Solutions Of A Class Of Quasilinear Parabolic Equations With Neumann Boundary Value Conditions

Parabolic equation has high research value in the field of partial differential equation.The research on it often needs to add boundary conditions,for instance Neumann boundary conditions,oblique derivative boundary conditions,Dirichlet boundary conditions,prescribed contact angle boundary conditions and and so on.In this text,we research the properties of solutions of equations with Neumann boundary conditions.This article is organized as follows:In part 1,mainly expounds the background and sources of the research problems in this paper;In part 2,lists the relevant symbols and theoretical knowledge of this paper;In part 3,the ut estimate and the C0 estimate of the solution are studied;In part 4,with the help of auxiliary functions,the internal gradient estimation of the solution is given,the global gradient estimation is obtained,and the longterm existence of the solution is obtained.Finally,we prove the asymptotic property of solutions of equations with Neumann boundary conditions over strictly convex bounded regions based on the maximum principle,prior estimation,Hopf lemma,Schauder’s theory.The dissertation concludes below:Theorem 1 Let Ω be a bounded domain in Rn and (?)Ω∈C3,n≥2.β>0,v is the inner unit normal.Suppose f,φ are functions defined on Ω×R and Ω respectively,f(x,τ)satisfies that fτ≥-k(k≥0)with |φ|C3(Ω)≤L2.The parabolic equation has a smooth solution u=u(x,t).Theorem 2 Assume Ω is strictly convex bounded domain region of Rn,n≥2.u(x,t)is the solution to the following problem where(u0)v=φ(x)and |φ|C3(Ω)is bounded.So u(x,t)will converge to something like a flat transfer of λ0t+ω,where ω is a suitable solution to the following equation where v is an inward unit normal vector to (?)Ω.