The Second Critical Exponents To Nonlinear Parabolic Equations

This thesis deals with the second critical exponents to nonlinear parabolic equations, as well as the critical Fujita exponents, and the life span of the non-global solutions. The second critical exponent describes the critical sizes of initial data required by global (non-global) solution via the decay order of the initial data at space-infinity in the coexistence region of global and non-global solutions, determined by the critical Fujita exponents. Four non-linear parabolic models are involved. At first, we consider the influence from the quadratically decaying potential and the weighted source, respectively, on the critical Fujita exponent and the second exponent in two nonlinear diffusion equations. Then, we establish the second critical exponent for the pseudo-parabolic equation, where the effect of the pseudo-parabolic is considered. Finally, we give the second critical exponent to a coupled high-order semilinear parabolic system.Chapter1describes the background and development of the related fields, and then briefly states the main results of the present thesis.Chapter2deals with the Cauchy problem to the fast diffusion equation ut=Δum-V(x)um+up with source and quadratically decaying potential. We obtain the critical Fujita exponent by using the extended Kaplan method and the super and sub-solution method. Comparing with the case without potential term, we find in what way the quadratically decaying potential affects the critical Fujita exponent, which is then il-lustrated by two graphs. In addition, we establish the second critical exponent by the comparison principle. It is observed that the potential term does not affect the second critical exponent.Chapter3studies the Cauchy problem to the nonlinear diffusion not in divergence form ut=upΔu+a(x)uq with weighted source a(x)uq. Firstly, we give the critical Fujita exponent by the L∞-norm estimate of solutions in the unit ball B1, and the extended Kaplan method. Then we obtain the second critical exponent through the existence estimate for the ODE problem with the comparison principle. The weight function in this model does contribute to the second critical exponent, without changing the critical Fujita exponent, quite different from the phenomenon in the model of Chapter2. Chapter4considers the second critical exponent and the life span of non-global solutions to the initial value problem of pseudo-parabolic equation ut—kΔut=Δu+up,where four numerical examples are given to show the effect of the pseudo-parabolic parameter k.Chapter5concerns the coupled higher-order semilinear parabolic system ut=-(-Δ)mu+|u|p,ut=-(-Δ)mu+|u|q. Notice that here the comparison principle does not hold any more. We introduce a new Majorizing kernel with the contractive mapping to obtain the global existence of solutions, and then prove the non-global existence of solutions by choosing appropriate test functions.