Diffusion Equations In Non-Divergence Form With Nonlinear Sources

Since the classical works by Oleinik, Kalashnikov and Zhou, nonlinear diffusion equations have been the subject of intensive study by many researchers, and hence have been thought of as an important branch of partial differential equations. As typicalexamples, the Newtonian filtration equation and the non-Newtonian filtration equation come from a variety of diffusion phenomena appearing widely in nature, such as filtration, phase transition, image processing, biochemistry and dynamics of biological groups. While, non-divergence form equations, as an another important class of nonlinear diffusion equations, have been proposed as mathematical models of physical problems in many fields such as the resistive diffusion of a force-free magnetic field, dynamics of biological groups, curve shortening flow, spread of infectiousdisease and so on. The study for such kind of equations has attracted a large number of mathematicians, such as Friedman, Passo, Giga, Wiegner, Mizoguchi, Pablo, Winkler etc. Different from classical diffusion equations in divergence form, induced by the special structure of non-divergence, the kind of equations might have some very different, even incredible features, which make some classical methods no longer work. Therefore, it must be necessary for us to develop some new ideas and approaches to treat the relevant problems, which, to a certain extent, will enrich the theory of partial differential equations. In this paper, we consider the equation of the following formwhere m∈R, p>1,q≥0, f(u,x,t) is a bounded and smooth function. The equation (1), is degenerate if m > 0, p≥2 or m≥0, p > 2; and is singular if m < 0, 1 < p≤2 or m≤0, 1 < p < 2; while for other cases, it has degeneracy as well as singularity. Each point (x, t) with u(x,t) = 0 orâ–½u(x,t) = 0 is the degenerate or singular point. Since that the equation (1) is degenerate at the points where u = 0 if m > 0, and at the points whereâ–½u = 0 if p > 2, while it is also singular at the points where u= 0 if m < 0, and at the points whereâ–½u = 0 if 1 < p < 2, there is no classical solution in general, and we therefore consider its weak solutions. Another peculiarity is that the equation is known as the non-divergence form equations, although, in the case m < 1, the equation (1) could be transformed into the following well known polytropic filtration equation with a source of the following formEquation (2), as a typical divergence form equation, is known as fast diffusive forλ(p-1)<1, and as slow diffusive forλ(p-1)>1.In the first chapter, we discuss periodic solutions for the equation (1) with the nonlinear source being the typical caseα(x, t)u^q, namely, we consider the following problemClearly, the above problem has a trivial periodic solution u≡0. Our purpose is to prove the existence of nontrivial and nonnegative periodic solutions. This chapter is mainly devoted to the case of m≥1.We also generalize these results to the case m<1. Motivated by the results of the special case m = 0, p = 2, it is reasonable to imagine that for every fixed m, there should be two curves, the singular curve q_s=f₁(p) and the critical curve q_c =f₂(p) with 0 < f₁(p) < f₂(p),such that for the three cases 0≤q < f₁(p), f₁(p) < q < f₂(p) and q > f₂(p) the results are very different from each other. In fact, we haveMore precisely speaking, we have showed that(1) If 0≤q < f₁(p), then the problem (3) admits at least a positive periodic solution;(2) If f₁(p) < q < f₂(p), then there exists nontrivial and nonnegative periodic solution;(3) If q≥f₂(p), then there is no nontrivial and nonnegative periodic solution at least for star shaped domains;(4) If q =f₁(p), then existence and nonexistence are all possible.For the conclusion (4), that is the critical case, we indeed find that there is no nontrivial and nonnegative periodic solution for smallα(x,t); While for large a, we guess that the existence of nontrivial and nonnegative periodic solutions with compact support is possible. We have testified the validity forαindependent of t. Although the result is not perfect, however, it is enough to show that q = f(p) is a singular curve.For the approach people adopted, it is worth to mention the proof for the case (2). As is known, the so called blow-up technique is a classical approach in the study of periodic problems. By translating the a priori estimates of solutions into the problem of the existence of solutions on R^N, one can obtain the boundedness of solutions, and then, combining with the topological degree method, get the existence of nontrivial and nonnegative periodic solutions. However, this method depends heavily on the Fujita exponents of the corresponding equation, one can only get the existence of nontrivial periodic solutions for q in some narrow set. Moreover, due to the linearity of the principal part of the heat equation