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

Quasilinear elliptic equations and parabolic equations are two classes of important partial differential equations,typical models of which are the p-Laplace equation in the fluid mechanics[1],the porous dielectric equation[2],the nonlinear elastic response d-iffusion equation[3]and so on.In recent years,with the research of partial differential equations becoming more deeply and widely,the form of differential operators has be-came more and more complicated and therefore the research on the quasilinear partial differential equations with p-Laplace operator has attracted a lot of attention of mathe-maticians both at home and abroad.At the beginning of the twentieth century,Soviet mathematician Sobolev introduced the concept of Sobolev space in[4,5].This kind of space plays an important role in the study of partial differential equations,especially in the study of p-Laplace equation.In this paper,we devote ourselves to studying the properties of the solutions to some elliptic and parabolic equations with p-Laplace operator.Our main interest includes the existence,uniqueness,regularity,blowing up,extinction and asymptotic behavior of the solutions.This paper is divided into five chapters.Chapter 1 is the introduction to the main content of this thesis.The background of the problems considered in this paper are described and the related works obtained by mathematicians both in China and aboard are briefly recalled.And we also state our main problems and the difficulties we encounter as well as the methods and techniques that we shall use.In Chapter 2,we study a class of singular p-Laplace elliptic equation coupled with Dirichlet boundary condition where Q(?)R~N(N?1)is a smooth bounded domain,?pu = div(|?u|P-2?u)is the standard p-Laplace operator,p>1,7>1 are real numbers,h(x)E L1(?)is a positive function(i.e.,h(x)>0 a.e.in ?).Due to the appearance of the strong singular term,the classical variational method(the critical point theory),the sup-subsolution method,the fixed point theorem and other frequently used methods have limitations to Problem(1)in some ways.Another difficulty arises from the nonlinearity of the p-Laplace operator,since one usually can not deduce from un?u weakly in W01,p(?)the convergence |?un|p-2?un?|?u|p-2?u weakly in Lp/p-1(?,R~N),which also brings great challenges to us.Moreover,when dealing with the singular elliptic problem,the nonhomogeneous term h(x)in the right hand of the equation plays a key role,which usually determines the methods and the complexity of the methods.h(x)in Problem(6)is a L1(?)function,with weak regularity,which also brings great difficulties to us.To overcome these difficulties stated above,we confine the corresponding functional on some special sets in W01,p(?)to restore integrability.And then we consider the constrained minimization problem on the sets of the specific choice of constraints.Then by Ekeland's variational method and combined with some analytical skills,a sufficient and necessary condition is given for the problem to admit at least one nonnegative weak solution.Finally,we prove that the weak solution is unique by making use of the monotonicity of the p-Laplace operator.Our main result of this chapter is as follows:Theorem 1.Let ?(?)R~N(N ? 1)be a bounded domain with smooth boundary,p>1,?>1,h(x)? L1(Q)is a positive function(i.e.,h(x)>0 a.e.in ?).Then Problem(6)admits a unique W01,p(?)solution if and only if there exists a u0?W01,p(?)such thatIn Chapter 3,we study a class of singular elliptic equation of p-Kirchhoff type coupled with Dirichlet boundary condition where ?(?)R~N(N?1)is a bounded domain with smooth boundary,p>1,0?q?p-1 and ?