Problems About The Nonlinear Degenerate Parabolic Equation

In this paper we study some nonlinear degenerate parabolic equations.In the first chapter,we discuss the existence and uniqueness of renormalized solutions for a class of degenerate parabolic equationsb(u)_t-div(a(u,â–½u))=H(u)(f+divg),where f∈L¹(Q), g∈(L^p'(Q))^N, p' =p/(p-1), a(u,â–½u) satisfies p - 1 powers increasing conditions for|â–½u|. These problems are motivated by control problems arising in chemical reactions. Under these assumptions, this problem does not admit, in general, a weak solution, since the fields a(u,â–½u) do not belong to (L_loc¹)^N and the meaning of the term H(u)(f + divg) is not clear. To overcome this difficulty, we use in this paper the framework of renormalized solutions. This notion was introduced by Lions and Di Perna for the study of Boltzmann equation. And many people applies this notion to evolution problems in fluid mechanics. In this paper, we first give a suitable formulation of the problem to overcome the difficulty that the term H(u)(f + divg) brings , then the existence and uniquess of weak solution are proved.In the second chapter, we discuss the asymptotic behavior of the solution to the p-Laplacian equationand the p-Laplacian equation with absorptionfor the Cauchy problem and the Dirichlet initial-boundary value problem as p→∞. For the Cauchy problem, when the initial value u₀(x) has compact support, the same problem has been studied by Evans et al.[21], where some refined results are obtained. For the case, when the initial value u₀(x) has no compact support, the following result was proved in [34], there exists a subsequence {u_pj} of {u_p} and a function u_∞∈C(R^N), such that for any compact set G (?) Q_T uniformly on G.In this chapter, we improve the above results and study Dirichlet problem. We proved the the asymptotic limit of the solution is uniquess and obtained the results:(?) up(x,t) = u_∞(x) uniformly on G.The third chapter is devoted firstly to the local existence of the solution to the Cauchy problem of the p-Laplacian equation with strongly nonlinear sources when the initial value u₀(x)∈L_loc¹(R^N),We proved whenas qp -1 +p/N,h>N/p(q- p+1), the solution to the Cauchy problem exists localy .In this chapter, we also discuss the global existence of the solution to the Dirichlet initial-boundary value problem of the p-Laplacian equation with particular coefficientThe following results we obtained . Let 1 < p < N,λ> 0, 0 <α< N, u₀(x)∈L^∞(Ω), onΩ, u₀(x)≥0. Denoteλ_N,p = ((N-p)/p)^p.Theorem 1 Let u₀∈W^1,p(Ω),λ<λ_N,p, for any 1 < p < N, then the problem (*) exists a global solution.Theorem 2 Letλ<λ_N,p ,1N,p ,2N/N+2-α< p < 2, there exists a finite time T*, depending only upon N,p,λ,|Ω|, such that the solution u(x, t) of the problem (*),u(.,t)=0, (?)t>T^* Theorem 4 Let Letλ≤μ_N,p,1N,p]=λ_N,p(s-1)(p/(p+s-2))^p,s=(N-α)(2-p)/(p-α)>2, there exists a finite time T^*, depending only upon N,p,λ,|Ω|, such that the solution u(x, t) of the problem (*),...