The Cauchy Problem For A Class Of The Damped Ibq Equation

This paper consists of five chapters. The first chapter is the introduction. In thesecond chapter, we will study the existence and uniqueness of the local solution for theCa,uchy problem of the damped IBq equation .In the third chapter, we will study theexistence and uniqueness of the the global solution to the Cauchy problem for the abovementioned equation. In the fourth chapter,we will discuss blow-up of the solution tothe Cauchy problem for the above mentioned equation and give sufficient conditions ofblow-up of the solution. In the fifth chapter, we will get some estimates, we also get thedecay property of the solution in the condition of small initial data, then we will provethe existence of the global solution . The details are these:In the second chapter ,we study the following Cauchy problem for a class of thedamped IBq equation :where v₀≥0,v₁≥0,v₀+v₁＞0 are constants. u(x, t) denotes the unknown function,f(u) is a given nonlinear function, subscripts t and x indicate the partial derivative withrespect to t and x.For this purpose, we will get the following equivalent form of the problem (1):where the operator I-(?) is invertible.Then, using the contraction mapping principle , we can prove the existence anduniqueness of the local solution for the Cauchy problem (1), (2). The main result is thefollowing: Theorem 1 Suppose that s＞1/2,φ∈H^s,ψ∈H^s,f∈C^[s]+1(R),and f(0)=0, then the Cauchy problem (1), (2) has a unique local solution u∈C¹([0,T⁰);H^s),where[0, T⁰) is the maximal time interval of existence of u(x, t). Moreover, ifthen T⁰=∞.In Chapter 3, we prove the existence and uniqueness of the global solution to theCauchy problem (1), (2). The main results are the following:Theorem 2 Suppose that s＞1/2,φ∈H^s,ψ_x∈H^s,f∈C^[s]+1(R),f(0)=0 and[0, T⁰) is the maximal time interval of existence of the corresponding solution u(x, t) tothe Cauchy problem(1),(2). Ifwhere M₂ is a constant, then T⁰=∞.Theorem 3 Suppose thatφ∈H¹,ψ∈H²,F(u)≥0,f∈C²(R) and f(u)=(?),F(u)=(?)=(?), satisfieswhere A, B a,re constants, then the problem (1), (2) has a unique global solution u(x, t)∈C¹([0,∞);H¹).In Chapter 4, with the help of a weighting function , the blow-up of the solutionto the Cauchy problem (1),(2) is proved by means of the concavity method. The mainresult is the following:Theorem 4 Suppose that u(x, t) is the solution to the Cauchy problem (1), (2),φ∈H¹,ψ∈H²,(?)∈L²,F(u)=(?),F(u)∈L¹,f(u)∈C²(R),(a+1)u²+2aF(u)+f(u)u≥0,a≥1. Then the solution u(x, t) of the Cauchy problem(1),(2) blows-up in finite time, if one of the following conditions is satisfied: In Chapter 5, we discuss the existence of the global solution to the Cauchy problem(1), (2) by small initial data. First,we discuss the following linear problemWe obtain some estimates , then using these estimates and contraction mapping principle, we get the existence of the global solution to the Cauchy problem (1), (2). The mainresults are the following:Theorem 5 Suppose thatφ∈H¹âˆ©L¹,ψ∈H²âˆ©L¹,h∈L²(0,T:H¹âˆ©L¹). thenthe Cauchy problem(4), (5) has a unique generalized u(x,t)∈C²([0,T):H¹),((?)＞0).Moreover, we haveTheorem 6 Let f(u)∈C²,satisfiesf(u)=O(|u|^1+α), as uâ†'0,α≥3. Then thereexist a constantδ＞0, such that for (?)∈H¹âˆ©L¹,ψ∈H²âˆ©L¹, satisfyingCauchy problem (1), (2) has a unique solution u(x,t)∈C²([0,∞);H¹). moreover,where the constant C₀ only depends on f and initial value.