| In this monograph, we consider the long behavior of solutions of nonlinear diffusion equations with nonlinear boundary sources. We mainly study the Newtonian filtration equation, the non-Newtonian filtration equation and the non-Newtonian polytropic filtration equation, these three types of equations are the representatives of the nonlinear diffusion equations, and they have been the subject of intensive study, enriching and perfecting the theory on nonlinear diffusion equations.Diffusion equations, as an important class of parabolic 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, image segmentation, biochemistry and dynamics of biological groups. In many cases the equations are nonlinear. Thus the study on the nonlinear problem is very important, it catches many internal and overseas mathematicians' eyes. Nonlinearity can come from reactive term, convection terms, diffusion terms, lower terms, boundary terms and so on, accordingly, the study on nonlinear problem has many contents and results, in which the research on nonlinear diffusion equations attract many mathematicians' attention, the classical works about the Newtonian filtration equation and the non-Newtonian filtration equation can be found in thepapers [3], [17], [18], [19]. The study on the problem with nonlinear sources is an important part in the theoretical study on nonlinear diffusion equations. According to the position of sources, we can define them by interior sources and boundary sources. The appearance of sources affects the properties of solutions deeply, especially the long behavior of solutions, it may make solutions to blow up or extinguish in finite time. Our main interests lie in the long behavior of solutions of nonlinear diffusion equations with nonlinear boundary sources, such as blow up in finite time and global existence in time.In the investigation on diffusion equations with nonlinear sources, such as the heat equation with sources: ut =Δu + f(x,t,u), for the case that the form of f(x, t, u) is more general, the studies are focused on the existence and uniqueness of solutions usually or the part behavior of the solutions (such as blow up and extinguish), it is difficult to describe the influence of sources on the large time behavior of solutions, see the references [10]-[14]. In the results related to the problems with nonlinear sources in the form of uq(q > 0), people found that the value of q has great effect on the solution, and it can describe the property of large time behavior of solutions accurately, so many people pay emphases of study on the problem with such kind of sources. It was Fujita who did the first work about this problem, in[2] he considered Cauchy problem on heat equation with interior sources onut = Δn + uq, he proved the following : (i) If 1 < q < 1 + 2/N, then no nontrivialnonnegative global solutions exist; (ii) If q > 1 +2/H, then there exist global positivesolutions if the initial values are sufficient small. The above results indicate thatthe value of the exponent p of nonlinear source has great impact on the propertiesof solutions, so people call the above result as Fujita type result, and qc — 1 + 2/N issaid to be the Fujita critical exponent. Since Fujita critical exponent can describe the behavior of solutions accurately, such problems attract more and more people'sattention. In the last twenty and thirty years, many mathematicians devoted themselves to the study on Fujita critical exponent, and expand Fujita's result in [2]. From the Fujita type results on different problems, it can be seen that the value of Fujita critical exponent depends on the shape of domains, the spacial dimension, the parameter in the equation and so on, the results on Fujita critical exponent in the last century can be found in the surveys [15], [16].Let us review some results on Fujita critical exponent of diffusion equations with interior nonlinear sources. First in the case of heat equations, Bandle and Levine [20] considered the Fujita critical exponent of the equation in [2] with Dirichlet boundary condition on D, which is a domain in RN with boundary, and it was shown that Fujita's statement held when D has bounded complement, this makes people to study more problems about critical exponent. It is remarkable that the critical exponent is qc = 1 in the case that D is bounded domain. Otherwise, for some domains, the critical exponent has the close relation on the parameters that describethe geometric shape of domain. For the domain with unbounded complement, Meier2 [21], [22] proved qc — 1 +-------- for the cases that D was an "orthant", that is,suppose k is a nonnegative integer in [0, N], defineDk = {xeRN\x1 >(),??? ,xk>0}.He found Fujita type results for other types of domains for which he could find the Green's function for the heat equation explicitly, see the reference [23].Along with the researches on critical exponent more deeply, mathematicians found that there are Fujita type results on degenerate equations. Additionally, Galaktionov[25], [26] considered the Cauchy problem on Newtonian filtration equation with interior sources: ut — div(um"1Vu) + uq and the non-Newtonian filtration equation with interior sources: ut = div(|V?