一类具耦合边界源快扩散方程组的临界指标
摘要
本文研究一类具耦合边界源快扩散方程组解的爆破性,此类方程组来源于渗流理论,人口生态学和生物群体动力学等领域中广泛存在着的扩散现象。
全文分为三章,分别对三种典型的非线性快扩散方程组,即Newton渗流方程组,非Newton渗流方程组和非Newton多方渗流方程组的非线性边值问题进行研究。本文的主要目的在于研究刻画这类问题解的爆破性的临界指标理论。我们在前两章研究Newton渗流方程组和非Newton渗流方程组,它们都是非线性快扩散方程组的典型情形,作为阶段性的研究,我们侧重于探索其中的方法和技巧。这里,我们需要构造一系列爆破的下解以及整体存在的上解等自相似形式的(弱)解,然后根据这些上下解的渐近性态来分析所考虑问题解的性质。由于这两类方程组在研究手法和理论结果上的差别,我们所构造的上下解也具有不同的结构。借鉴于前两章的研究,在第三章中,我们对非Newton多方渗流方程组构造出了恰当的上下解,得到了刻画解的爆破性的临界指标。与已有的有关快扩散方程组边值问题临界指标研究工作相比,本文所考虑的问题不仅含有耦合参数,也含有非耦合参数,耦合参数与非耦合参数的同时出现会给问题的研究带来一定程度上的困难。本文明确给出非耦合参数的临界指标以及耦合参数的临界曲线。
This paper is concerned with the nonlinear coupled boundary value problem for a class of fast diffusive system.We focus our attention to the discussions of the critical exponents theory,and try to give a complete characterization for these exponentsα,β,p,q.Comparing with former works for the fast diffusive system,the problem we considered not only include the coupled parameters,but also include the non-coupled parameters,which makes the study more challenging and involved to a certain extent.
As is well known,Newton Filtration system and Non-Newton Filtration system are two typical cases of the polypropic filtration system,and described different filtration processes,which both have something in common and some difference not only in the Study of Manner,but also the results.For instance, they both have singularity and degeneracy,but the reason caused that is different; and they both have no classical solution,so the weak solutions are considered,however,the definition and regularity are different for the two kinds of systems.In the first chapter and second chapter,we shall consider Newton Filtration system and Non-Newton Filtration system respectively.As two important classes of parabolic systems,which appear in many fields.So, to carry out research on the two kinds of systems have important academic significance and application value.We shall emphasize particularly on the approach and technic,and try to supply some thoughts for some general systems. In this paper,we mainly by using upper and lower solutions approach, and combining with comparison(see for example[11]-[15]),to discuss the blow-up property of solutions.This approach is often to be used to study the critical Fujita exponent,and the key is to construct suitable upper and lower solutions.In this paper,we mainly by construct a series of self-similar upper and lower weak solutions to study the large time behavior of solutions.Due to the nonlinearity caused both by diffusion and by the coupled boundary condition,which makes the study more difficulty and challenging.
In the third chapter,by using the approach employed in ChapterⅠand chapterⅡ,we shall consider more general case,that is the polypropic filtration system with coupled nonlinear boundary conditions,namely the case 0＜m_1(p_1-1)＜1,0＜m_2(p_2-1)＜1,α＞0,β＞0.We shall character the critical exponents and critical curves by the coupled parameters and noncoupled parameters precisely.We shall see that the critical exponents for the non-coupled parameters areα_c=(m_1+1)(p_1-1)/P_1,β_c=(m_2+1)(p_2-1)/p_2, which have the following properties:
(1) Ifα＞α_c orβ＞β_c,then the solution blow up for any p,q＞0;
(2) If 0＜α＜α_c and 0＜β≤β_c,then there exists p,q＞0 such that all nonnegative solutions will exist globally.
