Some Problems On Doubly Degenerate Parabolic Equations

With the rapid development of non-Newtonian fluid dynamics, the study on non-Newtonian polytropic filtration equation and doubly degenerate parabolic equation attracts more and more mathematicians, physicists and chemists. The same as Newtonian filtration equation and some other diffusion equations, these equations come from a variety of diffusion phenomena appeared widely in nature. They are suggested as mathematical models of physical problems in many fields such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations are nonlinear and possess degeneracy or singularity. The study of degenerate equations beginning from the classical works on second order equations degenerating on the boundary of the domain and with nonnegative characteristic form 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, constitutes an important branch of the theory of partial differential equations and has beendeveloped intensively.This thesis contains two parts, including three chapters.In the first part, we mainly investigate some properties, which include exis- tence, uniqueness and Blow-up, of the solution to the well-known non-Newtonian polytropic filtration equations with nonlinear source under different initial boundary conditionswhere p≥2, m≥1, Q_t =Ω×(0,T),Ωis a bounded domain in R^N with smooth boundary (?)Ω, 0 < T < +∞. The equation, which is used to describe the nonstationary flow in a porous medium of fluids with a power dependence of the tangential stress on the velocity of the displacement under polytropic conditions, is an important class of doubly degenerate parabolic equation, including as particular cases the porous medium equation and the degenerate evolution p-Laplace equation. This part contains Chapter One and Chapter Two.In Chapter One, we first discuss some properties of the solution to the equation (1) in a bounded domain of R^N under the following mixed boundary conditionswhere , and , n denotes the outer unit normal vector on the boundary, u₀(x)≥0 is smooth enough with u₀(x) (?) 0. The existence and uniqueness of local-in-time weak solutions in BV space and the nonexistence of global solutions, via blow-up, are obtained with the use of the regularized methods, Hopf's maximum principles and Gronwall inequality. Especially, the authors prove, under suitable conditions on nonlinear source f(u), a blow up result for solutions with vanishing or negative initial energy by using the energy inequality method to the first initial boundary value problem. At last, we prove that there cannot exist a global non-trivial non-negative solution to the Cauchy problem of (1) with nonlinear source h(x,t)u^α, where m(p- 1) <α< m(p - 1) +p/N. Owing to the degeneracy of (1), we first define the generalized solution of the problem and then state our main results.Definition 1 A function is called a generalized solution of the problem (1) - (4) on Q_T if and u satisfiesfor any , which vanishes for t = T and (x, t)∈(?)1×(0, T).Theorem 1 Assume that f > 0, f_t > 0, f_u≥0, and there exist real numbersα> 1,λ> 0, such thatwhere g(s)∈C¹(R). g(s) > 0. Assume also that at the point x where u0(x) = 0, there holdsand there exists a constantβ> 0 satisfyingwhere . Then any solution u(x,t) of the problem (1)-(4) must blow up within at a finite time T^* withwhereTheorem 2 There exists a T'∈[0,T] such that (1)-(4) has a solution in Q_T' which satisfiesTheorem 3 Suppose f(x,u,t)∈C¹(Ω×R×[0, T]) and there exists afunction h∈C¹(R) such that 0≤f_s(x,s,t)≤h(s). Then the solution of the problem (1)-(4) is unique. Theorem 4 Let f∈C(R) satisfying and , where q > p > 2, Then for any nonzero satisfyingthen the solution u(x, t) blows up in finite time.Corollary 1 If there exists t0≥0, for whichthen the solution u(x, t) either remains equal to zero for all time t≥t0.or blowsup in finite time t*> t0.Theorem 5 Assume that m(p -1) <α< m(p-1) + p/N, Then there cannot exist a global non-trivial non-negative solution to the Cauchy problem of(1).In Chapter Two, we study the existence of periodic solutions for equation(1) under the following Dirichlet boundary conditionAccording to the structure of the source, we will discuss the problem in three cases. Owing to the degeneracy of the equations considered, we adopt standard parabolic regularized methods to obtain the existence of the solution. But there are some differences in establishing a priori estimates of solutions among the three cases. According to the specialty of each case, We will apply the Leray-Schauder fixed point theorem or the topological degree theory to prove the proposition.In the first case of nonlocal term, we consider the periodic solution of the following equation This problem models some interesting phenomena in mathematical biology, where u = u(x,t) represents the density of the species at position x and time t,α= a(x,t) represents the maximal value of the natural increasing rate at location x and time t,Φ[u] is a bounded continuous functional, which denotes the actual increase rate subject to the quantity of the whole species, andα-Φ[u] denotes the actual increasing rate. Assume thatΦ[·] andα(x,t) satisfy the following conditions:(A1) satisfies , where 0 < C₁≤C₂ are constants independent of (A2) may change sign, but , where CT(Qt) is a class of functions which are continuous inΩ×R and T-periodic with respect to t.Definition 2 A function u is said to be a generalized solution of the problem (1)-(6), if and u satisfiesfor any with andDue to the degeneracy of the equation (1), we should consider the following regularized problemwhere ,εis a constant satisfying 0 <ε≤1/2 andηis a constant satisfying 0 <η≤1/2.Noticing the existence of the bounded continuous functionalΦ[uεη], we cannot ensure the problem (8), (5), (6) has a classical solution. Now we should define the strong generalized solution of the problem.Definition 3 A function uεηis called a strong generalized solution of prob-lem (8), (5)-(6), if and uεηsatisfies the Eq. (8) almost everywhere. Theorem 6 If the assumptions (A1)(A2) hold, then the problem (8), (5)-(6) admits a non-trivial nonnegative periodic solutionTheorem 7 If the assumptions (A1)(A2) hold, then a nontrivial nonnegative periodic solution solvesTheorem 8 If the assumptions (A1)(A2) hold, then the problem (1), (5), (6) admits a non-trivial nonnegative periodic solution u.In the second case of strongly nonlinear sources, we consider the periodic solution of the following equationwhere m(p - 1) > 1, 0 < h(x,t)∈CT(Q), Q =Ω×(0,T). Strongly nonlinear sources mean thatαsatisfies m(p - 1) <α< m(p - 1) + p/N.Definition 4 A function is said to be a solution of the problem (9), (5), (6), if andfor any withTheorem 9 The problem (9), (5), (6) admits at least one non-trivial non-negative solution which satisfies , provided that N > 1 and m(p - 1) <α< m(p - 1) +p/N.In the third case of weakly nonlinear sources, we consider the periodic solution of the following equationwhere is Holder continuous inΩ×R×R, T-periodic in t and satisfies for constants m(p -1). Weakly nonlinear sources are implied the above assumptions. Definition 5 A function is said to be a solution of the problem (10), (5), (6), if and u satisfiesfor any withTheorem 10 Under the assumptions given above, the problem (10), (5), (6) admits at least a solution u which satisfied In the second part of this thesis, namely Chapter Three, we investigate theexistence and uniqueness of the weak solution to the following generally nonlinear doubly degenerate parabolic equations with mixed initial-boundary conditions (2)-(4)where , and a(s), b(s) are smoothfunctions in R. In this chapter we extend some results of Chapter One. Usingthe parabolic, regularized methods, we obtain the existence and uniqueness of the generalized solutions of the problem (11), (2)-(4) in BV space.Theorem 11 The problem (11), (2)-(4) has a solution which satisfiesTheorem 12 Suppose that and there exists a function h∈G¹(R) such that . Then the solution of the problem (11), (2)-(4) is unique.