Maximum Principles And Blow-up Of Solutions For Nonlinear Partial Differential Equations

Nonlinear partial differential equations(PDE) are one of main research fields in nonlinear science. An intensive study of NPDE will accelerate the development of nonlinear analysis. The dissertation focuses on two important aspects of NPDE: maximum principles and blow-up of solutions.Maximum Principle is one of the basic theoretical problems in partial differential equations and has been a research question for ages. The dissertation is concentrated on some open questions of P-functional maximum principles for NPDE:(1) We considered various kinds of boundary-value problems ot second order elliptic equation in divergence formWe constructed two different functions P(u. u_k. x ) for this equation. By demonstrating the essence of maximum principle, we obtained the conditions that the functions satisfy classical maximum principles. Thus, we solve the problems remained in P.W.Schaefer[29](1977), [30](1978).(2) We considered the fourth order elliptic equation with non-constant coefficientswhere Lu: = a_ij(x}u_ij is a uniform elliptic operator. Based on some of suitable assumptions, we obtained some maximum principles tor certain functions which are defined on solutions of the equation and completely solved an open problem posed in P.W. Schaeier(57](1987).(3) We considered a series of boundary-value problems subject to second order uniformly parabolic equationUtilizing classical maximum principles and tensor analysis, we derived some conditions that the functions V( u. u_t) and P(x, t, u, k) satisfied maximum principles. Using our results, we can attain some bounds ot important physics quantities, such as. the solution oi the problems and the gradient of the solution and so on. In particular.we t^et growth estimations of the solutions. We claim that we solved the remained problems to some extent in R.P.Sperb[G7]( 1981).In the lust part, we considered blow-up of solutions for a series of initial-boundary value problems, such as nonlinear parabolic equation(/ f = Zi(fl(u)) + f[s.u).where the function / may he forced (i.e.. /(.r.O) > 0). Usiii", the functional niax-iniuin principles ot the nonlinear parabolic equation as main tool, combining com-p:uison principles and rchVctioii principles, we discuss the cunditions that the type of the problems creates blow-up and study blow-up time, blow-up rate and blow-up set. We provided a convenient and effective method for studying blow-up problem of general evolution equation.