Application Of Green’s Functions In Nonlinear Equations With Diffusion Mechanisms

This dissertation is mainly devoted to Green’s function method and its ap-plication in nonlinear equations with diffusion mechanisms. We consider two kinds of nonlinear diffusion equations. One of them is chemotaxis model, which possesses linear diffusion mechanism and nonlinear cross diffusion mechanisms. The competitive interplay between diffusion and cross diffusion is one key feature of the model, and also the main difficulty in our study. The other kind is scalar viscous conservation law. The corresponding linear equation around the shock wave possesses not only a linear term with variable coefficient, but also a diffu-sion term. We shall study the initial boundary value problem and the Cauchy problem with large perturbation of the two kinds of equations, respectively. The main results are generalized as follows.The Chapter 1 is the introduction. Here we show backgrounds of Green’s function, chemotaxis and viscous conservation law, and introduce some important results.In Chapter 2, we study the well-posedness of one attraction-repulsion chemo-taxis model (ARC) in the whole space. This Chapter is consist of two sections. In Section 1, we consider the pointwise estimates of solutions. By the Green’s function and the precise estimates for the nonlocal terms, we obtain the point-wise estimates of solutions for the Cauchy problem with small initial data, and we get further the Ws.p decay rates of solutions. The results suggest that the long time behavior of solutions coincides with the heat equation. In Section 2, we continue to study the ARC model with large initial data. This case is different from the former case, because here the attraction may prevail over the repulsion and the diffusion, and thereby leads to blow up. This depends on the relations of parameters of the equations. We prove that when the repulsion prevails over the attraction, the problem always admits a unique global solution, and the decay estimate is obtained. Here the methods employed are mainly the energy esti-mates and the Moser-Alikakos iteration technique. Particularly, with the help of the half-order derivative method, we prove the smoothness of solutions under the weak assumptions. Then when the attraction prevails over the repulsion and under some conditions, we can use the moment method to prove that the finite time blow-up may occur.In Chapter 3, we concern the initial boundary value problem of Keller-Segel model in the half space xn> lt. In order to keep the mass conserved, we propose here a conservative boundary condition. Under the condition, we study the global solvability, regularity and long time behavior of solutions. With the help the Fourier-Laplace transformation and complex analysis, we first construct the Green’s function of the linear equations and study its properties. Then by the Green’s function, we give the implicit formula of solutions. We prove that when the initial data is small enough, the half space problem is always globally and classically solvable. Moreover, we obtain decay estimates of solutions when the parameter l have different signs. We state that the sign of l will make important influence on the long time behavior of solutions. For l<0, the time decay of solutions is (1+t)-n/2 which coincides the heat equation. While for l> 0, we have different results which also depends on the space dimension n. When n> 2, the time decay is (1+t)-(n-1)/2. While when n= 1, the solutions will converge to a conserved stationary solution with exponential time decay. We state that the conserved stationary solution will not be 0 if and only if the initial mass is not 0. This is a very interesting result, which tells us that the boundary will essentially influence the long time behavior of solutions in some sense.In Chapter 4, we are interested in nonlinear stability of large perturba-tion around the planar shock wave for scalar viscous conservation law in two dimensions. Because of the special structure of equation and the loss of small-ness assumption, we cannot just employ Green’s function method or L2 energy method to derive the global existence and decay rates of solution. Fortunately, we can obtain the maximal principle and the contraction property. With them and classical energy estimates, We prove that the problem always admits a global classical solution if the initial perturbation is in L1 ∩H4(R2). Furthermore, for large perturbation problem of certain types of the 2-d scalar viscous conservation law, we obtain nonlinear stability of the shock profile for weak shock, and estab-lish the L2 decay rate t-1/4 and L∞ decay rate t-1/2 of solution toward the planar shock wave. The idea of the proof is using a technique combining the semigroup approach and the energy method to get some smallness estimates, and then using the energy inequality to obtain the asymptotic behavior.