>1 are real numbers,h(x)?L1(?)is a positive function(i.e.,h(x)>0 a.c.in?),k(x)?L?(?)is a non-negative function,B:R+?R+ is a C1 continuous function with positive lower bound.For Problem(7),in addition to the strong singularity(?>1),another notable feature is that the coefficient of the second order term is related to ??|?|pdx.Usu-ally,??|?u|pdx is called nonlocal term,and thus equation(7)is called nonlocal equa-tion.Usually,one can not deduce from un(?)u weakly in W01,p(?)the convergence(?)and this is the biggest difficulty when dealing with problems of p-KirchhofF type.Similar to Chapter 2,we still need to choose spe-cial sets of proper constraints in W01,p(?)and consider the the constrained minimization problem on the sets of the specific choice of constraints.By Ekeland's variational method and some analytical techniques,we obtain some results about the strong convergence of the minimized sequence in W01,p(?).And we also give a sufficient and necessary condition for Problem(7)to admit a nonnegative W01,p(?)solution.Furthermore,we prove that the weak solution is unique under some proper conditions.In Part 2 of Chapter 3,we consider the special case of p = 2 and k(x)?0 in Problem(7),where we also assume that B satisfies the following conditions (B1)B'(s)?0,(?)s?0.(B2)There exist constants ?>0,?>0 and M>0 such that B(s)??s?,(?)s?M,where B(s)= ?0s B(?)d?.The main result of this part is as follows:Theorem 2.Let ?(?)R~N(N ? 1)be a bounded domain with smooth boundary,p=2,?>1,k(x)= 0,h(x)? L1(?)is a positive function(i.e.,h(x)>0 a.e.in ?),B:R+.?R+ is a C1 continuous function with positive lower bound and satisfies(B1)and(B2).Then Problem(7)admits a unique H01(?)solution if and only if there exists a u0 ? H01(Q)such that??h(x)|u0|1-?dx<+?.In Part 3 of Chapter 3,we apply the same methods and skills used in Part 2 to extend the results to the general case with p>1 and k(x)(?)0,where we also assume that B satisfies the following conditions(B3)B'(s)?0,(?)s ? 0.(B4)There exist constants ? ?1+q/p,?>0 and M>0 such thatB(s)??s?,(?)s ? M,where B(s)= ?0s B(?)d?.When ? = 1+q/p,we also require that ?>?k??pq+1/pSq+1/q+1,where S>0 is the bestSobolev embedding constant from W01,p(?)to Lq+1(?).The main results of this part are as follows:Theorem 3.Let ?(?)R~N(N ? 1)be a bounded domain with smooth boundary,p>1,0?q?p-1,?>1,h(x)? L1(?)is a positive function(i.e.,h(x)>0 a.e.in ?),k(x)? L?-(?)is a nonnegative function,B:R+ ? R+ is a C1 continuous function with positive lower bound and satisfies(B3)and(B4).Then(7)admits at least one W01,p(?)solution if and only if there exists a u0?W 01,p(?)such thatTheorem 4.If Problem(7)admits W01,p(?)solution,it is unique when k(x)?0.In Chapter 4,we study a class of singular p-Laplace elliptic equation with degenerate coercivity and lower order term coupled with Dirichlet boundary condition where ?(?)R~N(N?p)is a bounded domain,p>1,B,???>0 are real numbers,f?Lm(?)(m?1)is a nonnegative function.Notice that 1/1+|u|??0,when |u|??.Therefore,(?)is not coercive in W01,p(?).Based on the truncation method,we approximate the degenerate coercive term(?)and the singular term(?)by non-degenerate coercive and non-singular operators respectively.Then by choosing appropriate test function,we obtain the a priori estimates for the approximation equation.And then by a sequence of limit process,we get some necessary convergence,based upon which we prove the existence and regularity of solution to Problem(8).The main results of this chapter are as follows:Theorem 5.Assume 0<?<1,f?Lm(?)is a positive function with m?pN/pN-?(N-p),then there exists a function u ? W01,p(?),strictly positive in such that(?)and for every ??W01,p(?)?L?(?).Theorem 6.Assume 0<?<1.f?Lm(?)with N/pN-?(N-1)<m<pN/pN-?