|p2Vu) + uq on M.N, here2 12m > l,p > 2,9 > 1, he proved that qc = m + — and qc — (p - 2)(1 + —) + 1 + —.In 1998, Galaktionov and Levine [27] considered the more general equation: ut =1 2div(|Vu|p-2Vum) + uq, and they proved: qc = {p - 2)(1 + —) + m 4- —.Paralleling to the study on critical exponent of heat equations with source uq instead of ^l^u^see the reference [28], [29]), Qi[30] considered the Cauchy problemon Newtonian filtration equation: ut = Aum + tk\x\auq on R^, it was shown that., 1N 2 + 2fc + crqc = m + k{m - I) -\--------—------.Besides the above results on critical exponent of the problem only with interior sources, many scholars abroad discussed the critical exponent of the equations with source term and lower term together. The following problem was considered:ut = Aum + a- Vup + uq, (x,t) G Dx (0,oo), (1)u{x, t) = 0, (x, t) edDx (0, oo), (2)u(x, 0) = uo(x) > 0, x ￡ D, (3)where m,q > 1, a = a.(x) € KN. If D is an unbounded domain, several results of Fujita type have been established for certain ranges of m and q. First, consider the case m = p — 1. In [33], Meier showed that the critical exponent qc is related to the growth of solutions of a linear equation. Specially, this number is qc = 1 + 1/s*, where s* is the maximal decay rate for solutions of wt = Aw + a ? Vw. Bandle and Levine[43] noted that if a is a constant vector and D = RN, then qc — 1 + 2/AT, hence the presence of the convection term does not affect the large time behavior of solutions. However, the situation is different in cones. Depending on the direction of a, heat is either transported into the cone from infinity or carried away to infinity. They showed that if a-.x > 0 for all x in the cone, then qc = 1. Next consider the casem = \,p > 1. Aguirre and Escobedo[34] showed that when D — RN, the critical/ 2 2p \ exponent is: qc — min ( 1 H-----, 1 H----------I. The case m > l,p > 1, comparingto the heat equation which can be expressed by Green's function on some domains,the study on the problem (l)-(3) is more difficult. A partial Fujita type result was established by Suzuki[35] for the case that D — RN. He set2 p-m+V " N N+land proved the following: If p > m — 1 and max(m, p) < q < g^p, or if q > p > m + I/TV and q = q^p, then all non-negative solutions are non-global whereas if q > m + 2/N, there are nontrivial global solutions. (Remark: the above result is incomplete because the case for which m —1 < p < m + l/N and q^p < q < m+2/N is not included in that).The foregoing statements are Fujita type results on the equations with interior sources, there are Fujita type results in the case that nonlinear source uq appears as a boundary source, that is the interest of this thesis, the results on this problem are not too much, in which the more classical one is the work by Galaktionov and Levine [1]. In [1], the authors discussed the one-dimension case of the heat equation, the Newtonian filtration equation and the non-Newtonian filtration equation: ut = uxx, ut = u?x, ut — (\ux\p2ux)x for x > 0, t > 0 with nonlinear boundary conditions:-u3x=0= uq(0,t), -{um)x = uq{0,t), -\ux\p-2ux = uq(0,t), the initialx=0x=0functions are assumed to be bounded, smooth and to have compact supports. They proved that for each problem there exist positive critical values qo,qc {qo < Qc) such that for q G (O,qo\, all solutions are global while for q G (qo,Qc\ any solution u ^ 0 blows up in a finite time and for q > qc small data solutions exist globally in time while large data solutions are nonglobal. They obtained qc = 2, qc = m + 1and qc = 2(p — 1) for each problem, while q0 = I, qo = —-— and qo = ----------2 prespectively. This indicates that the large time behavior of solutions can be described suitably by qQ and qc. The self-similar solutions, sup-solutions and sub-solutions in the paper [1] offer very important reference for the after correlative works.In the investigations on Fujita critical exponent of the multi-dimensional equa-tion with nonlinear boundary sources, some results have been established only for the case of heat equation. Problem on the heat equation in [1] can be extended to higher dimensions. Deng, Fila and Levine[38] discussed the following problem in a half space R+ x R^"1 x [0, oo), namelyut = An, (x, t) eR+ x R^1 x (0, oo), (4)-uxi(o,xut) = u"{o,x\,t), OM) e ^Nl x (o,oo), (5)u(x, 0) = uo{x) > 0, xeR+ x RNl. (6)They showed that the critical exponent is: qc — 1 + 1/iV. Note that this number is smaller than that for the problem on heat equation with interior source: qc = 1 + 2/N (see [2]). Such a result is to be expected on physical grounds. Roughly speaking, it is more difficult for a boundary source to influence the interior growth of the solution than it is for a source term in the equation to influence this growth. Hence we might expect global, nontrivial solutions over a wider range of q for (4). Recently, Hu and Yin extended the above result when the nonlinearity appears in the boundary condition on a convex cone-type domain, the critical exponent is the same as that in half space[38]. They revisited this problem in a sectorial domain in R2 and an orthant domain in R^. For a sectional domain in R2 with an opening angle 60, they showed that qc — 1 + l/(2 + 7r/20o)- Let D be an unbounded domain in MN with piecewise smooth boundary 3D = dDy U dDi. For an orthant domain D = D.