Study Of Solutions To Elliptic And Parabolic Equations With P-Laplace Operator

Author:Yanchao GaoMajor:Applied MathematicsSupervisor:Prof. Wenjie GaoIn this paper, we mainly study the properties of solutions to some elliptic and parabolic equations with p-Laplace operator, which include the existence and unique-ness of weak solutions, regularities, finite time blow up and extinction. The whole paper is divided into four chapters.In Chapter1, we first introduce the background of the problems under consideration and briefly recall some related results obtained by the researchers both in China and aboard. Then we describe our problems and state some methods and techniques that will be used in this paper.In Chapter2, we mainly study the following nonlinear degenerate parabolic problem where Ω (?) R RN(N≥1) is a bounded domain,0<T<∞, QT=Ω×(0,T], ΓT denotes the lateral boundary of the cylinder QT, a(x,t,u)=|u|σ(x,t)+d0with the assumption that d0is a positive constant and the source f(x,t,u) satisfiesf(x,t,u)=b(x,t)-b0u(x, t), u∈(-∞,+∞), x∈Ω, t>0(2) with b0>0,b(x,t)≥0,(x,t)∈QT.It will be assumed throughout the paper that the exponents p(x,t),σ(x,t) are continuous in Q=(?) with logarithmic module of continuity: whereThe model may describe some properties of images restoration in space and time. Especially when the source f(x,u)=b(x,t)-b0u,the functions u(x,t),b(x,t)represent a recovering image and its observed noisy image,respectively. Since the coeffcient of the diffusion term α(x,t,u)may not be bounded from the above,this brings us some difficulties.To overcome these difficulties,we applied the method of parabolic regular-ization and Galerkin’s method to prove the existence of weak solutions to the problem. We obtain the existence and uniqueness of weak solutions not only in the case when σ(x,t)∈(2,2p+/p+-1),p+≥2,but also in the case when σ(x,t)∈(1,2),1<p-<p+≤1+(?).Furthermore,we apply energy estimates and Gronwall’s inequality t0obtain the extinction of solutions when the exponent p-and p+belong to different intervals.To the best of our knowledge,such results seldom seen for the problems with variable exponents. Finally,we get the long time asymptotic behavior of the solutions by using some techniques in convex analysis.Our results are as follows.Theorem1.Let the function f(x,t,u) and the exponents p(x,t),σ(x,t) satisfy Con-ditions (2)-(5).If the following conditions hold then Problem(1) has at least one weak solution u satisfying||u||∞,QT≤||u0||∞,Ω.Theorem2.Suppose that the conditions in Theorem1are fulfilled and the following condition is satisfied Then the bounded nonnegative solution of Problem(1) is unique within the class of all bounded nonnegative weak solutions.Theorem3.Suppose that the conditions in Theorem1are fulfilled and the following condition is satisfied Then the nonnegative solution of Problem(1) is unique within the class of all non-negative weak solutions.Theorem4.Suppose that f(x,t,u) and the exponents p(x,t),σ(x,t) satisfy Con-ditions(2)-(5),b(x,t)(?)0and2N/N+2≤p-<p+<2.Then any bounded nonneg-ative solution of Problem(1) vanishes in finite time for any nonnegative initial data0(?)u0∈L∞(Ω)∩W1,p(x)(Ω) ane satisfies the following estimates and C1is a positive constant.Theorem5.Suppose that f(x,t,u) and the exponents p(x,t),σ(x,t) satisfy Con-ditions(2)-(5),b(x,t)-0,1<p-<2N/N+2and1<p+<Np-/N-p-.Then any bounded nonnegative solution of Problem(1) vanishes in finite time for any nonnegative initial data0(?)u0∈L∞(Ω)∩W1,p(x)(Ω) and satisfies the following estimates where and C2is a positive constant.Theorem6.Suppose that f(x,t,u) and the exponents p(x,t),σ(x,t) satisfy Condi-tions(2)-(5),b(x,t)-0and2≤p-<p+. If the following conditions are satisfied (H5)there exists a positive continuous function g(t) such that the following inequality holds Then for all0<σ-, the solution to problem (1) satisfiesIn Chapter3, we consider the following quasilinear degenerate parabolic problem where QT=Ω x (0,T], ΓT denotes the lateral boundary of the cylinder QT, b(x,t,▽u)=b(x,t)Vu describes the diffusion of mass and f is a continuous function which satisfies the following condition|f(s)|≤a0|s|q(x,t)-1,0<a0=const.