Study On The Extinction Of Solutions To Some Fast Diffusion Equations (Systems) With Nonlinear Sources

Author:Haixia LiMajor:Applied MathematicsSupervisor:Prof. Wenjie GaoAs an important class of partial diffusion equations, reaction diffusion equa-tions come from a variety of diffusion phenomena that appear widely in nature. They arise from many applied fields such as chemistry, physics, dynamics of bio-logical groups, finance and economics ect.In the past few years, a lot of famous mathematicians both in China and aboard have devoted themselves to reaction d-iffusion equations with suitable initial and boundary conditions, and remarkable progress has been made on the local existence and uniqueness of solutions, global existence, regularity, blow-up, extinction and quenching. In particular, the singu-larities of solutions caused by nonlinear diffusions, sources, absorptions, convection terms, boundary terms as well as the coupling among them have attracted the inter-est of many people. Until now, the study on the singularities of solutions to reaction diffusion equations is still a very active research area. In this paper, we are mainly concerned with the finite time extinction of solutions to some nonlinear reaction diffusion equations (systems) that arise from some physical problems, and investi- gate the combined effect of the nonlinear diffusion, nonlinear sources and nonlinear absorption terms on the extinction properties of solutions to these problems. This paper is divided into four chapters.In Chapter 1, the background of the problems considered in this paper is de-scribed and the related works obtained by mathematicians both in China and aboard are briefly recalled. Then we state our problems as well as some methods and tech-niques that we shall use.In Chapter 2 we will devote ourselves to the extinction properties of solutions of the following fast diffusion problem with nonlinear sources and an absorption term where 0< m< 1, a, b, q,r>0,Ω is a bounded domain in RN(N> 1) with smooth boundary (?)Ω, and u0∈∞(Ω) is a nonnegative nontrivial function.When the absorption term is linear (r= 1), under the transformation that v(x,t)= ebtu(x,t), Problem (4) can be changed into one without absorption terms. However, when the absorption tern is a nonlinear function, it is much more com-plicated to determine whether the solutions will extinct in finite time or not. Only partial answers can be given by combining the methods of integral estimates with the famous Gagliardo-Nirenberg’s interpolation inequality. In order to draw a much more clear picture of the combined effect of the diffusion terms, the nonlinear non-local source terms and the absorption terms on the extinction of the solutions to (4) and to show the critical extinction exponents, we first establish a weak com-parison principle for our problem. However, since the nonlinearity is non-Lipschitz when 0<q<1,the weak comparison principle can only be established for sub and super-solutions that fulfil some special conditions.We then investigate whether the solutions will vanish or not on the basis of this weak comparison principle,by constructing non-vanishing sub-solutions or vanishing super-solutions.The main results in this chapter are as follows:Theorem 1. Assume one of the following holds:(i) q> m.(ii) min{q,1}>r, or q=r< 1 with a|Ω|<b. Then the solutions of(4) vanish in finite time for suitably small initial data.Theorem 2. Assume q< m. If q< r, or q=r with b< aγ, then Problem (4) admits at least one non-extinction solution for any non-negative initial data.Here γ=∫Ωφ1q/m(x)dx, and φ1(x)>0 (x∈Ω) is the first eigenfunction of the following eigenvalue problem which satisfies ‖φ1‖L∞(Ω)=1.Theorem 3. Assume that q=m.(â…°) If aμ<1, then all the solutions of (4) vanish in finite time for any non-negative initial data;(â…±) If q< r< 1, then the solutions of (4) vanish in finite time for any non-negative initial data when aμ=1, and vanish in the sense of limtâ†'+∞ ‖(·,t)‖2=0 when a|Ω|≤λ1.Here λ1 is the first eigenvalue of -Δ in Ω with homogeneous Dirichlet boundary condition;(â…²) If q< r with aμ> 1, or q<1≤r with aμ=1, then Problem (4) admits at least one non-extinction solution for any positive initial data.Here μ=∫Ωφ(x)dx, and φ(x) is the unique positive solution of the following elliptic problem-Δφ(x)=1, x∈Ω;φ(x)=0, x∈(?)Ω.In Chapter 3, we will extend the results obtained in Chapter 2 and investi-gate the extinction properties of solutions to a class of non-Newtonian polytropic filtration equations with nonlocal sources and absorptions where a,b,m,q,r> 0,0<m(p-1)< 1,Ω is a bounded domain in RN(N≥1) with smooth boundary (?)