Global Existence, Blow-up And Quenching Phenomena Of The Solution For Two Nonlinear Parabolic Equations

This dissertation is devoted to the global existence and quenching phenomena of the solution to a nonlinear parabolic equation of the mean curvature type and the blow-up conditions to another nonlinear parabolic equation. As we all known, there are not always continuous solutions for nonlinear parabolic equations with time passing: some have global solutions, some have the solutions blowing up on finite time and others have the solutions that approach to zero on finite time. Therefor it is necessary theoretically to make sure which conditions ensure the global solution and which ensure the blow-up.The dissertation contains three parts: In Chapter 1, the basic application background, advanced studies and the main idea of this dissertation is introduced. In Chapter 2, 1 consider a nonlinear parabolic equation of the mean curvature type with convection term:where Ω is a bounded domain in Rn with smooth boundary (such asC2); is a function just is a nonlinear vector-valued function,satisfying are some constants the initial apply compact theory and Moser technique to obtain the global existence and quenching phenomena of the solution.In Chapter 3 ,1 consider the nonlinear parabolic equation:where a bounded domain with smooth boundary in ; v is outward normal vector on is a positive function satisfying some compatibility conditions focus myattention on the case of m > 1, to obtain the blow-up conditions of the positive solution using the method of subsolution and supersolution. Then I consider a correlative problem:where m,θ > 0 , Ωs is a bounded domain with smooth boundary in ; v is outward normal vector on ; f(s) is continuous function and satisfies some increasing conditions; u0(x)is a positive function satisfying some compatibility conditions, to obtain the blow-up conditions of the positive solution using the method of subsolution and supersolution, extend the result of Song and Zheng.