(N-p),is a positive function?then there exists a function u ?W01,p(?),with?=mN(p-?)/N-?m,strictly positive in ?,such that(?)and for every ? ? C01(?).Theorem 7.Assume 1??<p,?>?-1,f? Lm(?)is a nonn,egative function with m?pN/pN-?(N-p)and satisfies for every compactly contained subset ?(?)(?)?,then there exists a function u? W01,p(?),strictly positive in ?,such that |?u|p/u??L1(?)and for every ?? W01,p(?)? L?(?).Theorem 8.Assume 1??<p,?>?-1,f? Lm(?)is a nonnegative function with N/pN-?(N-p)<m<pN/pN-?(N-p),and satisfies for every compactly contained subset ?(?)(?)?,then there exists a function u? W01,?(?),with strictly positive in ?,such that |?u|p/u??L1(?)and for every ?? C01(?).In Chapter 5,we study the properties of solutions to some thin-film equations with p-Laplace operator by introducing a family of modified potential wells.We first study the following thin-film equation of p-Laplace typewhere ?(?)R is an open interval,f(u)=|u|q-1u-/|?|??|u|q-1udx,T?(0,+?),p>1,q>max{1,p-1},u0?H.HereWe first establish the corresponding Lyapunov functional J(u)and Nehari functional I(u)to Problem(11)and introduce a family of potential wells,where By analyzing the properties of some relative functionals and the potential wells together with Galerkin method and convex method,the threshold results of global existence,blowing up in finite time and extinction in finite time of the weak solution are given.Furthermore,we prove that the global solution is unique.And then the corresponding asymptotic behavior of the global solution is also obtained.Finally,a numerical example is given to illustrate the blowing up of solution in finite time.The main results of this part are as follows:Theorem 9.Assume p>1,q>max{1,p-1},u0?H.If J(u0)<d and I(u0)>0,then Problem(11)admits a unique global weak solution u ? L?(0,?;H2(?))with ut?L2(0,?;L2(?)).Moreover,u does not vanish in finite time,andTheorem 10.Assume p>1,q>max{1,p-1} and u is the weak solution of Problem(11)with u0 ? H.If J(u0)<d and I(u0)<0,then there exists a finite time T such that u blows up at T,that is,Theorem 11.Assume p>1,q>max{1,p-1},uo? H.If J(u0)= d and I(u0)? 0,then Problem(11)admits a unique global weak solution u ? L?(0,?;H2(?))with ut ? L2(0,?;L2(?))and I(u)? 0.Moreover,if I(u)>0 for every t>0,the solution does not vanish in finite time,and If not,the solution vanishes in finite time.Theorem 12.Assume p>1,q>max{1,p-1},and u is the weak solution of Problem(11)with u0 E H.If J(u0)= d and I(u0)<0,then there exists a finite time T such that u blows up at T,that is,In the next part of this chapter,we extend the results obtained above to the following thin-film equation of p-Kirchhoff type where ?(?)R is an open interval,Similarly,we establish the corresponding Lyapunov functional J(u)and Nehari func-tional I(u)of Problem(12),where We obtain the results paralleled to Problem(11)by the theory of potential wells.The main results of this part axe as follows:Theorem 13.Assume p>1,q>2p-1,u0? H.If J(u0)<d and I(u0)>0,then Problem(12)admits a unique global weak solution u ? L?(0?;H2(?)))with ut ?L2(0,?;L2(?)).Moreover,u does not vanish in finite time,andTheorem 14.Assume p>1,q>2p-1,and u is the weak solution of Problem(12)with u0 ? H.If J(u0)<d and I(u0)<0,then there exists a finite time T such that u blows up at T,that is,Theorem 15.Assume p>1,q>2p-1,uo ? H.If J(u0)= d and I(u0)? 0,then Problem(12)admits a unique global weak solution u ? L?(0,?;H2(?))with ur?L2(0,?;L2(?))and I(u)? 0.Moreover,if I(u)>0 for every t>0,the solution does not vanish in finite time,andIf not,the solution vanishes in finite time.Theorem 16.Assume p>1,q>2p-1,and u is the weak solution of Problem(12)with u0 ? H.If J(u0))=d and I(u0))<0,then there exsts finite time Tsuch that u blows up at T,that is,(?).