with similar nonlinear sources, it is easy to obtain the boundedness of periodic solutions by using the first eigenfunction of the Laplacian operator, which is not suitable to the equation with nonlinear principal part. Fortunately, we find that the critical exponent of the heat equation with nonlinear sources is exactly the same as the blow-up exponent of the corresponding elliptic equation. By virtue of this phenomenon, we make some improvement for the classical blow-up technique. Exactly, using the moving plane method, Liouville theorem and topological degree method, we obtain the existence of nontrivial and nonnegative periodic solutions for the case f₁(p)2(p). Summing up, for any value of the exponent q without no gaps, we have completely determined the existence and nonexistence of periodic solutions.Finally, we extend the results to the case m<1, which supplement some gaps in the previous works.In the second chapter, we discuss the asymptotic behavior of solutions of the initial and boundary value problem for the equation (1), including the Fujita exponent,the existence of steady states, the stability of the steady states or periodic solutions and the existence of periodic attractor.Firstly, consider the critical Fujita exponent for the following equationMany mathematicians have paid their emphases on the study of the same kind of problem. It was Fujita who first considered, in [85] (1966), the Cauchy problem of the heat equation with nonlinear sources on R^N, that is the case of m = 0, p = 2for the equation (4). He obtained the following results:(i) If 11+2/N, then there exist global positive solutions if the initial valuesare sufficient small.From then on, the same kinds of results as above are thought of as the Fujita typeresults, and q_c=1+2/N is said to be the critical Fujita exponent. The advantageof critical Fujita exponent lies in that it can be used to describe accurately the asymptotic behavior of solutions. That is why such kind of problems have attracted much attention by more and more mathematicians. In the past forty years, many mathematicians have devoted themselves to the study on critical Pujita exponent, and obtained many significant results.see for example [86]. Usually, for the divergenceform equation, there exist two exponents, the one, denoted by q₁, is the global existence exponent, and the another, denoted by q₂, is the Fujita exponent as mentioned above, and q₁≤q₂ in general. In other words, all solutions exists globally if 0≤q≤q₁, all nontrivial solutions blow up if q₁2, while if q > q₂,both blow-up at finite time and global existence are possible. For the equation (4), we find that q₁=q₂, namely, there is no interval I, such that for q∈I, all nontrivial solutions blow-up at finite time. Denote by q^* the same exponent q₁ or q₂. Then what we obtain is the following results:(1) If q*, then solutions exist globally for any nonnegative initial datum;(2) If q>q^*, then there exist both blow-up solutions and global existence solutions;(3) If q=q^*, then the first eigenvalueλ₁ of the p-Laplace equation with homogeneousDirichlet boundary value condition will play a crucial role. More precisely speaking, ifλ≤λ₁, then the solution exists globally, while ifλ>λ₁,then all positive solutions blow up at finite time.Whereafter, in Section 3 and Section 4, we study the asymptotic stability of solutions, including both the problem without periodic sources, that is (4), and the following problem with periodic sourcesWe aim to study the asymptotic stability and the attractability of the steady state solutions or the periodic solutions. It should be noticed that, due to the special nondivergencestructure of this kind of equations, the weak solutions of the problem (4) or (5) might not be uniquely determined by the initial data. Indeed, from [13], we see that there exist a group solutions which vanish after some finite time. Therefore, we consider the asymptotic behavior of the so-called maximal solution.Firstly, it is necessary to review the history in this aspect. In the early years, Matano [87] (1979) studied the asymptotic behavior of a semilinear parabolic equation,see [88] and [89]. While, it is worth to mention the work [90] (1985) of Sacks, who studied the asymptotic behavior of the homogeneous Dirichlet boundary problemfor the following filtration equation with nonlinear sourceLetλ₁ be the first eigenvalue of the Laplace equation with homogeneous Dirichlet boundary value condition,ρ₁ be the corresponding eigenfunction. Combining with [91], [92], Sacks obtained that:(i) If q1, then all solutions of (6) goes to 0 as t tends to∞;(iii) If q = m andλ=λ₁,then all solutions of (6) goes toθρ₁^1/m,whereθdepends only on u₀;(iv) If q = m andλ>λ₁,then all positive solutions of (6) blow up at finitetime;(v) If q > m, solutions of (6) may or may not exist for all time. In this case, 0 is an asymptotically stable equilibrium solution of (6), while any positive equilibrium solution is unstable.Thereafter, Bandle, Pozio and Tesel [93] (1987) considered asymptotic behavior of solutions for a nonrepresentational equation. Yin, Wang [94] (2001) studied the asymptotic behavior of solutions for the homogeneous Dirichlet boundary value problem of the following filtration equation with periodic sources,For 1≤q1,all solutions of the equation (4) with initial and boundary value tend to 0 for any initial datum;b) Ifλ=λ₁,all solutions of the equation (4) with initial and boundary value tend to 0 or k(?) for any initial datum, where A; is a constant;c) Ifλ>λ₁,all positive solutions blow up.