The Global Existence Of Nonlinear Rod Equations, Asymptotic Behavior And The Bursting

In the paper we study the initial boundary value problem for the nonlinear beam equationwhere x∈(0, l),t > 0. Under the suitable conditions on noninear term and initial data, we prove there exists a global weak solution to problem (P), and the solution decays exponentionally to zero as tâ†'∞. We show the solution to (P) blows up in finite time under suitable conditions.This dissertation consists of five sections.The first and second sections are introduction and preliminaries respectively.In the third section, we show the existence and asymptotics of the solution to problem (P), and give the main result as following,Theorem Assume that: i)σ∈C³(R),σ"'(s) is locally Lipchitz continuous,σ'(0) =σ"(0) = 0, |σ(s)|≤b|s|^m,s∈R, s∈R.ii) u₀∈W∩H⁴ ,u₁∈H₀² such thatThen for anyT＞0, problem (P) exists a unique generalized solution u∈C([0, T), H⁴∩H₀²)∩C¹([0,T]; H₀²âˆ©C²([0, T]; L²), andWhen we prove the existence of the solution, we use the Galerkin approximate method, and we define the energy functional to prove that the solution decays exponentionally to zero as tâ†'∞. In the process of the proof, we define the energy functional Then for any T > 0, problem (P) admits a unique generalized solution u∈C([0, T], H⁴∩H₀²)∩C¹([0, T]; H₀²)∩C²([0, T]; L²), andIn the forth section, we show the blowup of the solution, and give the following results:Theorem Assume that: i)σ∈C¹(R),σ(s)s < k (?)σ(r)dr < -kβ|s|^m+1,s∈R, where k > 2,β> 0 are constants.ii) u₀∈H₀² , u₁∈L², such that E(0) < 0, and E(0) is as shown in the forth theorem.Then the solution u blows up in finite time (?), that is whenμ> 0,and whenμ=λ= 0,where (?) is different for different conditions.We apply the proper auxiliary functionalto prove that the solution blows up in finite time, that is there is a (?) , we proveIn the fifth section, we give two examples related to the theorems, so we can comprehendthem easily.