The Semi-bound Problem For The Generalized IMBq Equation

In this paper, we study the semi-bound problem for a class of the generalized IMBq equation. This paper consists of four chapter. The first chapter is the introduction. In the second chapter, we will study the existence and uniqueness of the local solution to the initial boundary value problem for the generalized IMBq equation. In the third chapter,we will prove the existence and uniqueness of the global solution to the problem mentioned in Chapter two by integral estimates. In the last chapter, we will discuss the unexistence of the solution to the problem mentioned in Chapter two.In the second chapter, we study the existence and uniqueness of the local solution to the following initial boundary value problem for the generalized IMBq equation on the half-space H²(R⁺)∩H₊0¹(R⁺).Where u(x,t) denotes the unknown function, f(s) is the given nonlinear function, u₀(x) and u₁{x) are given initial value functions, and the subscript t,x indicate the partial derivative with respect to t, x.For this purpose, we first convert the problem (0.1),(0.2) to the following equationWhereSo the problem (0.1),(0.2) are equivalent to the equationThen, using the contraction mapping pinciple, we can obtain the existence and uniqueness of the local solution. The main result is the following: Theorem 1. Assume that , then the problem (0.1)-(0.3) has a unique local solution u(x,t)∈C([0,T₀),H²{R⁺)∩H₀¹(R⁺)), where [0,T₀) is a maximal time inteval. Moreover, ifthen T₀ =∞.In Chapter three, we first get energy identity by Fourier sine transform, and then get some integral estimates. At last, the existence and uniqueness of the global solution to the problem (0.1)-(0.3) are proved. The main result is the following:Theorem 2. Assume that , and satisfiesthen the problem (0.1)-(0.3) has a unique global solutionWhere , F_s^-1 and F_s indicate respectly Fourier sine inverse transform and Fourier sine transform.In Chapter four, the blow-up of the solution to the problem (0.1)-(0.3) is proved by means of concavity method. The main result is the following:Theorem 3. Suppose that , , and , there exists a canstantα> 0, so that for all u∈R,Then the solution of the problems (0.1)-(0.3) blows-up in finite time if and only if one of the following conditions holds:(1). E(0)<0, (2). E(0)=0, (3). E(0) > 0,...