Blow-up Phenomena For Several Kinds Of Nonlinear Diffusion Equations

This thesis studies blow-up phenomena for four kinds of nonlinear diffusion equations. For example, the conditions of global existence and blow-up; the upper-lower bounds of blow-up time; blow-up rate; blow-up profile; blow-up set, etc.The thesis consists of four chapters:Chapter 1 studies the blow-up phenomena of the solutions to a semilinear parabolic equation with nonlocal source and inner absorption under homogeneous Dirichlet or Neumann boundary conditions. Applying modified differential inequality techniques and in the conditions of blow-up, the lower bound estimates are derived.Chapter 2 investigates the initial-boundary value problem for a quasilinear parabolic equation with time coefficient inner absorption and nonhomogeneous Neumann boundary condition. Under appropriate conditions, using auxiliary function and modified technique of differential inequality, the global existence of solutions, the occurrence of blow-up phenomenon and upper bound for blow-up time are obtained. Under somewhat more restrictive conditions, a lower bound for blow-up time is also derived.Chapter 3 considers the blow-up phenomena of the solutions to initial boundary value problem for a porous medium equation with localized source and weighted linear nonlocal boundary. By using the method of upper-lower solution, an influence of weighted function in boundary condition and a competitive relation between localized source and inner absorption terms in determining whether the nonnegative solutions blow up or not are found. Meanwhile, under proper conditions, the estimates of blow-up rates are obtained. Furthermore, this chapter studies the initial-boundary value problem for localized porous medium equation with variable coefficients. The estimates of uniform blow-up profiles are obtained.Chapter 4 focuses on a localized nonlinear fast diffusion equation in a ball with homogeneous Dirichlet boundary condition. This chapter studies the interaction between local source and localized source. The complete classification of total blow-up phenomena and single point blow-up phenomena for the radial solution in ball when fixed point as origin or non-origin are established.