Heat Equation And The Study Of Equations With Singular Coefficient

In this paper, a class of heat equation and equations with singular coefficients has been studied. We have discussed the existence of the integral solutions for the heat equations, and we have given out of the nonnegative non-trivial solutions set for the equations in a certain special circumstances. Finally, in the fourth part of the thesis we have discussed the extinguishing properties for the solution of a class of initial-boundary heat equation with singular coefficients. This paper has four chapters:1. By means of the fixed point theorem, semi-group theory and Gronwall inequality, we have discussed the existence of solutions of the initial value problem of the semi-linear heat equations system. We have obtained that there are at least one non-negative global solution of the heat equations. Because the right end of the equations do not satisfy the local lipschtz conditions in this paper and the discussion method about the existence of the global solution is different from the general developing equations, we transform the equations by constructing lipschitz function. Then we certificate that the original equations exist local solutions through proofing the solution's convergence of the system to which we have constructed. And we used Gronwall inequality to give the local solutions continuations, thus we get the existing conclusion of the global solutions of the system.2. By means of semi-group theory and several inequalities, we have discussed non-negative non-trivial solutions of the initial value problem of a class of heat equations. We have verified that the non-negative non-trivial solution of the equations is bounded, and we gain its upper and lower bounds for further, thus we obtained the non-negative non-trivial solution set for the equations.3. The diffusion equation studied in this part is containing the diffuse item and absorb item. We know that the solution of this kind of equation maybe extinguish, also may be overall existing.In this thesis we discuss the coefficient heat equation with initial-boundary value problem by means of maximum modulus estimation, energy method, and some important soblev inequalities. Then we get the extinction sufficient conditions of the equation solution, and we also give out the extinction time estimation formula under certain conditions. Meanwhile we also discuss the conditions on which the heat equation's solution doesn't happen to extinguish, and we get the sufficient conditions on which the equation's solution doesn't happen to extinguish.