Quenching Problems Of Nonlinear Heat Equations

This paper studies the quenching of nonlinear heat equations, including the prob-lem of single initial boundary value and that of equations with coupled conditions inmultidimensional space.In actual application, the problem of single initial boundary value is consideredthe phenomenon of heat reaction in a fixed region. Quenching means termination ofsuch reaction in finite time. This paper studies the impact of convection on quenchingfor the first time. By the analysis on steady solution, the conclusion has been drawnthat the convection has essential impact on the quenching.In actual application, the problem of equations with coupled boundary conditionsis considered the phenomenon of reaction between two materials on the boundary ofa fixed region. The duration of such reaction (whether it reaches infinity) and whichmaterial has more impact on the sudden termination has also been studied. This paperpoints out that the quenching phenomenon will take place inevitably and gives outthe comparison principal as well as the representation formula. The multidimensionalquenching system are studied in this paper for the first time and conclusion has beendrawn for various conditions of quenching.With further study on the equations that is radial symmetric, more detailed resultsfrom the quenching phenomenon and the condition for various quenching could beobtained. The scope of the quenching rate could also be estimated.Lastly, this paper studies the phenomenon of reaction between two materials onthe interior of a fixed region. The main purpose is to study simultaneous and non-simultaneous quenching in finite time, i.e. the termination caused by interior couplingreaction of the two materials. The coupling reaction strength and the impact of ini-tial condition on the entire system are presented based on auxiliary function and theanalysis on the quenching point.