%k - {x e R^|xi > 0, ??? ,xs > 0,xs+1 > 0, ??? ,xk > 0} with dDi = Usj=l{x e DsMX]=0} and dD2 = UJU+1{z € D8ik\Xi=0}, they showed that qc — 1 + 2/(N + k — s), here k — s is the number of sides of Ds,k where the Dirichlet boundary condition is taken, see [40].It is difficult to find a sup-solution and sub-solution of a nonlinear equation with nonlinear boundary conditions because the value of the function on the boundaryis not explicitly given. For the case of heat equation, some problems can be solved by Green's function, but for the nonlinear equations, due to the complexity of the equation and the indeterminacy of the boundary condition, the Fujita type results on the nonlinear equations with nonlinear boundary sources are not abundance as that on the problem with interior sources. Specially, as we know that the Fujita type results for multidimensional cases are not perfect as that for one dimension problem in [1] .This thesis is concerned with the nonlinear diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball, they are the Newtonian nitration equation, the non-Newtonian filtration equation and the non-Newtonian polytropic filtration equation. For the case of Newtonian filtration equation, the problem can be expressed as the following:Mt = Aum + a ? Vum, \x\ > 1,0 < t < oo, (7)Vum-n = uq, \x\ = 1,0 < t < oo, (8)u{x,O) = uo{x), \x\>l, (9)here m > 0,q > 0 are constants, a = a(rc) e R^, n denotes the inner normal to the boundary of the unit ball, Uo(x) > 0 is assumed to be bounded, smooth and with compact support, it also satisfies the compatibility condition. It is known that there is local solution in small time by the classical theory on the degenerate parabolic equations, moreover, the solution is non-negative, see [4], [5], [44]. Our interest lies in the large time behavior of the solution and the critical exponent of the problem(7)-(9).Since there are more than one nonlinearities in problem (7)-(9), for example the nonlinearity of diffusion term, convection term, even boundary sources and so on, these nonlinearities appear at the same time, that brings much difficulties in the study. Noticing the radial symmetry of the exterior domain of the unit ball, andthe fact that the radial solutions can be used as the sub-solutions and sup-solutionsfor the non-radial problem, we first discuss the radial solution of the problem (7)-Xx (9). Specially, we consider the case that a(x) = —— (A > 0), then we analysis the\x\behavior of the solutions of problem the (7)-(9) by using the results we obtain for the radial solutions.By the transformation r = \x\, and replacing A + iV - 1 by a, then when uo(x) can be expressed as the function of |x|, the problem (7)-(9) is transformed to one-dimensional problem with lower term and nonlinear boundary conditions. Taking the space variable still by x, we haveut = (um)xx + umx,{x,t)eQ,(10)0 < t < oo,(11)1 < X < OO,(12)u(x,0) = uo(x),where a > 0. If we can get the Fujita type results on the problem (10)-(12), then the corresponding results on the problem (7)-(9) can also be shown for that the radial solutions are solutions of the non-radial problem.In this thesis, we mainly consider the large time behavior of solutions of the radial problems of the Newtonian filtration equation, the non-Newtonian filtration equation and the non-Newtonian polytropic filtration equation in three chapters respectively, based on the comparison principle, using upper and lower solutions theory then we obtain the Fujita type result on multi-dimensional nonlinear diffusion equations with nonlinear boundary sources. The upper and lower solutions theory is the primary tool in the study on Fujita critical exponent, the key step in that is constructing suitable sup-solutions and sub-solutions, by this method, we make that studying Fujita type result on the multidimensional problem using the radial solutions to be possible.In the first chapter, we discuss the one dimensional problem (10)—(12). We show that not only the value of the exponent q influence the behavior of solution, but also the value of the coefficients of the lower term a, moreover, there is a "threshold" for a, which makes the properties of solution change greatly, denoted by ac. We find ac = 1 for the above problem, this result indicates that the characters of solutions on the problem with lower terms are distinctly different from that without lower terms. For the case of a > ac for problem (10)—(12), it is shown that the existence exponent q0 and the critical exponent qc are the same one, that is: qo = qc = m- Furthermore, for the case en < ac, we also obtain the partial behavior of the solutions, although we can't find the exact value of the critical exponent in this case, we obtain the range of them. From the research for the case a > 0, it is easy to see that the existence exponent becomes more large on the problem with lower terms than that without lower terms, that is to say, the presence of the lower term plays the positive role in the balance of blowing up which arose from nonlinear boundary sources. However, the solutions of the one dimensional problem may induce the radial solution on multi-dimensional problem. Using these solutions as sup-solution and sub-solution, we obtain the results on the multi-dimensional problem: if A > 0, N > 1 or A = 0, N > 2, then the existence exponent and critical exponent for the multi-dimensional problem (7)-(9) are the same one, i.e., q0 = qc = m. The above result is smaller than that for the problem only with interior sources (see [25]), that is the range of q that make the solutions exist globally is wider, this phenomenon is consistent with that in [38],In the second chapter, we discuss the non-Newtonian filtration equation with nonlinear boundary source ug. We consider the problem in one dimensional form:ut = (\ux\p-2ux) + -\uxr2ux. (13)XSimilar to the research in the last chapter, in the discussion on the large time be-havior of the solutions for the