Critical Exponents Of A Class Of Degenerate And Singular Parabolic Equation In Non-Divergence Form

In this monograph, we consider the long behavior of solutions of the following non-divergence parabolic equation with nonlinear sourcewhere m≥1, q∈(?), p > 0, k > 0, and 0≤u₀∈C₀((?)^N) n W^2,m+1((?)^N). This is a typical fully nonlinear parabolic equation. When q = 0, this is dual porous medium equation, while when m= 1, this is a class of typical parabolic equations in non-divergence form: u_t = u^qΔu + u^p, which come from filtration theory. This class of equations possess the nature of singular or degenerate. WhenΔu = 0, it may degenerate, while when u = 0, it have the properties of degenerate or singular, which depends on q > 0 or q < 0. Due to their profound physical background and structural complexity, they possess important theoretical value and are worth further research. and our researches in this field can be referred to in explanation of certain physical phenomena.The study on the problem with nonlinear sources plays an important part in the theoretical study on nonlinear diffusion equations. The appearance of sources affects much the nature of solutions, especially the long behavior of solutions. It may cause the solution's blow up or extinguish in finite time. From the results related to the problems with nonlinear sources in the forms of u^p(p > 0), people found that the value of p has great effect on the solution, and it can describe the property of large time behavior of solution 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 [1] he considered the Cauchy problem of the semi-linear equationIt was shown that the problem does not have any nontrivial, non-negative global solution if 1c=1 + 2/N, whereas if p > p_c, there exist both global (with small data) and non-global (with large initial data) solutions. People call p_c the critical Fujita exponent and such results the blow-up theorems of Fujita type. Since Fujita critical exponent can describe the behavior of solutions accurately, such problems attract more and more people's attention. From then on, there have been a lot of works on the critical exponents of Fujita type for various nonlinear evolution equations and systems (see. e.g.. the survey papers [4, 5, 6]). From the research on diffusion equation with nonlinear source, people find that in many cases there are two critical parameters for p, they are global existence and critical blow-up exponents: (1) If 0 < p < p₀, then any nonnegative solution exists globally; (2) If p₀ < p < p_c, then no nontrivial nonnegative global solutions exists; (3) If p > p_c, then there exist global positive solutions if the initial values are sufficient small.Along with the researches on critical exponent more deeply, mathematicians found that there are Fujita type results on parabolic equations with non-divergence form. One of the results derived in [9, 10] is about forced porous medium equationAfter the transformthis equation translates towith q =σ/(σ+1)∈(0,1), and p = (σ+β)/(σ+1)∈(1,∞), and oneof the results reads as follows: For thereare no global (positive) solutions. For p > p_c, there are both global solutions and solutions blowing up in finite time. As the complementation of the results of [9, 10], Winkler [8] explores the conditions of q∈(1,∞) in (1), there he considers the following Cauchy problem with general positive pwith u₀∈C⁰((?)^N)∩L^∞((?)^N) positive and q≥1 as well as p≥1, and the main results are formulated as follows: For 1≤p < q + 1(resp. 1≤p < 3/2), if q = 1, all positive solutions of (2) are global but unbounded, provided that u₀ decreases sufficiently fast in space. For p = q +1, all positive solutions blow up in finite time. For p > q+1, there are both global and non-global positive solutions, depending on the size of u₀.Another motivation for this paper comes from [2], in which Galaktionov investigated the blow-up and critical exponents for the fully nonlinear dual porous medium equationwith given nonnegative bounded initial data u₀(x), for which a simple explicit expression for the critical exponent p_c does not exist. It turns out to be a "transcendental" algebraic nature, i.e., cannot be obtained from simple algebraic dimensional equations for parameters. Galaktionov [2] gave the approximation of p_c: p^* =m(1+(2(1+λ+μ)/(N-2μ)) which is obtained by the classical PDE tech-niques based on traditional embedding and multiplicative estimates as well as on the nonlinear capacity approach. As a consequence, p_c vary according to the functional setting associated with nonlinear integral multipliers involved.This paper is going to explore the two critical exponents of q: the global existence exponent and the critical blow-up exponent. We find that the exponent q of diffusion coefficient exerts great influence on the two critical exponent of p. In fact, different from the former research on the critical exponents of p in the equation with nonlinear source u^p(p > 0), we find there exists an critical exponent of m, that is m = N/2. When m≥N/2, there exist three critical exponents for qq₀ = 0, q₀ = 1, q₂ = m.While when m < N/2, q has four critical exponents q₀=(N-(N + 2)m)/(N-2m) q₁=0, q₂=1, q₃=mFirstly by the method of nonlinear capacity method proposed in [2]we calculate the Fujita exponent for cases: q≤0,0 1. For q≤0, we havefor 0 < q≤1, we havewhile for q > 1, we have Furthermore, we get the global existence exponent of p for the cases of q > m and q < m by the method of sub and super-solution: when q > m,p₀ = q + m;while when q < m,p₀ = 1.This monograph is arranged as follow: In section 2, we investigate the blow up exponent for doubly degenerate parabolic equation. In section 3, we consider the blow up exponent for degenerate and singular parabolic equation. Next, we research the global existence exponent for doubly degenerate parabolic equation in section 4. Finally, we give the global existence exponent for degenerate and singular parabolic equation.