Analysis Of The Singular Solutions For Two Evolution Equations

This thesis deals with analysis of the singular solutions for evolution equations. Firstly, we consider the quenching behavior of solutions for quasilinear parabolic system with coupled nonlinear absorptions. Secondly, we study the blow-up phenomenon for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. The thesis composes of the following three chapters:Chapter 1 mainly summarize the background of the related issues(the quenching and blow-up phenomenon in the reaction-diffusion equation, the nonlocal diffusion equation) and state the main results of the present thesis.Chapter 2 deals with quenching behavior for a quasilinear parabolic system with coupled nonlinear absorptions u_t = (φ(u_x ))_x - v^-p , v_t = (φ( v_x ))_x- u^-q, p , q> 0 The sufficient conditions for finite-time quenching is obtained. Then under some appropriate hypothesis, we distinguished simultaneous and non-simultaneous quenching. Finally, the estimates of quenching rate is given.Chapter 3 mainly discuss blow-up behaviors of solutions to non-local diffusion equation with exponential reaction term and Neumann boundary condition u_t (x , t ) =∫_ΩJ ( x - y )(u ( y , t ) - u ( x , t ))d y +δe^{u (x,t)}, ( x , t )∈Ω×[0, T) Firstly, the local existence and uniqueness of the solution are obtained. Furthermore, we prove that the solution of the equation blows up in finite time. Under appropriate hypotheses, we obtain the estimate of the blow-up rate, and prove that the blow-up set is a single point x = 0 for radially symmetric solution with a single maximum at the origin. Finally, some numerical experiments are performed, which illustrate our results.