Some Problems On Evolution P-Laplace Systems

The aim of this thesis is to study the problems of the generalized solutionsto the evolution p-Laplace system, i.e. the non-Newtonian filtration system.This thesis consists of three chapters.In Chapter one, we consider the non-Newtonian filtration systemwhere Rn is a bounded domainwith smooth boundary ??. The system models non-Newtonian ?uids andnonlinear filtration, etc.Since the system is coupled with nonlinear terms, it is in general di?cultto study the system. We consider some special cases by stating some con-strains to the nonlinear functions. We discuss the cases that the nonlinearfunctions are monotone or quasimonotone. Our method is based on theresults for single equations satisfying comparison principle. We mainly usethe method of regularization to construct a sequence of approximation so-lutions with the help of monotone iteration technique and hence obtain theexistence of solutions to a regularized system of equations. Then we obtain the existence of solutions to the system by a standard limiting process. Theuniqueness of the solution is also given.Owing to the degeneracy of (1), we study the existence and uniquenessof the generalized solution in the following sense:Definition 1 A function u = (u1,u2) is called a generalized solution ofthe systems (1.1)(1.3), if ui∈L∞(T)∩Lpi(0,T;W01 ,pi()), uit∈L2(T),for any i∈W1,∞(T),i(x,T) = 0,i(x,t) = 0, for (x,t)∈×(0,T),i = 1,2.To prove the existence of the solutions, we need to assume the following:(H0) fi(x,t,u1,u2)∈C(×[0,T]×R2), and there exists a nonnegativefunction g(s)∈C1(R) such that|fi(x,t,u1,u2)| min{g(u1),g(u2)}.Our main results are the following:Theorem 1 Let fi be monotonically nondecreasing and (H0) be satisfied,and ui0∈L∞() W01 ,pi(). Then there exists a constant T1∈(0,T] suchthat (1)-(3) has a solution u = (u1,u2) in the sense of Definition 1 with Treplaced by T1. In addition, if f = (f1,f2) satisfies the Lipschitz condition,then the generalized solution of (1)-(3) is unique.If fi is monotonically nonincreasing, similar results can also be achieved.Theorem 2 Let fi be quasimonotonically nonincreasing and assume(H0) and the Lipschitz condition. ui0∈L∞() W01 ,pi(). Then there exists a constant T1∈(0,T] such that (1)-(3) has a solution u = (u1,u2) inthe sense of Definition 1 with T replaced by T1. Also, the solution is unique.If fi is quasimonotonically nondecreasing, we can also obtain the similarresults.In Chapter two, we study the global existence and uniqueness results andblow-up for the degenerated systems of m equations.We study the initial and boundary value problemwhere pi > 2, i = 1,2,···,m, T > 0 is arbitrary, Rn is an openconnected bounded domain with smooth boundary .We consider some special cases by stating some constrains to the nonlin-ear functions in this chapter. The method we are using in this chapter isdierent from that in Chapter one. First, we regularized the problem (5)-(7). The initial and boundary data are approximated by smooth positivefunctions, and since term fi(u) could be superlinear for large u, we willapproximate it by a sequence of linear maps for large u. Then we prove theexistence of the generalized solutions to the regularized problem. Second,we will prove some uniform estimates for the solution of the regularizedproblem to get the global existence of the solution to the regularized prob-lem. Then we obtain the existence and uniqueness of the solutions to thesystem (5)-(7) by a standard limiting process.We make the following assumptions:(A0) If ui 0, i = 1,2,···,m, fi(u) = fi(u1,···,um) are smooth in R+mand fi satisfies the following quasi-positive condition: fi(u) 0 for every We obtained the following results:Theorem 3 If max{j}{αij} < pi 1, whenever cij > 0 and ui0∈L∞()∩W01 ,pi(), for every T > 0, there exists a generalized solution u =(u1,···,um) of problem (5)(7) in T. In addition, if f = (f1,f2,···,fm)satisfies the Lipschitz condition, then the solution is unique.Theorem 4 Assume that fi(u1,u2) satisfies the Lipschitz condition. Letu = (u1,u2) and u = (u1,u2) are the generalized subsolution and superso-lution of (5)-(7) respectively satisfying u0 = (u10,u20) and u0 = (u10,u20),ui0 ui0. Then ui(x,t) ui(x,t), i = 1,2.In Chapter three, we study the existence of periodic solutions for degen-erated quasilinear systems, i.e. evolution p-Laplace systemswhere pi > 2,ω> 0, pi,qi 2, fi(t) > 0, fi(t +ω,u1,u2) = fi(t,u1,u2), i =1,2. Rn is a connected bounded open domain with smooth boundary.We add some constraints to the nonlinear sources, and define a Poincar′eMapping. Then we prove the existence of a periodic solution to the systemby monotone iteration technique.The definition of a periodic solution is the following: Definition 2 A nonnegative vector function u = (u1,u2) is called a gener-alized solution of the systems (8)(10), if ui∈L∞(T)∩Lpi(0,T;W01 ,pi()),uit∈L2(T), T > 0, i = 1,2, and satisfiesuit i | ui|pi2 ui i + fi(t,u1,u2)i dxdt = 0. (11)We get the following results:Theorem 5 Let pi > 2, m1,n2 0, m2,n1 > 0, (p1 1 m1)(p2 1 n2) m2n1 > 0. fi is quasimonotone and satisfies the Lipschitz condition,and there exists nonnegative functions ci1(t) and ci2(t), s.t. ci2(t)u1m iu2nifi(t,u1,u2) ci1(t)um1 iun2 i, cij(t) = cij(t +ω),i = 1,2,j = 1,2. Then thereexists a nontrivial nonnegative periodic solution to the problem (8)(10).