Ω, and the initial datum u0(x) is a nonnegative nontrivial function such that v0m∈L∞(Ω)∩W01,p(Ω).We first apply Leary-Schauder’s fixed point theorem to the study of local ex-istence and uniqueness, global existence and global boundedness of weak solutions to this problem,and then investigate the extinction properties of the solutions by using the sub and super-solution methods. Similarly to the cases in Chapter 2, we can only compare some special sub arid super-solutions with each other. Based on the weak comparison principle, we give a much more complete classification of the exponents and coefficients regarding whether the solutions vanish in finite time or not, by constructing vanishing super-solutions or non-vanishing sub-solutions. The main results in this part are as follows:Theorem 4. Assume that one of the following holds:(â…°)q>m(p-1).(â…±) min{g,1}> r, or q=r< 1 with a |Ω|< b. Then the solutions of (5) vanish in finite time for small initial data.Theorem 5. Assume that q< m(p-1). If q<r, or q-r with b< aγ, then Problem (5) admits at least one non-extinction solution for any non-negative initial data.Here γ=∫Ωφ1q/m(x)dx, and φ1(x)> 0 (x∈Ω) is the first eigenfunction of the following eigenvalue problem which satisfies ‖φ1‖L∞(Ω)=1.Theorem 6. Assume that q= m(p-1).(i)If ak<1, then the solutions of (5) vanish in finite time for any non-negative initial data;(ii) If q< r< 1, then the solutions of (5) vanish in finite time for any non-negative initial data with ak=1, and vanish in the sense of limtâ†'+∞‖u(·,t)‖m+1= 0 when a|Ω|≤λ1-Here λ1>0 is the first eigenvalue of p-Laplace operator in Ω;(iii) If q<r with ak> 1, or g<1≤r with ak=1, then Problem (5) admits at least one non-extinction solution for any positive initial data.Here k=fΩφp-1(x)dx,and φ(x) is the unique positive solution of the following elliptic problemIn Chapter 4 we study the extinction properties of solutions to a system of fast diffusion equations with nonlinear sources where p, q>1, m, n, a,/3> 0, m(p-1), n(q-1)< 1,Ω is a bounded domain in RN (N> 1) with smooth boundary (?)Q and u0(x), v0(x) are nonnegative nontrivial functions with u0m∈ L∞(Ω)∩ W01,,p(Ω),v0n∈L∞(Ω)∩W01,1(Ω).It should be pointed out that the sub and super-solution methods used in the Chapters 2 and 3 are not applicable to Problem (6) for two reasons. One is that it is quite difficult to construct a suitable super-solution that vanishes in finite time when αβ> mn(p-1)(q-1). The other difficulty that excludes the possibility of applying super and sub-solution method is that the nonlinearities in (6) may be non-Lipschitz, which makes the comparison principle invalid (even if for a very weak form). In order to overcome these difficulties, we combine the modified integral estimate methods with the invariant domains of ordinary differential equations to show that the solutions of (6) vanish in finite time when the nonlinear sources are in some sense weak and when the initial data u0 and u0 are "comparable".Moreover, we also obtain a non-extinction result for some special cases by using the monotone iteration process, which, to our best knowledge, seems to be new for quasilinear parabolic systems of this kind. The main results in this chapter are as follows:Theorem 7. Assume that mn(p-1)(q-1)<αβ.(â… )If αβ≤1 and the initial data (u0,v0) satisfy, for some 0<δ1<1, that then every solution of(6)vanishes in finite time;(â…¡)If αβ>1 and the initial data (u0,v0) satisfy, for some 0<δ2<1, that then every solution of (6) vanishes in finite time for sufficiently small initial data. Here ai,bi,a’i,b’i>0(i=1,2), s,r,s’,r’>1,0<α1<α and 0<β1≤β are constants to be defined in the process of the proof.Theorem 8.Assume that mn(p-1)(q-1)= αβ and [Ω] is suitably small. Then there exists a solution of(6) vanishing in finite time for suitably small initial data.Theorem 9.Assume that p=q>1,0<a< n(p-1),0<β≤m(p-1) and αβ< mn(p-1)2. Then Problem (6) admits at least one non-extinction solution for any smooth positive initial data (u0,v0). estimate methods with the invariant domains of ordinary differential equations to show that the solutions of (6) vanish in finite time when the nonlinear sources are in some sense weak and when the initial data u0 and u0 are "comparable".Moreover, we also obtain a non-extinction result for some special cases by using the monotone iteration process, which, to our best knowledge, seems to be new for quasilinear parabolic systems of this kind. The main results in this chapter are as follows:Theorem 7. Assume that mn(p-1)(q-1)<αβ.(â… )If αβ≤1 and the initial data (u0,v0) satisfy, for some 0<δ1<1, that then every solution of(6)vanishes in finite time;(â…¡)If αβ>1 and the initial data (u0,v0) satisfy, for some 0<δ2<1, that then every solution of (6) vanishes in finite time for sufficiently small initial data. Here ai,bi,a’i,b’i>0(i=1,2), s,r,s’,r’>1,0<α1<α and 0<β1≤β are constants to be defined in the process of the proof.Theorem 8.Assume that mn(p-1)(q-1)= αβ and [Ω] is suitably small. Then there exists a solution of(6) vanishing in finite time for suitably small initial data.Theorem 9.Assume that p=q>1,0<a< n(p-1),0<β≤m(p-1) and αβ< mn(p-1)2. Then Problem (6) admits at least one non-extinction solution for any smooth positive initial data (u0,v0).