High Order Diffusion Equations With Periodic Potentials Or Sources

In this monograph, we investigate high order diffusion equations with periodic potentials or sources in one spatial dimension. We are interested in time periodic solutionsand asymptotic estimates or behavior of the solutions to the initial boundary value problems. 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. Prom the early 19th century so far, diffusion equations have been widely investigated, among them periodic problems have been paid much attention. As far as we know, the researches on second order periodic diffusion equations are extensive,and many profound results have been obtained, see for example [5, 34, 58, 66]. But there are only a few investigations devoted to high order periodic diffusion equations. Notice that high order diffusion equations can be used to describe modelswith periodic factors, such as biological groups, the diffusion and migration of population (see [23, 53]) and so on. We can also find many corresponding numericalresults which provide the references to explain certain practical problems, see [2, 13, 73, 74]. On the other hand, the studies on stationary solutions and the stability of solutions ([1, 15, 33, 37]) indicate that the research on such equations needs various mathematical tools, and the solutions may take on multiple asymptoticstates under different conditions. On all accounts, the study on high order periodic diffusion equations has practical significance and academic value.We are engaged in the investigation of high order periodic diffusion equations, especially Cahn-Hilliard type equations and viscous diffusion equations. This monographis divided into three chapters to consider the Cahn-Hilliard type equations with periodic concentration dependent potentials and sources, the Cahn-Hilliard type equations with periodic gradient dependent potentials and sources and the pseudoparabolic equations with periodic sources.In the first chapter, we consider the following Cahn-Hilliard type equationwhere Q=Ω×R,γ>0 denotes the mobility, (?)(s,t) = H'_s(s,t) = a(t)_s³ - b(t)s. The equation (1) is a typical nonlinear fourth order diffusion equation. It was propoundedby Cahn and Hilliard [21] in 1958 as a mathematical model describing the diffusion phenomena in phase transition. Such equations can also be used to portray competition and exclusion of biological population [23], diffusion and migrationof population which are sensitive to environment [53], moving process of river basin [38], diffusion of oil film over a solid surface [69] and many other diffusion phenomenon. Prom the appearance of Cahn-Hilliard equation, many researchers have made profound investigations. The classical works can be found in the papers [7, 8, 32]. For the studies on the existence, uniqueness, regularity and asymptotic behavior, see [29, 49, 76, 77, 78] and [26, 50, 51,56, 57, 71, 74, 80]. The surveys in [37, 44, 48, 59, 73] were devoted to the research on the stationary solutions and the stability of solutions. Furthermore, some authors paid attention to periodic prob- lerns, such as spatial periodic problems [11, 25, 39, 54, 75] and periodic boundary problems [17, 24, 30, 51, 79]. But as far as we know, only a few papers deal with time periodic problems until now. In this chapter, we are interested in Cahn-Hilliard type equations with time periodic potentials and sources.When we consider the diffusion phenomenon such as population growth and migration, a class of more extensive equations—the viscous Cahn-Hilliard type equations have been proposed and widely studied [3, 6, 9, 14, 15, 31, 36, 55]. This type of equations can be used to describe the state of population growth and migrationmore factually. Thus, besides the equation (1) we also discuss the following viscous Cahn-Hilliard type equationwhere k > 0 is viscosity coefficient.For the above equations (1) and (2), two typical boundary value conditions are generally considered, that is Dirichlet boundary value conditions (see [4, 9, 31])and Neumann boundary value conditions (see [32, 37, 48])We develop our study basing on the first boundary value conditions. The equations also satisfy the following initial value conditionu(x,0) = u₀(x), x∈Ω.Moreover, we assume that the coefficients a(t), b(t) and the source f(x, t) in equations(1) and (2) areω-periodic in t. Then another important kind of solutions are time periodic solutions, which satisfy the equation (1) or (2), boundary value conditions (3) and the following periodic conditionwhereωis a positive constant.In the first chapter, we consider the Cahn-Hilliard type equation (1) and the viscous Cahn-Hilliard type equation (2). We are interested in the asymptotic estimatesof the solutions to the initial boundary