(iii) When q > m+p-1,0 is an asymptotically stable steady state of (4), while any positive steady state is unstable.Here,λ₁ is the first eigenvalue of the homogeneous Dirichlet boundary value problem of the p-Laplace equation, (?) is the corresponding eigenfunction.In the last chapter of this paper, we are mainly interested in a particular class of solutions to (1), the traveling wavefronts. By a traveling wave solution, we mean a solution u(x, t) of (1) in Q = {(x,t);x∈R^N,t > 0} of the form u(x, t) =(?)(γ·x+ t) withγbeing a given vector, and (?) a one-dimensional function. Our purpose is to discuss the existence, uniqueness and regularity of smooth traveling wavefronts of the equation (1). Without loss of generality, we consider the case of f(u,x,t)≡f(u), that isWe study the above equation both for the case m≥1 and for the case m < 1. As adapted by many authors, we restrict ourselves mainly to two typical cases of f(s), the one with unchanging sign and the other with changing sign.Firstly, for the case of f(s) with unchanging sign, we find an interesting phenomenonthat there exist thresholds m_c,q_c for both the exponent m and the exponentq. More precisely speaking, we have the following conclusions. (1) For the case m≥m_c, there is no minimal wave speed, that is, for any wave speed c> 0, there always exists a smooth wavefront with wave speed c;(2) For the case m < m_c, if q≥q_c, then there exists a minimal wave speed c^* > 0 such that the equation (1) admits a unique smooth traveling wavefront if and only if c≥c^*; while if q < q_c, there is no smooth traveling wavefront.It is worth to mention that m_c =1,which is exactly the same as the critical value of the rigorous non-divergence equation and the equation could be transformed into divergence form. There is a great difference between the traveling wavefronts of non-divergence equations and divergence equations. For the divergence form equation, there exists a critical speed c^* such that smooth traveling wavefront exists if c≥c^*, and there is no mooth traveling wavefront if 0 < c < c^*. While for the rigorous non-divergence equation, there is no such a critical speed, that is for any speed c > c^*, there always exist a smooth traveling wavefront. In addition, the non-existence results we obtained for the divergence form equation is also very interesting.For the case of f(s) with changing sign, we may obtain the similar result as divergence form equations, that is there is at most a unique speed c^* such that the equation admits a smooth traveling wavefront corresponding to the speed c^*.In the last section of this chapter, we discuss the regularity of smooth traveling wavefronts. For the divergence form equation, a distinct characteristic between the equation with degeneracy or singularity and the equation without degeneracy or singularity is the appearance of sharp waves. Many researchers had ever paid their attention to the study of the existence of sharp waves, that is, the solution which decreases to 0 at a finite spatial position, at which, (?)'(t) is no longer continuous in spatial variable. All these results show that for most equations with degeneracy, the sharp wave appers only if c = c^*, where C^* is the critical speed. In fact, the appearance of sharp waves closely related to the diffusion coefficient vanishing at equilibrium point. However, there are also sharp waves corresponding to some other speeds, see for example [56], [46]. For non-divergence equation, the results of regularity, in fact, have already included the sufficient and necessary conditions. Indeed, we find that, for the equation what we consider, the appearance of sharp waves is not related to the critial speed, but only correlates to the exponents m, p, q. Due to the degeneracy and singularity as well the special non-divergence structure, the classical upper and lower solutions approach or pure phase plane arguments is not work again. Moreover, the method based on higher order terms of Taylor series as well as the Center Manifold Theorem will be more complicated. In this paper, we first reduce the equation into a system by using variational method, then by an approximation combining with phase plane analysis, and finally obtain the existence of smooth traveling wavefront.