Study On Explosive Solution Of Reaction Diffusion Equations

In mathematics,we usually put semi-linear parabolic equations with the following forms(?)u/(?)t= D(x,u)?u + f(x,u,gradu),((x,t)? ? × R+)(0-1)called reaction diffusion equations,among them ?(?)Rn,n,m? 1,x =(x1,...,xn),u =(u1,...,um),?u =(?u1,...,?um),gradu =(gradu1,...,gradum),D(x,u)=(dij(x,u))(i,j =1,2,...,m),In(1),D and f can depend on t,D(x,u)?u can also be replaced by a nonlinear elliptic operator,boundary conditions can also be non-linear,f can also be a functional,and so on.According to different backgrounds,we can study its initial value problem,which is ?(?)Rn;the equation satisfies the initial condition u(x,0)= u0(x),x ? Rn;It is also possible to study various different boundary value problems as required,which is ?(?)Rn has bound,(?)? is the bound of ?,and if the function satisfies u = g(x,t),(x,t)?(?)?×R+,this boundary condition is called the Dirichlet boundary condition,or if its partial derivative satisfies(?)u/(?)t=g(x,t),(x,t)?(?)?×R+,we call this the Riemannian boundary condition under this kind of boundary condition.The reaction diffusion equation under these two kinds of boundary conditions is the most discussed at present.The formulation of the reaction-diffusion equation model has strong practical significance,at present,some practical problems encountered in practical life,especially in the fields of physics,chemistry and other fields,often require the establishment of models to solve these problems,most of the models established by the problem can meet the requirements of the reaction diffusion equation,therefore,it is very important to study the reaction diffusion equation to solve the problems in real life.At present,most of the researches that researchers have studied are the problems of the global nature and the blasting nature of the solution of the diffusion equation with time integral and space integral under different initial boundary conditions.The global properties and blow-up of the solution of the diffusion equation have strong research value.In the physical,chemical,and biological systems,the global solution of the nonlinear diffusion equation corresponding to the actual problem represents the steady state of the system,while the blow-up solution corresponds to the unstable state of the entire system,and further,the burst rate of the solution can show the rate of change of the system instability.In practical work,often the steady state of the system is of great significance to the operation of the entire system,at this time,we need to study the global solution of the corresponding diffusion equation;sometimes,in practical problems,we also need to understand the conditions of a system instability,even the rate of change under its instability,in this case we will explore the explosion solution of the corresponding nonlinear diffusion equation.It can be seen from this that the properties of the solution of the nonlinear reaction-diffusion equation are of great significance.In this thesis,we mainly use the self-similar method of upper and lower solutions to study the global and explosiveness of the solutions of the two types of exponential nonlinear reaction-diffusion equations with time integration.In our study we considered the Dirichlet boundary condition,and we take a positive value for its initial value.Referring to other related literature,in the layout of the article,we first briefly introduced the origin of the nonlinear reaction diffusion equation and its practical application in real life,followed by the literature at home and abroad that are similar to the equations studied in this paper,listed at the same time,the theoretical basis involved in this article is also described.Different from previous studies on the reaction-diffusion equation in power function form,in this study,we mainly conducted relevant research on the following non-negative initial-time equations(1-12)and(1-13).Its boundary conditions are homogeneous Dirichlet boundary conditions;(?)where ? is a bounded domain in Rn with C2 boundary(?)?,and u0(x)is nonnegative continuous initial data vanishing on(?)?,we discussed the explosiveness of its solution and obtained the conditions for its solution to blasting.The blasting rate is explored with reference to the previous power function form research method on blasting rate and adding appropriate assumptions.Since the blasting rates for equations(1-12)and((1-13)have not yet been derived in this paper,inspired by reference[13],we have tried to simplify equations(1-12)and((1-13),resulting in a simplified nonlinear reaction-diffusion equation with non-negative initial value for homogeneous Dirichlet boundary conditions:(?)at the same time when the solution of its blasting was obtained,the blasting rate was also obtained.