For Case(2),we need to make further discussion.We aimed to study the global existence and the blow up properties of solutions under the assumption that 0＜α＜α_c,0＜β≤β_c.Precisely speaking,we showed that if 0＜α≤α_c and 0＜β≤β_c,the critical curve of global existence is and the Fujita curve is where
Summing up,we discussed the critical exponents characterizing the blowup and global existence properties for a class of fast diffusive system with nonlinear coupled boundary sources,and obtain the critical exponents or critical curve given by the non-coupled parameters and the coupled parameters. Comparing with the already known works,the problem we considered not only include the coupled parameters,but also include the non-coupled parameters, the appearance of which bring more challenges and make the problem more complicated.Especially,we need to search the critical exponent for the noncoupled parameters,which is the essential differences between the problems we considered and the known works.
全文分为三章,分别对三种典型的非线性快扩散方程组,即Newton渗流方程组,非Newton渗流方程组和非Newton多方渗流方程组的非线性边值问题进行研究。本文的主要目的在于研究刻画这类问题解的爆破性的临界指标理论。我们在前两章研究Newton渗流方程组和非Newton渗流方程组,它们都是非线性快扩散方程组的典型情形,作为阶段性的研究,我们侧重于探索其中的方法和技巧。这里,我们需要构造一系列爆破的下解以及整体存在的上解等自相似形式的(弱)解,然后根据这些上下解的渐近性态来分析所考虑问题解的性质。由于这两类方程组在研究手法和理论结果上的差别,我们所构造的上下解也具有不同的结构。借鉴于前两章的研究,在第三章中,我们对非Newton多方渗流方程组构造出了恰当的上下解,得到了刻画解的爆破性的临界指标。与已有的有关快扩散方程组边值问题临界指标研究工作相比,本文所考虑的问题不仅含有耦合参数,也含有非耦合参数,耦合参数与非耦合参数的同时出现会给问题的研究带来一定程度上的困难。本文明确给出非耦合参数的临界指标以及耦合参数的临界曲线。
This paper is concerned with the nonlinear coupled boundary value problem for a class of fast diffusive system.We focus our attention to the discussions of the critical exponents theory,and try to give a complete characterization for these exponentsα,β,p,q.Comparing with former works for the fast diffusive system,the problem we considered not only include the coupled parameters,but also include the non-coupled parameters,which makes the study more challenging and involved to a certain extent.
As is well known,Newton Filtration system and Non-Newton Filtration system are two typical cases of the polypropic filtration system,and described different filtration processes,which both have something in common and some difference not only in the Study of Manner,but also the results.For instance, they both have singularity and degeneracy,but the reason caused that is different; and they both have no classical solution,so the weak solutions are considered,however,the definition and regularity are different for the two kinds of systems.In the first chapter and second chapter,we shall consider Newton Filtration system and Non-Newton Filtration system respectively.As two important classes of parabolic systems,which appear in many fields.So, to carry out research on the two kinds of systems have important academic significance and application value.We shall emphasize particularly on the approach and technic,and try to supply some thoughts for some general systems. In this paper,we mainly by using upper and lower solutions approach, and combining with comparison(see for example[11]-[15]),to discuss the blow-up property of solutions.This approach is often to be used to study the critical Fujita exponent,and the key is to construct suitable upper and lower solutions.In this paper,we mainly by construct a series of self-similar upper and lower weak solutions to study the large time behavior of solutions.Due to the nonlinearity caused both by diffusion and by the coupled boundary condition,which makes the study more difficulty and challenging.
In the third chapter,by using the approach employed in ChapterⅠand chapterⅡ,we shall consider more general case,that is the polypropic filtration system with coupled nonlinear boundary conditions,namely the case 0＜m_1(p_1-1)＜1,0＜m_2(p_2-1)＜1,α＞0,β＞0.We shall character the critical exponents and critical curves by the coupled parameters and noncoupled parameters precisely.We shall see that the critical exponents for the non-coupled parameters areα_c=(m_1+1)(p_1-1)/P_1,β_c=(m_2+1)(p_2-1)/p_2, which have the following properties:
(1) Ifα＞α_c orβ＞β_c,then the solution blow up for any p,q＞0;
(2) If 0＜α＜α_c and 0＜β≤β_c,then there exists p,q＞0 such that all nonnegative solutions will exist globally.
