Qualitative Study To A Class Of Quasilinear Degenerate Parabolic Systems

With the rapid development to the study of non-Newtonian fluid dynamics, the study on non-Nnewtonian filtration equation and system attracts more and more mathematicians, physicists and chemists. The same as Newtonian filtration problem, these problems come from a variety of diffusion phenomena appeared widely in nature. for example, filtration, phase transition, biochemistry and dynamics of biological groups. The equations and systems are nonlinear and possess degeneracy or singularity. In the middle of the twentieth century, the study of degenerate equation was started, including the classical works on second order equations degenerating on the boundary of the domain and with nonnegative characteristicform by G. Fichera and O.A. Oleinik, and on degenerate quasilinear parabolic equations by O.A. Ladyzhenskaja, V.A. Solonnikov, N.N. Ural'tzeva and Prof Zhou Yulin. with the typical examples the Newtonian filtration equation and the non-Newtonian filtration equation. These constitute an important branch of the theory of partial differential equations in later several ten years. An exhaustive exploration on the non-Newtonian filtration equation has been presented in Degenerate parabolic equations by E. DiBenedetto in 1993. However, the research of systems is just beginning.In this paper, we study the following initial boundary value problem of non-Newtonian filtration system where p, q > 2,Ωis a bounded domain in Rⁿ with a smooth boundaryαΩ, T > 0 is given. Denote Q_T =Ω×(0.T).S_T =αΩ×(0,T).This theses contains three parts. According to the different forms of source functions, we investigate the existence, uniqueness, global existence and blow-up of the weak solutions to system (1) respectively.The system degenerates when (?)or V(?). In general, there would be no classical solutions and hence we study the generalized solutions.Definition 1 Function (u.v) is called a generalized solution of the system. (1) if u∈L^∞(Q_t)∩L^p(0, T; W₀^1,p(Ω)), v∈L^∞(Q_t)∩L^q(0,T; W₀^1,p(Ω)), u_t,v_t∈L²(Q_t), and satisfiedfor anyφ,φ∈(?),(?) and (?),(x,t)∈αΩ×(0, T) .We discuss the global existence and blow-up of generate solutions in the sense of L^∞norm.Definition 2 We say the solution (u, v) of the problem (1) blows up in finite time if there exists a positive constant T^* <∞such thatWe say the solution (u,v) exists globally if for any positive constant T <∞. In the first part, namely Chapter One, we discuss the system (1) where the source functions are quasimonotone nondecreasing and have some special properties about the initial value, that is(H1) f(x,t,0,0)≠0, g(x,t,0,0)≠0.(H2) There are fore constants (?) satisfied (?) and(H3) (?), where (?).(H4) f(x, t, u. v) is nondecreasing with respect to v and g(x. t, u. v) to u in∑.By a monotone iteration technology, the regularization method and an integral method, the existence and uniqueness of the weak solution to the system are obtained.Theorem 1 Let (H1)-(H4) be satisfied, and u₀∈L^∞(Ω)∩W₀^1,p(Ω),v₀∈L^∞(Ω)∩W₀^1,q(Ω), then the generalized solution of (1) exists and is uniqueness.In the second part, it's discussed that the ignition model where the source functions are exponential functions,Using the regularization method and an integral method. the existence and uniqueness of the weak solution are obtained under some assumptions. Also the exact time when the weak solution exists is calculated by taking advantage of the special form of the source functions. Theorem 2 Let u₀∈L^∞(Ω)∩W₀ ^1,p(Ω), v₀∈L^∞(Ω)∩W₀ ^1,q (Ω), when T < S. the generalized solution of (1) exists and is uniqueness, whereThen by establishing the comparison principle, the conditions are got under which the weak solution exists globally and it blows up in finite time.Theorem 3 If (u₀,v₀) is sufficiently small, and there is a solution to the following system of inequalitiesthen the solutions of problem (1) exist globally, where M₁,,M₂ are respectively the upper bounds of the positive solutionsΦ₁(x),Φ₂(x) of the following elliptical equations that is M₁ = max _x∈ΩΦ₁(x),M₂ = max _x∈ΩΦ₂(x).Theorem 4 If (u₀,v ₀) is sufficiently large, then the solutions of problem (1) blow up in finite time.In the last part, the following ignition model with a nonlocal source is discussedUnder some hypotheses, it's received that the existence and uniqueness of the weak solution. the time when the solution exists and the conditions under which the solution exists globally and blows up in finite time. Theorem 5 Let u₀∈L^∞(Ω)∩W₀ ^1,p(Ω), v₀∈L^∞(Ω)∩W₀ ^1,q (Ω) ,when T < S. the generalized solution of (1) exists and is uniqueness, whereTheorem 6 If (u₀,v₀) sufficiently small and the measure ofΩis also sufficiently small, then the solutions of problem (1) exist globally.Theorem 7 If (u₀,v₀) is sufficiently large, then the solutions of problem (1) blow up in finite time.