existence of the solution of the additive eigenvalue problem and the uniqueness of the solution:On this basis,the large-time behavior and convergence of the solution of the diffusion equation with a prescribed contact angle boundary condition are derived.Finally,the largetime behavior of the fully nonlinear equation with oblique derivative boundary condition is considered.The section arrangement is as follows:In section 1,this part mainly makes a brief introduction to the historical development of the problem studied in this paper.In section 2,we summarize relevant symbols and the required knowledge.In section 3,we first gives the convergence of the solution of the additive eigenvalue problem of Poisson’s equation with a prescribed contact angle boundary condition.In Section 4.we mainly proves the large-time behavior and convergence of the solution of the diffusion equation with a prescribed contact angle boundary condition,In section 5,we prove the large-time behavior of fully nonlinear equations with oblique derivative boundary conditions according to Scha.uder theory and maximum principle.The main conclusions of this paper are as follows:Theorem 1 Assume Ω (?) Rn(n≥ 2)be a bounded doma in with a smooth boundary,ν is the inner normal vector of ?Ω.For any (?),and | cos θ|≤b<1,there exist a unique τ ∈R and a smooth function (?)solving.(?)Moreover,the solution to(1.1)is unique up to a constant.Theorem 2 Let Ω (?) Rn(n≥ 2)be a bounded domain with a smooth boundary.Assume (?),then the unique smooth solution u(x,t)to equation (?)converges to the translating soliton w+τt,that is to say that (?)where(w,τ)is a suitable solution to(1.1).The constant τ depends only on Ω and θ.Theorem 3 Let Ω (?) Rn(n≥ 2)be a bounded domain with a smooth boundary.If F satisfies(F1)-(F4),(?),then the smooth solution u(x,t)of equation (?) converges to U+τt,namely,? D (?)(?)Ω,ζ<1 和 0<α<1,(?)where(U,τ)is a suitable solution to (?)The constant τ depends only on Ω,φ and F.The solution to(1.6)is unique up to a constant.