与p-Laplace算子相关的发展型方程组的一些问题
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摘要
本文主要研究了发展型p-Laplace方程组广义解的一些问题.
第一章主要研究了下面具有耦合非线性源的非牛顿渗流系统Rn,是具有光滑边界的有界区域.研究了fi满足不同单调条件下广义解的局部存在性与唯一性.
第二章主要研究了下面由m个方程构成的具有耦合非线性源的方程组这里是一个开的有界连通区域,并且具有光滑边界??.这里用了与第一章不同的正则化方法,得到了fi在满足某种特殊结构条件下广义解的全局存在性与唯一性.并证明了广义解满足比较原理.
第三章研究了具有非线性源方程组的周期解问题.通过对源函数加些特殊限制,定义一个Poincar′e映射,利用比较原理和单调迭代方法得到了周期解的存在性.
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).
第一章主要研究了下面具有耦合非线性源的非牛顿渗流系统Rn,是具有光滑边界的有界区域.研究了fi满足不同单调条件下广义解的局部存在性与唯一性.
第二章主要研究了下面由m个方程构成的具有耦合非线性源的方程组这里是一个开的有界连通区域,并且具有光滑边界??.这里用了与第一章不同的正则化方法,得到了fi在满足某种特殊结构条件下广义解的全局存在性与唯一性.并证明了广义解满足比较原理.
第三章研究了具有非线性源方程组的周期解问题.通过对源函数加些特殊限制,定义一个Poincar′e映射,利用比较原理和单调迭代方法得到了周期解的存在性.
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).
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[2] L. K. Martinson, K. B. Pavlov, Unsteady shear ?ows of a conducting ?uidwith a rheological power law, Magnitnaya Gidrodinamika, 2 (1971), 50-58.
[3] J. R. Esteban, J. L. Vazquez, On the equation of turbulent filteration inone-dimensinal porous media, Nonlinear Anal., 10 (1982), 1303-1325.
[4] A. Constantin, J. Escher, Z. Yin, Global solutions for quasilinear parabolicsystem, J. Di?erential Equations, 197 (2004), 73-84.
[5] F. Dickstein, M. Escobedo, A maximum principle for semilinear parabolicsystems and application, Nonlinear Anal., 45 (2001), 825-837.
[6] M. Pierre, D. Schmidt, Blowup in reaction-di?usion systems with dissipationof mass, SIAM J. Math. Anal., 28 (1997), 259-269.
[7] J. Zhao, Existence and nonexistence of solutions for ut = div(| u|p?2 u)+ f( u,u,x,t), J. Math. Ana. App., 172 (1993), 130-146.
[8] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear andQuasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI,1968.
[9] A. Friedman, Partial Di?erential Equations of Parabolic Type, Prentice-Hall,INC. Englewood Cli?s, N. J., 1964.
[10] E. Coddington, N. Levinson, Theory of Ordinary Di?erential Equations,McGraw-Hill, New York, 1955.
[11] 伍卓群, 尹景学, 王春朋, 椭圆与抛物型方程引论, 科学出版社, 2003.
[12] G. Astrita, G. Marrucci, Principles of non-Newtonian ?uid mechanics,McGraw-Hill, (1974).
[13] H. Chen, Global existence and blow-up for a nonlinear reaction-di?usionsystem, J. Math. Anal. Appl., 12 (1997), 481-492.
[14] M. Escobedo, M. A. Herrero, Boundedness and blow up for a semilinearreaction-di?usion system, J. Di?erential Equation, 89 (1991), 176-202.
[15] M. Escobedo, M. A. Levine, Fujita type exponents for reaction-di?usion sys-tem, Arch. Rational Med. Anal., 129 (1995), 47-100.
[16] J. Zhang, Boundedness and blow-up behavior for reaction-di?usion systemsin a bounded domain, Nonlinear Anal., 35 (1999), 833-844.
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