Cauchy Problems For A Class Of N+1 Dimensional Product Form Of Partial Differential Equation

Partial differential equation is built up gradually in the course of discussing natural phenomenon, especially in the phenomenon of physics.Up to today partial differential equation has developed into a subject with rich theories and wide range of applications, but it is still far from perfection. The product of partial differential equation is an important subject which can be used to interpret many phenomena of acoustic and engineering mechanics, and is practically significant in materials engineering, oil exploration, and mining geophysics that appeals to more and more scientists. The paper promotes and improves some existed results, mainly discusses Cauchy problem of n +1 dimensional product of partial differential equations. Uniform convergence of solution on the equation is testified to prove the existence of the solution firstly; then the uniqueness and continuous dependence of the solution is testified by using energy method. The paper contains five parts to interpret these problems.In part one the author introduces the research background about the product of partial differential equation, current research situation and research significance.In part two there are some preliminaries that will be used late in the paper.In part three the author studies the Cauchy problem of n+1 dimensional product form of hyperbolic-parabolic equation.In the first section the author discusses the solution's existence of the Cauchy problem of n+1dimensional product form of hyperbolic-parabolic equation. Firstly the author introduces an unknown function, and then divides the original question into two questions. Finally, the two questions are discussed respectively to testify the convergence of formal solution, and so the solution's existence of the original question is testified by the means of this paper.In the second section the author discusses the solution's uniqueness and continuous dependence of the Cauchy problem of n+1dimensional product form of hyperbolic-parabolic equation. First the energy mode estimates of the two questions are reasoned out; then the uniqueness and continuous dependence of the solution is testified through Energy mode estimate.In part four the author studies the Cauchy problems of the product of n +1 dimensional hyperbolic-parabolic equation.In the first section the questions, relevant hypothesis and requirements are clarified. In the second section the existence of the questions'solution are studied. First the author divides the original question into two questions; then the solution in the series form of the original question is reasoned out through the two questions. And with the testimony of original question's convergence, the solution's existence of the original question is worked out.In the third section the author studies the solution's uniqueness and continuous dependence. By discussing the two questions'energy mode estimate, the energy mode estimate of the original question is reasoned out. Then the author uses the energy mode estimate to testify the solution's uniqueness and continuous dependence to initial function and free function.