The Initial Boundary Value Problems For A Class Of Generalized Imbq Equations With Damping Terms

In this paper,we study the following initial boundary value problems for aclass of generalized IMBq equations with damping termswhere u(x, t) denotes an unknown function, g(s) and f(s) are the given nonlinearfunctions, u₀(x) and u₁(x) are the given initial value functions, a, b＞0,α＞0,β＞0,γ＜0 andδare all constants , and the subcripts x and t indicatepartial derivatives.This paper consists of four chapters.The first chapter is the introduction.Weintroduce the physical phenomena and the relative results about the model equa-tion.In the second chapter we prove the existence and the uniqueness of the localclassical solution for the initial boundary value problems for a class of generalizedIMBq equations with damping terms.In the third chapter we prove the existenceand the uniqueness of the global classical solution for the problem (1)-(3).In thefourth chapter we give the sufficient conditions of blowup of the solution forthe problems (1)-(3) by virtue of the concavity method.The main results are asfollows:Theorem 1 Suppose that u₀(x),u₁(x)∈C²[0,1], and satisfy the boundaryconditions (2), f(s)∈C²(R),g(s)∈C¹(R), then the problem(1)-(3)has a uniquelocal classical solution u(x,t)∈C²([0,T₀);C²[0,1]), where [0, T₀) is the maximal time interval of existence of u(x, t) ;Furthermore,ifthen T₀=∞.Theorem 2 Suppose that u₀(x),u₁(x)∈C²[0,1], and satisfy the boundaryconditions (2),f(s)∈C³(R),|f'(s)|≤C,g(s)∈C¹(R), andwhereG(s)=(?),A,B,C＞0are constants, then the problem (1)-(3)hasa unique global classical solution u(x,t)∈C²([0,∞);C²[0,1]) .Theorem 3 Suppose that f(s)=0,g(s)∈L¹(R),G(s)=(?),u₀,u₁∈H¹(0,1), and there exists a constantη＞0, such thatthen the classical solution u(x, t) of the initial boundary value problem (1)-(3)blows up in finite time if one of the following conditions holdsand where...