For Case(2),we need to make further discussion.We aimed to study the global existence and the blow up properties of solutions under the assumption that 0＜α＜α_c,0＜β≤β_c.Precisely speaking,we showed that if 0＜α≤α_c and 0＜β≤β_c,the critical curve of global existence is and the Fujita curve is where
Summing up,we discussed the critical exponents characterizing the blowup and global existence properties for a class of fast diffusive system with nonlinear coupled boundary sources,and obtain the critical exponents or critical curve given by the non-coupled parameters and the coupled parameters. Comparing with the already known works,the problem we considered not only include the coupled parameters,but also include the non-coupled parameters, the appearance of which bring more challenges and make the problem more complicated.Especially,we need to search the critical exponent for the noncoupled parameters,which is the essential differences between the problems we considered and the known works.
引文
[1]A.de Pablo,F.Quiros and J.D.Rossi,Asymptotic simplification for a reaction-diffusion problem wish a nonlinear boundary condition.IMA J.Appl.Math.67(1)(2002),69-98.
[2]A.S.Kalashnikov,Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations,Russian Math.Surveys,42(2)(1987),169-222.
[3]A.W.Leung and Q.Zhang,Finite extinction time for nonlinear parabolic equations with nonlinear mixed boundary data.Nonlinear Anal.31(1-2)(1998),1-13.
[4]B.Hu and H.M.Yin,On critical exponents for the heat equation with a nonlinear boundary condition,Ann.Inst.H.Poincare Anal.Non Lineaire 13(6)(1996),707-732.
[5]B.Hu and H.M.Yin,On critical exponents for the heat equation with a mixed nonlinear Dirichlet-Neumann boundary condition,J.Math.Anal.Appl.,209(1997),683-711.
[6]C.Bandle,Blow up in exterior domains.Recent advances in nonlinear elliptic and parabolic problems(Nancy,1988),15-27,Pitman Res.Notes Math.Ser.,208,Longman Sci.Tech.,Harlow,1989.
[7]C.H.Jin and J.X.Yin,Critical exponents and non-extinction for a fast diffusive polytropic filtration equation with nonlinear boundary sources,Nonl.Anal.,67(2007),2217-2223.
[8]D.Andreucci and A.F.Tedeev,A Fujita type result for a degenerate Neumann problem in domains with noncompact boundary,J.Math.Anal.Appl.231(2)(1999),543-567.
[9]E.DiBenedetto,Degenerate parabolic equations,Springer-Verlag,New-York,1993.
[10]F.Quiros and J.D.Rossi,Blow-up sets and Fujita type curves for a degenerate parabolic system wish nonlinear boundary conditions,Indiana Univ.Math.J.50(1)(2001),629-654.
[11]F.Andreu,J.M.Mazon,J.Toledo and J.D.Rossi,Porous medium equation with absorption and a nonlinear boundary condition.Nonlinear Anal.49(4)(2002),541-563.
[12]H.A.Levine,The role of critical exponents in blow-up theorems.SIAM Review,32(1990),262-288.
[13]H.Fujita,On the blowing up of solutions of the Cauchy problem for u_t=△u+u~(1+α),J.Fac.Sci.Univ.Tokyo Sect.I 13(1966),109-124.
[14]J.D.Rossi,The blow-up profile for a fast diffusion equation with a nonlinear boundary condition,AMS.35(1)(2000);40-65.
[15]J.R.Anderson,Local existence and uniqueness of solutions of degenerate parabolic equations,Comm.Partial Differential Equations 16(1)(1991),105-143.
[16]J.Zhou and C.L.Mu,The critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux,Nonl.Anal.,22(2006).
[17]K.Deng and H.A.Levine,The role of critical exponents in blow-up theorems:the sequel,J.Math.Anal.Appl.,243(1)(2000),85-126.
[18]M.Fira and B.Kawohl,Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition,Adv.Math.Sci.Appl.11(1)(2001),113-126.
[19]N.Su.Extinction in finite time of solutions to degenerate parabolic equations with nonlinear boundary conditions,J.Math.Anal.and Appl.,246(2)(2000),503-519.
[20]O.A.Oleinik.E.V.Radkevie,Second order differential equations with nonnegative characteristic form,American Mathematical Society,Rhode Island and Plenum Press,New York,1973.(中译本:O.A.奥列尼克,E.B.拉德克维奇著,具非负特征形式的二阶微分方程,科学出版社,1986)
[21]P.Meier,Blow-up of solutions of scmilincar parabolic differential equations,Z.Angew.Math.Phys.39(2)(1988),135-149.
