A Study On The Well - Posedness Of The Solution Of Two Shallow Wave Equations

In this paper, we investigate some well-posedness problems for the generalized two-component Camassa-Holm system and a generalized Degasperis-Procesi equa-tion on the circle. These problems include local well-posedness, blow-up scenario and the sufficient conditions for blow-up solutions and sufficient conditions for the existence of global solutions.In Chapter1, we study the generalized two-component Camassa-Holm system which can be derived from the theory of shallow water waves moving over a linear shear flow. A sufficient condition to guarantee the solution blow up in finite time is given for the system, i.e. Theorem1.3.2, the main content of the result is:Let (u0,p0)∈Hs(S)×Hs-1(S)(s>3/2) and T>0be the maximal time of existence of the corresponding solution (u, p) to the above system. Assume there exist x1,x2∈S such that u0,x (z1)=inf u0,x (x), ρ0(x1)=0and When0<σ<1,1<σ≤3,σ>3, σ<0, providing suitable conditions on m1(0)+m2(0) respectively, we can show that the corresponding solution (u, ρ) to system (0.3) blows up in finite time.Especially, we get Theorem1.3.4, the main content of the result is:Let (u0,p0)∈Hs(S)×Hs-1(S)(s>3/2) and T>0be the maximal time of existence of the corresponding solution (u, p) to system (0.3). Assume that∫s u0dx=(α0)/2. If there exist x3, x4∈S such that u0,x (x3)=inf u0,x (x), ρ0(x3)=0and When0<σ≤1,1<cσ<3, σ>3, σ<0, providing suitable conditions on m.i(0)+m2(0) respectively, we can prove that the corresponding solution (u,p) to system (0.3) blows up in finite time.In Chapter2, we study the following generalized nonlinearly dispersive equation known as μ-Degasperis-Procesi equationWe first demonstrate some conditions on the initial data that lead to finite time blow-up of certain solutions for the above system, i.e. Theorem2.3.2and Theorem2.3.3.The main content of Theorem2.3.2is:Let u0∈Hs(S), s>3/2, u0≠0and T>0be the maximal time of existence of the corresponding solution u(t, x) to (0.4) with the initial data uq. If where μ0,μ2are defined by (2.15) and (2.16), respectively, then the corresponding solution u(t, x) to (0.4) must blow up in finite time.The main content of Theorem2.3.3is:Let ε>0. Assume the initial profile u0∈Hs(>), s>3/2has at some point x0∈S a slope which is less than-(1+∨)P(T1). Then wave-breaking for the corresponding solution of (0.4) occurs and the maximal time of existence is estimated above by T1.We also give some conditions for global strong solutions to the equation, i,e. Theorem2.4.1, the main content of the result is:If the initial potential value m0∈H1(S) satisfies that m0+3/2k does not change the sign, then the solution u(t) to the initial-value problem (0.4) exists permanently in time.We finally establish a blow-up rate result, i.e. Theorem2.5.1, the main content of the result is:Let u(t, x) be the solution to the initial-value problem (0.4) with initial data u0∈Hs(S), s>3/2. Let T>0be the maximal time of existence of the solution u(t,x). If T<∞, we have while the solution remains uniformly bounded.