Optimal Energy Decay for the Damped Klein-Gordon Equation

Abstract

In this dissertation we study the long time dynamics of damped Klein-Gordon and damped fractional Klein-Gordon equations using $C_0$- Semigroup theory and its application. The $C_0$-semigroups are used to solve a large class of problems commonly known as evolution equations. Such models arise from delay differential equations and partial differential equations in many disciplines including physics, chemistry, biology, engineering, and economics. Water waves, sound waves and simple harmonic motion of strings are few important models of evolution equations. The Klein-Gordon equation is a relativistic version of the Schr\"odinger equation. It was named after Oskar Klein and Walter Gordon who proposed it to describe quantum particles in the framework of relativity. It describes the motion of spinless composite particles. Indeed, one of the most fundamental questions that should be asked when studying these equations is whether the solution (if it exist) goes to equilibrium (stable) state or behaves erratically as time evolves. Understanding these properties can help determine how robust a system is, as well as provides insight on the characteristics of the corresponding phenomena it is modeling. In the first part we consider a one dimensional damped Klein-Gordon equation on the real line. It is well known fact that if there is no external force (i.e damping) acting in the system, the wave will oscillate forever in time since the energy is conserved in the system. An interesting question to ask is at what rate the energy starts leaving the system when we introduce damping force? This question was intensely studied in the last ten years. In this direction, Burq and Joly have proved that the energy decays at exponential rate if the damping force $\gamma(x)$ satisfies the geometric control condition (GCC) in a sense that there exist $T$, $\epsilon>0$, such that $\int_0^T\gamma(x(t))dt\geq \epsilon$ along every straight line unit speed trajectory. However, GCC does not provide an optimal condition to ensure exponential rate of energy decay. We address this problem in chapter 2 and provide optimal conditions on the damping coefficient $\gamma$ under which the exponential decay holds in one-dimensional setting. In addition, we derive simple to verify necessary and sufficient conditions for such exponential rate of decay. In the second part we relate the energy decay rate for the fractional damped wave equation to the order of its fractional derivative. In fact we prove that the energy decays at a polynomial rate if the order of derivative lies between $00$, such that $\int_0^T\gamma(x(t))dt\geq \epsilon$ along every straight line unit speed trajectory. However, GCC does not provide an optimal condition to ensure exponential rate of energy decay. We address this problem in chapter 2 and provide optimal conditions on the damping coefficient $\gamma$ under which the exponential decay holds in one-dimensional setting. In addition, we derive simple to verify necessary and sufficient conditions for such exponential rate of decay. In the second part we relate the energy decay rate for the fractional damped wave equation to the order of its fractional derivative. In fact we prove that the energy decays at a polynomial rate if the order of derivative lies between $0<s<2$ and at an exponential rate when $s\geq 2$ provided the damping coefficient is non-zero and periodic. An important ingredient of the proof is the derivation of a new observability estimate for the fractional Laplacain. Such important estimate has potential applications in control theory.