Researches On Blow-up Of Parabolic Equations With Non Local Terms Under Different Boundary Conditions

Parabolic equation is an important part of partial differential equation,and the parabolic equation with non-local term not only appears in the combustion model of compressible gas,but also has great applications in the thermal elastic system of physics.Therefore,it is of great significance to study the blow-up phenomenon of parabolic equation in mechanics and other fields.In this thesis,parabolic equations with non-local reaction terms under different boundary conditions are studied.By constructing different auxiliary functions,using sub-solutions and comparison principle,using differential inequality technique and Sobolev inequality,the existence problem of blow-up solutions to equations is studied,and the upper and lower bounds of blow-up time are obtained on the premise of ensuring the blow-up happening.It provides different methods for studying parabolic equations with non-local terms,enriches existing mathematical theories,and provides corresponding mathematical models for studying problems and phenomena in physics,biology and other disciplines,which has extensive research significance and application value.The main work of this paper is as follows:In the first part,the blow-up problem of parabolic equations with non-local terms under nonlinear boundary conditions is studied.Under appropriate assumptions,by means of auxiliary functions and differential inequality techniques,the sufficient conditions for the blow-up of the solution to the equation are obtained and the upper bound of the blow-up time is obtained.Furthermore,the lower bound of computing blow-up time is given under nonlinear boundary conditions.Finally,the feasibility of the conclusion is verified by corresponding example.In the second part,the blow-up phenomenon of parabolic equations with non-local terms is studied by constructing reasonable auxiliary functions under Robin boundary conditions.With the help of H(?)lder inequality and Young inequality,it is proved that(u(x,t),v(x,t))will blow-up in finite time,and the upper and lower bounds of the blow-up time are obtained when the blow-up occurs.Finally,an example is given to verify the rationality of the conclusion.In the third part,the existence of blow-up solutions for reaction-diffusion equations with non-local terms is studied.Considering the processing of boundary conditions,the upper bound of blow-up time is estimated by constructing upper and lower solutions of the equation and using auxiliary functions.Meanwhile,the lower bound of blow-up time is obtained by using auxiliary function method,divergence theorem and differential inequality.