[22]P.Meier,Existence and nonexistence of global solutions of a semilinear heat equation:extension of a theorem of Fujita,C.R.Acad.Sci.Paris Ser,I Math.303(1986),no.13,635-637.
[23]S.N.Zheng,X.F.Song and Z.X.Jiang,Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux,J.Math.Anal.Appl.,298(2004),308-324.
[24]S.Wang,C.H.Xie and M.X.Wang,The blow-up rate for a system of heat equations completely coupled in the boundary conditions,Nonl.Anal.,35(1999),389-398.
[25]V.A.Galaktionov,Conditions for global nonexistence and localization for a class of nonlinear parabolic equations,U.S.S.R.Comput.Math.and Math.Phys.,23(1983),35-44.
[26]V.A.Galaktionov and H.A.Levine,A general approach to critical Fujita exponents and systems,Nonlinear.Anal.TMA.,34(1998),1005-1027.
[27]V.A.Galaktionov and H.A.Levine,On critical Pujita exponents for heat equations with nonlinear flux conditions on the boundary,Israel J.Math.,94(1996),125-146.
[28]V.A.Galaktionov,On unbounded solutions of the Cauchy problem for a parabolic equation u_t=(?).(u~σ▽u)+u~β,Dokl.Akad.Nauk SSSR.,252(6)(1980),1362-1364.
[29]王明新,《非线性抛物方程》,科学出版社,1993.
[30]伍卓群,赵俊宁,尹景学,李辉来,《非线性扩散方程》,吉林大学出版社,长春,1996.
[31]叶其孝,李正元,《反应扩散方程引论》,科学出版社,1994.
[32]Z.J.Wang,J.X.Yin and C.P.Wang,Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition,Appl.Math.Lett.,20(2007),142-147.
[33]Z.P.Li and C.L.Mu,Critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux,J.Math.Anal.Appl.,346(2008),55-64.
[34]Z.P.Li,Z.J.Cui and C.L.Mu,Critical curves for a fast diffusive polytropic filtration equations coupled through boundary,Applicable Analysis,87(9)(2008),1041-1052.
[35]Z.Q.Wu,Degenerate quasilinear parabolic equations,(Chinese) Adv.in Math.(Beijing),16(2)(1987),121-158.
[2]A.S.Kalashnikov,Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations,Russian Math.Surveys,42(2)(1987),169-222.
[3]A.W.Leung and Q.Zhang,Finite extinction time for nonlinear parabolic equations with nonlinear mixed boundary data.Nonlinear Anal.31(1-2)(1998),1-13.
[4]B.Hu and H.M.Yin,On critical exponents for the heat equation with a nonlinear boundary condition,Ann.Inst.H.Poincare Anal.Non Lineaire 13(6)(1996),707-732.
[5]B.Hu and H.M.Yin,On critical exponents for the heat equation with a mixed nonlinear Dirichlet-Neumann boundary condition,J.Math.Anal.Appl.,209(1997),683-711.
[6]C.Bandle,Blow up in exterior domains.Recent advances in nonlinear elliptic and parabolic problems(Nancy,1988),15-27,Pitman Res.Notes Math.Ser.,208,Longman Sci.Tech.,Harlow,1989.
[7]C.H.Jin and J.X.Yin,Critical exponents and non-extinction for a fast diffusive polytropic filtration equation with nonlinear boundary sources,Nonl.Anal.,67(2007),2217-2223.
[8]D.Andreucci and A.F.Tedeev,A Fujita type result for a degenerate Neumann problem in domains with noncompact boundary,J.Math.Anal.Appl.231(2)(1999),543-567.
[9]E.DiBenedetto,Degenerate parabolic equations,Springer-Verlag,New-York,1993.
[10]F.Quiros and J.D.Rossi,Blow-up sets and Fujita type curves for a degenerate parabolic system wish nonlinear boundary conditions,Indiana Univ.Math.J.50(1)(2001),629-654.