(7) Here q(x,t) satisfies the same conditions (4) and (5) as p(x,t) in Q=QT.The difficulty of this problem is that the existence of weak solutions depends seriously on the summability of the coefficient b(x, t) and the relationship between p(x, t) and2. In this chapter, we first construct suitable function spaces that the solutions belong to and apply Galerkin method to reduce Problem (6) to ODEs. The existence of solutions to the ODEs are then proved by applying Peano’s theorem. Furthermore, we use interpolation inequalities, embedding theorem and a modified Gronwall’s inequality to obtain some necessary a priori estimates which guarantee the existence of solutions to Problem (6) via a standard weak convergence argument. Subsequently, we choose a suitable test function and argue by contradiction to prove the uniqueness of solutions under suitable conditions. At the end of this chapter, we construct a suitable functional and apply energy estimate method to prove the nonexistence of global solutions. Our main results are the followingTheorem7. Suppose that the continuous function satisfy Conditions (7) and the exponents p(x,t),q(x,t) satisfy Conditions (4)-(5). If the following conditions hold then Problem (6) has at least one weak solution.Theorem8. Suppose that the conditions in Theorem7are fulfilled and b(x,t)≥0. Then the bounded solution of Problem (6) is unique provided that one of the following conditions holds(H8) f(s)∈C1(R);(H9) the function f(s) is decreasing in s∈R.For convenience, we assume that b(x,t) and f(s) satisfy the following two conditions, respectively. b(x, t)≥0, bt(x, t)≤0,(?)(x, t)∈QT; f(u)∈C(R), f(u)u-p+G(u)≥0,(?)u∈R,(9) where G(u)=∫u0f(s)ds.Next, we consider the case when the variable exponent p is independent of time t. Our main result is the followingTheorem9. Fix T>0. Assume that u∈W(QT) satisfies all the conditions of Theorem7. If (8),(9) are fulfilled and u0∈W1,p+0(Ω),p+>2such that then there exists a T*∈(0, T] such thatIn the case when the variable exponent p depends on the time t, we obtain a new energy estimate and then apply this estimate to prove the following result.Theorem10. Fix T>0. Suppose that u∈W(QT) satisfies the conditions of The-orem7. If(8),(9) and u0∈W1,p+0(Ω),p+>2, pt≤0hold and the following relations remain true then there exists a T*∈(0,T] such thatIn the fourth chapter, we study the existence and regularity of solutions to the following quasi-linear elliptic problem where Ω is a bounded domain in RN(N≥1) with smooth boundary (?)Ω.f≥0,f(?)0,f∈L1(Ω), p>1, α>0.The problem may describe many physical phenomena such as the diffusion of non-Newtonian flows, chemical heterogeneous catalysts, nonlinear heat equations etc. In this paper, we discuss separately the properties of solutions to the problem when α=1,>1and0<α<1. To obtain the existence of solutions, we apply the method of regularization and Leray-Schauder fixed point theorem as well as a necessary compactness argument to overcome the difficulties caused by the nonlinearity of the differential operator and the singularity of the nonlinear terms. Moveover, some estimates on the L∞norm are obtained by constructing some suitable iterative sequences. Our main results are as follows:Theorem11. Let α=1and f be a nonnegative function in L1(Ω)(f(?)0). Then Problem (12) has a solution u∈W1,p0(Ω) satisfying Moreover, suppose that f∈Lm(m≥1), then the solution u of Problem (12) satisfies the following propertiesTheorem12. Let0<α<1and f be a nonnegative function in Lm(Ω)(f(?)0) with m≥m*>1. Then Problem (12) has a solution u∈W1,P0(Ω) satisfying Moreover, the solution u of Problem(12) satisfies the following propertiesTheorem13.Suppose that0<α<1and f∈Lm(Ω) with1≤m<m*.Then for any Np/N+p<q≤q*,Problem(12) has a weak solution u∈W1,q(Ω) satisfying if the following assumptions holdTheorem14.Let α>1and f be α nonnegative function in L1(Ω)(f(?)0).Then Problem(12) has a solution u with up+α-1/p∈W1,p0(Ω) satisfying Moreover,suppose that f∈Lm(m≥1).Then the solution u of Problem(12) has the following propertiesTheorem15.If1<α<2and f(x)∈L∞(Ω),the Problem(12) has a unique positive solution in W1,p0(Ω).Theorem16.If α>2,f(x)∈L∞(Ω) and f(x)≥Φ1/P-1+α in (?),then Problem(12) has no solutions in W1,p0(Ω),where Φ∈C((?)) is an eigenfunction corresponding to the first eigenvalue λ of the following problem...