Asymptotic Analysis To A Nonlinear Parabolic Equation With A Weighted Localized Source

This paper deals with a nonlinear parabolic equation u_t =Δ_u + a(x)e^{pu(x,t)+qu(0,t)} with a more complicated source term, which is a product of localized source e^qu(0,t), local source e^pu(x,t), and weight function a(x). We investigate how the three factors influence the asymptotic behavior of solutions. It is shown that the blow-up set consists of single point {x = 0} if p > 0. When p≤0 with p + q > 0, the blow-up take place everywhere in B. Moreover, the blow-up rate estimate is established with more precise coefficients determined.In the introduction, we give the background to the parabolic equation with localized source. In Chapter 2, we give some basic knowledge to be used in this paper. In Chapter 3, we describe the main results of the paper, and then prove the theorems in Chapter 4. In the last chapter, we discuss all the conclusions obtained in this paper.