non-Newtonian filtration equation in one dimensional case, we not only obtain the influence of the boundary source exponent q in describing the behavior of the solutions, but also find the threshold value for a, that is ac = p - 1. It is seen that the value of a also affect the characteristics of solutions. Specially, for the case that a > ac, it is shown that the values of the existence exponent and critical exponent for the equation (13) with nonlinear boundary source uq are the same one, that is qo = Qc — P — 1- We also study the case that a < ac, obtain the partial behavior of solutions and the ranges qo and qc. From the above results we can find that the range of q which make solutions exist globally for the problem with lower term (a > 0) is wider than that for the problem without lower term(a: = 0) (see [1]). This indicates that the lower term in the non-Newtonian filtration equation also influence the character of the solutions, furthermore, it balances the blowing up. The appearance of threshold value ac on a indicates the great role of the coefficient of lower term a in the fluence for the behavior of solution. Consider the following non-Newtonian filtration equation with boundary condition: | Vu|p2Vu ? n = uq(x, t)ut = div (\S/u\p-2Vu) + X ■\Vu\p-2Vu, (14)\x\lnoticing the fact that the radial solution on one dimensional problem (13) can induce the solutions on multi-dimensional equation (14), so we obtain the Fujita type result on multi-dimensional problem by using the radial solutions as sup-solution and sub-solution: if A + N > p, then the values of existence exponent and critical exponent are the same, i.e., qQ = qc = p — 1. Other results on multi-dimensional problem can be found in § 4 of the second chapter.In the third chapter, we study the behavior of solutions on the non-Newtonian filtration equation with nonlinear boundary source uq. Paralleling to the researchesin the last two chapters, for the one dimensional case:Ut = (|fum)T|p fum)1.) H—l(Mm)Tlp (um) (15)* xwe find the threshold value on a for the non-Newtonian filtration polytropic equation is the same as that for the non-Newtonian filtration equation, i.e., ac = p - 1.In the discussion for the case a — 0, we show that: q0 = ------------------ andP qc = (m + l)(p— 1) . In fact, we have not find the Fujita type result on this equationby now, thus the above result not only suits for the special cases that p = 2 and m = 1 in [1], but also fill the vacancies of the Fujita type result on above equation. We obtain the ranges of q0 and qc for the case 0 < a < ac. For the case a > ac, it is shown that the values of q0 and qc for the problem with lower terms are same again, Qo = 2Vum, (16)by the result on one dimensional equation (15), we have the following: if A + 7V > p, then q0 and qc for the multi-dimensional case both are m(p — 1).From the research on radial problems, it is shown that there is obvious difference in the character of solutions on the equation with lower terms from that without lower term. Next taking Newtonian filtration equations which is discussed in the first chapter as an example to make a comparison. The case 0 < a < ac can be taken as a "little" disturbance to the case a — 0, but our results indicate that this brings the essential change on the large time behavior of solutions. First of all, the range in which solutions exist globally is more wider, this is easy to see from thefollowing estimation on the ranges of q0 and qc:m---------------------- < q < m, m H---------< qc < m +-------?Specially, qo = —-—, qc = m + 1, if a = 0. This result is consistent with that in [1]. When a exceeds the threshold value, namely, if a > ac, then q0 = qc — m, this not only indicates the values of the existence exponent and critical exponent are the same, that is very different from the case that without lower term, but also indicates the solutions jumping at the threshold value. But it is a pity that the exact values of qo, qc for the case 0 < a < ac remain open.Next, in our investigation, we need to construct a series of blowing-up sub-solutions and global sup-solutions in self-similar form, then using these sup and sub solutions to analyze the character of the solutions. When there is no lower terms in the equation such as [1], the construction of self-similar solutions can be solved by the discussion on the corresponding ordinary differential equation or inequality, however the existing sup-solutions and sub-solutions are no longer suitable in our study, so we need to construct the new one. Since the order with irregularity brought by the lower term, the problem we meet not only normal ordinary differential equation, but the ordinary differential equation with parameter variable. For the case q > m, using the suitable magnifying and minifying for the inequalities, the problem is transformed to the ordinary differential equation with parameter T, then we construct a class of suitable self-similar sup-solutions in every case, furthermore, we show the blowing up of the solutions to the problem (10)-(12). Unfortunately, the method as above doesn't work for the case q < m, this causes us to guess that the solutions are global in this case, by introducing more than one parameter into our study, we find a suitable global sup-solution in this case, thus, it is shown that the solutions are global, the above guess is confirmed.In the investigation of the existence of global solutions to problem (10)—(12),...
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