A Class Of Degenerate Staggered Diffusion Equations With Boundary Layer Traveling Wave Solutions Exist

This thesis is composed of two parts.Our main result is in the first part. In this part, we investigate the existence of the traveling wave solutions for a certain degeneracy biological model with cross-diffusion and transition layers. Forwhere u and v are the density of two biological species.The rest states p_- = (u_-, v_-) at z = +∞and P₊ = (u₊, v₊) at z = -∞satisfyingHere, the typical examples of nonlinearities are given by the competitive dynamics:and the prey-predator dynamics:where all the coefficients a_i, b_i, c_i, e_i are positive constants.When S(β^*) > 0 with S(β^*) = (?)2sf(s,β^*)ds, by using singular perturbation method which is based on the implicit function theorem and with the help of center manifold theorem, we prove the existence of the traveling wave solutions with a speed c(ε)ε, and transition layers connecting two equilibrium points P_- and P₊, which extend the results of [23] to the case with cross-diffusion. Here we impose a restriction on the wave speed c that c = O(ε) and henceforth writeεc (c =O(1)) in place of c. Assume (u(z,ε),v(z,ε)), z = x-ct are traveling wave solutions for (1) with speed cε, which join two rest states (u_-,u_-) and (u₊,v₊). From (1), we have the equations for (u(z), v(z)) asand boundary conditionshere, ' denotes (?).The main results of this section are:Theorem 1 Assume f and g satisfy (A.1)-(A.4), which will be given in the introduction of part one. Then for the system (1), we have for small positiveε, there exists a traveling wave solution (u(z,ε), v(z,ε)) with a speed c(ε)εand transition layers satisfying that for small positiveρ,Furthermore, the speed c(ε)εsatisfieswhere, c^* is called the singular limit velocity and we call ((?)(z ,ε), (?)(z ,ε)) a singular limit traveling wave solution of (4) and (5), which becomes the lowest order approximation uniformly in R. The definition of ((?)(z ,ε), (?)(z ,ε)) as follows: here, u₀⁺(z,β^*), u₀^-(z,β^*), v₀(z,β^*) are the approximation solutions of (4) and (5) with slow scale. y⁺((?),β^*), y^-((?),β^*) are the approximation solutions of (4) and (5) with fast scale, which will be given in 2.1 and 2.2 especially.In the second part, we study the decays with algebraic rate of two class scalar reaction-diffusion equation.First we consider the following p-degree Fisher-type equationhere, p > 1 is a positive constant. We have known that there exist a positive constant c_*(p), if and only if c≥c_*(p), this Fisher-type equation has a traveling wave solution u(z), z = x- ct connecting two equilibrium points u = 0 and u = 1. In this part, by using center manifold theorem, we prove the traveling wave solution with non-critical speed c> c_* decays to zero at z = +∞with algebraic rate. And we can obtain: Theorem 2 For this p-degree Fisher-type equation, if c > c_*(p), the traveling wave solution u(z), z = x-ct connecting two equilibrium points u = 0 and u = 1 decays to zero at z = +∞with algebraic rate (?)Second we discuss the following viscous balance lawwhereε> 0 is called the viscosity parameter.Assume f, g∈C²(R), f" is bounded on any bounded interval of R, andAlso assume f and g satisfy other conditions, we know this equation has a traveling wave solution u(z), z = x - ct connecting two equilibrium points u = 0 and u =1. By using center manifold theorem, we prove the traveling wave solution with non-critical speed decays to zero at z = +∞with algebraic rate. And we get:Theorem 3 For the viscous balance law (7), under the above conditions, if c is a non-criticalspeed, the traveling wave solution u(z), z = x-ct connecting two equilibrium points u = 0 and u = 1 decays to zero at z =+∞with algebraic rate (?) ? If we also assume g satisfying g^(k-1)(0) = 0, g^k(0)≠0 with k > 1 is integer, the traveling wave solution decays to zero at z =+∞with algebraic rate (?)...