value problem and the asymptotic behavior of the solutions to the viscous equation (2) as the viscosity coefficient tends to zero. We first obtain the existence of solutions to the initial boundary value problembasing on Leray-Schauder fixed point theorem, then prove the uniqueness of the solutions with conjugate method. By virtue of the energy F[u], we obtain the upper bound of periodic solutions, then prove the existence of nontrivial periodic solutions. Furthermore, for the equation (1), we prove that if the mobilityγis properly large, then for any initial data, there exists a point T^* such that for any time larger than T^*, the L² norm of solutions to the initial boundary value problems can be bounded by the upper bound of the periodic solutions. For the viscous equation (2), after obtaining some similar conclusions to the equation (1), we prove that when the viscositycoefficient tends to zero, the solutions of the initial boundary value problems and the periodic solutions converge to the corresponding solutions of the equations without viscosity almost everywhere.In the second chapter, we study another important class of Cahn-Hilliard type equations with periodic gradient dependent potential and sourceswhere Q =Ω×R,γ>0 denotes the mobility, k≥0 is the viscosity coefficient,Φ(s,t) = a(t)|s|^α-1s-b(t)s,A(s,t) =c(t)|s|^β-1s-d(t)s,α≥2,β>1. This kind of equation was proposed by King etc (see [46]) in 2003 for the case k = 0. Furthermore, Murray in [53] put forward a similar model when nonlocal effects and long range diffusion are taken into account. The investigations on this type of equations can also be found in [16, 22].Similar to the equations (1) and (2), we also consider the equation (5) with Dirichlet boundary value conditionsu=â–³u=0, x∈(?)Ω, t>0.(6)The equation (5) satisfies the initial value conditionu(x,0)=u₀(x), x∈Ω. (7)We also discuss the periodic solutions which satisfy the periodic conditionu(x,t)=u(x,t+ω), (x,t)∈Q. (8)We investigate the existence and uniqueness of the asymptotically stable nontrivialperiodic solutions of the equation (5) in the cases k = 0 and k > 0 respectively,and discuss the limit process when the viscosity coefficient k→0.We first use Leray-Schauder fixed point theorem to obtain the existence of nontrivial periodic solutions and the existence and uniqueness of the solutions to the initial boundary value problems. In the following, we show that if the mobilityγis properly large, there exists a unique periodic solution, which is asymptotically stable and attracts the solutions of the initial boundary value problems for any initial value. Exactly speaking, in the cases k=0 and k>0, the corresponding periodic solution attractsthe solutions of the initial boundary value problems in the norms of L² and H¹ respectively. At last, we prove that when k→0, the periodic solutions and the solutions of initial boundary value problem are respectively convergence to the corresponding solutions of the equations without viscosity. In the last chapter, we consider a type of viscous diffusion equation, which can be used to describe many mathematical and physical models, such as in the theory of seepage of homogeneous fluids through a fissured rock [10, 12,18], two-temperature theory for heat conduction in nonsimple materials [20, 68], and a model for populationswith the tendency to form crowds [61, 62]. This type of equations can also be called pseudoparabolic equations [65, 70], Sobolev type equations [45, 64] or Boussinesque type equations [42]. On the other hand, according to experimentalresults, some researchers have recently proposed modifications to Cahn's model which incorporate out-of-equilibrium viscoelastic relaxation effects. Then in such special situation, we can also arrive this kind of equations, see [59]. In this chapter, we discuss the following pseudoparabolic equation with nonlinear sourcesWe assume that the above equation satisfies boundary value conditionu(x, t) = 0,(x, t)∈(?)Ω×R (10)and initial value conditionu(x,0) = u₀(x),x∈Ω. (11)If we further suppose that the coefficient m(x, t) is periodic in variable t₁ then the equation (9) admits periodic solutions, i.e. satisfies (9), (10) and the periodic conditionu(x,t) = u(x,t+ω), (x,t)∈Q. (12)We investigate the existence of nonnegative nontrivial periodic solutions to the equation (9) and the asymptotic behavior of the solutions as the viscosity coefficientk tends to zero. In the case q>1, we apply topological degree method to prove the existence of nonnegative nontrivial periodic solutions to the problem (11)-(12). Then we discuss the existence and uniqueness of the solutions to the initial boundary value problems. Next, we prove the limit process of the solutions as k tends to zero. In another case 0