[11]F.Andreu,J.M.Mazon,J.Toledo and J.D.Rossi,Porous medium equation with absorption and a nonlinear boundary condition.Nonlinear Anal.49(4)(2002),541-563.
[12]H.A.Levine,The role of critical exponents in blow-up theorems.SIAM Review,32(1990),262-288.
[13]H.Fujita,On the blowing up of solutions of the Cauchy problem for u_t=△u+u~(1+α),J.Fac.Sci.Univ.Tokyo Sect.I 13(1966),109-124.
[14]J.D.Rossi,The blow-up profile for a fast diffusion equation with a nonlinear boundary condition,AMS.35(1)(2000);40-65.
[15]J.R.Anderson,Local existence and uniqueness of solutions of degenerate parabolic equations,Comm.Partial Differential Equations 16(1)(1991),105-143.
[16]J.Zhou and C.L.Mu,The critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux,Nonl.Anal.,22(2006).
[17]K.Deng and H.A.Levine,The role of critical exponents in blow-up theorems:the sequel,J.Math.Anal.Appl.,243(1)(2000),85-126.
[18]M.Fira and B.Kawohl,Large time behavior of solutions to a quasilinear parabolic equation with a nonlinear boundary condition,Adv.Math.Sci.Appl.11(1)(2001),113-126.
[19]N.Su.Extinction in finite time of solutions to degenerate parabolic equations with nonlinear boundary conditions,J.Math.Anal.and Appl.,246(2)(2000),503-519.
[20]O.A.Oleinik.E.V.Radkevie,Second order differential equations with nonnegative characteristic form,American Mathematical Society,Rhode Island and Plenum Press,New York,1973.(中译本:O.A.奥列尼克,E.B.拉德克维奇著,具非负特征形式的二阶微分方程,科学出版社,1986)
[21]P.Meier,Blow-up of solutions of scmilincar parabolic differential equations,Z.Angew.Math.Phys.39(2)(1988),135-149.
[22]P.Meier,Existence and nonexistence of global solutions of a semilinear heat equation:extension of a theorem of Fujita,C.R.Acad.Sci.Paris Ser,I Math.303(1986),no.13,635-637.
[23]S.N.Zheng,X.F.Song and Z.X.Jiang,Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux,J.Math.Anal.Appl.,298(2004),308-324.
[24]S.Wang,C.H.Xie and M.X.Wang,The blow-up rate for a system of heat equations completely coupled in the boundary conditions,Nonl.Anal.,35(1999),389-398.
[25]V.A.Galaktionov,Conditions for global nonexistence and localization for a class of nonlinear parabolic equations,U.S.S.R.Comput.Math.and Math.Phys.,23(1983),35-44.
[26]V.A.Galaktionov and H.A.Levine,A general approach to critical Fujita exponents and systems,Nonlinear.Anal.TMA.,34(1998),1005-1027.
[27]V.A.Galaktionov and H.A.Levine,On critical Pujita exponents for heat equations with nonlinear flux conditions on the boundary,Israel J.Math.,94(1996),125-146.
[28]V.A.Galaktionov,On unbounded solutions of the Cauchy problem for a parabolic equation u_t=(?).(u~σ▽u)+u~β,Dokl.Akad.Nauk SSSR.,252(6)(1980),1362-1364.
[29]王明新,《非线性抛物方程》,科学出版社,1993.
[30]伍卓群,赵俊宁,尹景学,李辉来,《非线性扩散方程》,吉林大学出版社,长春,1996.
[31]叶其孝,李正元,《反应扩散方程引论》,科学出版社,1994.
[32]Z.J.Wang,J.X.Yin and C.P.Wang,Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition,Appl.Math.Lett.,20(2007),142-147.
[33]Z.P.Li and C.L.Mu,Critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux,J.Math.Anal.Appl.,346(2008),55-64.
[34]Z.P.Li,Z.J.Cui and C.L.Mu,Critical curves for a fast diffusive polytropic filtration equations coupled through boundary,Applicable Analysis,87(9)(2008),1041-1052.
[35]Z.Q.Wu,Degenerate quasilinear parabolic equations,(Chinese) Adv.in Math.(Beijing),16(2)(1987),121-158.