First Derivative Estimation Of Solutions Of Some Parabolic Equations Neumann Problems

Derivative estimation of solutions plays an extremely important role in the study of partial differential equations.The Neumann problem is one of the important edge value problems in partial differential equations,and the a priori estimation of the solution of the edge value problem is the key to studying the existence of the edge value problem solution.This article discusses the following three types of parabolic equations:first derivative estimation of Neumann's problem solution.In this paper,the extreme value principle and differential method are used to discuss the first derivative estimation of equation solutions.Generally,it is necessary to discuss the C0 estimate first and then discuss the C1 estimate.The C1 estimation of u is divided into C1 derivative for x and C1 derivative for t.The C1 estimates for this are proved in three cases:Case 1:If x0 ?(?)?,| Du |(x0)is bounded by the Hopf lemma;Case 2:If x0 ?(?)??0??,it can be attributed to internal gradient estimation;Case 3:If x0 ???0,the principle of maximum value can be used to prove that| Du |(x0)is bounded.In the proof of case 3,the first derivative of parabolic equation is estimated by introducing a special frame,constructing a suitable auxiliary function,using the principle of extreme value,the nature of the basic symmetry function and the nature of the function at the maximum value point.Combining the results of the three cases,discussing and drawing an C1 estimate of the three types of parabolic equation solutions.