Blow up and long time behavior of solutions of some types of partial differantial equations

In the first chapter, the historical progressions and various studies in the literature concerning the blow up of solutions and energy decay are discussed. In the second chapter, essential theorems, definitions and inequalities required for the thesis are presented. In the third chapter of this study, several fundamental lemmas and proofs concerning the blow-up of solutions and energy decay are provided. These are used in the fourth, fifth and sixth sections of the thesis. In the fourth chapter, the global existence, blow up of solutions and energy decay of the generalized Klein-Gordon equation system with source and nonlinear damping terms are studied. We examine the following problem: where is a bounded domain with smooth boundary in , and Let with ; and One can easily verify that Firstly, we find energy functional related to this system. is a nonincreasing function for Then, using some inequalities and lemmas, the energy decay for and are established by using Nakao's inequality. The blow up of the solution with negative initial energy was proved by the method introduced by Georgiev-Todorova in 1994. We suppose that the solution exists for all time and we find in a contradiction. Set , then and Define where is a small value to be determined later and Our purpose is to demonstrate that satisfies a differential inequality in the following format As a result, the solution of this system blows up within finite time , and where and are given above. In the fifth chapter, we study the blow up of solutions of the generalized Rosenau equations with a hydrodynamically damping term. The following problem is being examined: where are constants and . The blow up of solutions was proved by the Concavity method introduced by Levine in 1974. Firstly, the solution satisfies , with a corresponding evolution of as follows: Energy function of this equation satisfy Now we let where , and positive numbers. Let's suppose that , a positive function and twice differentiable, satisfies the following inequality on where is a constant. If and , thus there exists a positive constant such that as In the sixth chapter, we study the global existence, decay estimates of the energy function and blow-up of solutions for a system of p-triharmonic with strong and nonlinear damping term. The following initial-boundary value problem is being examined: where is a bounded domain with smooth boundary in , and . Let with ; and It is confirmed that Firstly, we find energy functional related to this system. So, is a nonincreasing function Then, using some inequalities and lemmas, the energy decay for and are derived using Nakao's inequality. The blow up of solutions was proved by the method introduced by Li and Tsai in 2003. We consider problem for the blow up of solutions. A solution with is termed as blow up if there exist a finite time such that We attain the blow up of solutions in three different ranges of the initial energy: , and . Keywords: Global existence, Blow up, Energy decay, Nakao inequality, Klein-Gordon equation, Rosenau equation, p-triharmonic equation