Qualitative analysis of solutions of some of hyperbolic type equations with variable exponent

The first chapter thoroughly reviews the historical development of studies concerning energy decay and blow-up of solutions. In the second chapter, the fundamental concepts, main theorems, and important inequalities that will be utilized throughout the thesis are presented in detail. In the third chapter, the basic lemmas related to energy decay and blow-up of solutions, along with their proofs, are provided, forming the essential analytical tools employed in the subsequent analysis. In the fourth chapter, the energy decay of solutions to a Lame-type inverse problem involving nonlinear source and damping terms is studied. The following problem is examined: $\begin{align} & {{u}_{tt}}-{{\Delta }_{e}}u-div(|\nabla u{{|}^{r(x)-2}}\nabla u)+\beta {{u}_{t}}+h(x,t,u,\nabla u)+a|{{u}_{t}}{{|}^{m(x)-2}}{{u}_{t}}+b|u{{|}^{p(x)-2}}u \\ & =f(t)w(x),(x,t)\in \Omega \times (0,\infty ), \\ & u(x,t)=\frac{\partial u}{\partial v}(x,t)=0,\qquad (x,t)\in \partial \Omega \times (0,\infty ), \\ & u(x,0)={{u}_{0}}(x),{{u}_{t}}(x,0)={{u}_{1}}(x),\qquad x\in \Omega , \\ & \int_{\Omega }{u}(x,t)w(x)dx=\phi (t),\qquad t>0, \\ \end{align}$ where $\Omega $ is a bounded domain with smooth boundary $\partial \Omega $ in ${{\mathbb{R}}^{n}}(n\ge 1)$, $\beta ,b,a>0$ and $w(x)$, $h(x,t,u,\nabla u)$ are real functions. Also, ${{\Delta }_{e}}$refers to the elasticity operator, which is a differential operator of size $n\times n$ and is defined as follows; \[{{\Delta }_{e}}u=\mu \Delta u+(\alpha +\mu )\nabla (div\ u),\quad u={{({{u}_{1}},{{u}_{2}},...,{{u}_{n}})}^{T}},\] where $\mu $ and $\alpha $ are the Lame constants such that, \[\mu >0,\quad \alpha +\mu \ge 0.\] This problem originates from elasticity theory and represents a model frequently encountered in structural analysis. It is demonstrated that the energy functional of the solutions decreases over time and converges to zero under certain conditions. The analysis carried out under the Luxemburg norm is supported by Poincaré-type inequalities and differential energy estimates. In Theorem 4.3.1, it is proved that the energy functional of the solutions decays exponentially with respect to time. In the fifth chapter, blow up of solutions of inverse problem for the wave equation with source and nonlinear damping terms is studied. The following problem is examined: \[\begin{align} & {{u}_{tt}}-div({{\left| \nabla u \right|}^{r(x)-2}}\left| \nabla u \right|)+a{{\left| {{u}_{t}} \right|}^{m(x)-2}}{{u}_{t}}-b{{\left| u \right|}^{p(x)-2}}u=f(t)w(x),(x,t)\in \Omega \times (0,T), \\ & u(x,t)=0,\partial \Omega \times (0,T), \\ & u(x,0)={{u}_{0}}(x),{{u}_{t}}(x,0)={{u}_{1}}(x),\Omega , \\ & \int\limits_{\Omega }{u(x,t)w(x)dx}=\phi (t),t>0, \\ \end{align}\] where $\Omega $ is a bounded domain with smooth boundary $\partial \Omega $ in ${{\mathbb{R}}^{n}}(n\ge 1)$, and a unit outer normal ν. Also, $a$ and $b$ are positive constants and $w(x)$ and \[\phi (t)\]are real valued functions. The exponents $p(x)$, $m(x)$, and $r(x)$ are measurable and continuous functions on $\bar{\Omega }$ such that satisfy the following inequality; \[2\le {{r}_{2}}<{{m}_{1}}\le m(x)\le {{m}_{2}}<{{p}_{1}}\le p(x)\le {{p}_{2}}\le {{r}_{*}}(x),\] with $\begin{align} & {{p}_{1}}:=ess\underset{x\in \bar{\Omega }}{\mathop{inf}}\,p(x),{{p}_{2}}:=ess\underset{x\in \bar{\Omega }}{\mathop{\sup }}\,p(x), \\ & {{r}_{1}}:=ess\underset{x\in \bar{\Omega }}{\mathop{inf}}\,r(x),{{r}_{2}}:=ess\underset{x\in \bar{\Omega }}{\mathop{\sup }}\,r(x), \\ & {{m}_{1}}:=ess\underset{x\in \bar{\Omega }}{\mathop{inf}}\,m(x),{{m}_{2}}:=ess\underset{x\in \bar{\Omega }}{\mathop{\sup }}\,m(x), \\ \end{align}$ and ${{r}_{*}}(x)=\left\{ \begin{array}{*{35}{l}} \frac{Nr(x)}{esssu{{p}_{x\in \bar{\Omega }}}(N-m(x))}, & {{r}_{2}}0$ and $0<\delta <1$. Energy decay has been proven using the Nakao inequality. First, an $E(t)$ function associated with this system of equations is derived, and it is shown that the function $E(t)$ is non-increasing. Then, energy decay is demonstrated in two different cases: for $\rho =1$ and for $1<\rho <\frac{3}{2}$, using certain inequalities and lemmas. The blow-up of the solutions has been shown for three different initial energy conditions: $E(0)>0$, $E(0)=0$, and $E(0)<0$, in the case where $m(x)=2$ is the given function using the method found by Li and Tsai in 2003. Additionally, for $m(x)\ne 2$, the blow-up of the solution in a finite time ${{T}^{*}}$ for a solution $u(x)$ is shown using the method developed by Georgiev-Todorova in 1994. The final and seventh chapter of the thesis contains the discussion and conclusion sections. Here, the innovations of the results presented in the thesis are discussed, as well as the potential future studies that could benefit from these findings. Keywords: Global existence, Blow-up of solutions, Energy decay, Variable exponent, Inverse problem, Wave equation, Viscoelastic equation