Analysis of solutions of parabolic type equations in sobolev spaces with variable exponents

This thesis focuses on the analysis of solutions equations of parabolic type in Sobolev spaces with variable exponents. Partial differential equations with variable exponents offer a more advanced approach to modelling physical phenomena compared to equations with fixed exponents. For this reason, there has recently been considerable interest in mathematically modelling parabolic equations with variable exponents. For example, modelling the behavior of electro-rheological fluids helps to understand how the viscosity of the fluid responds to an electric field. Similarly, modelling liquids with temperature-dependent viscosity is very important for science and engineering applications. Studying filtration processes in porous media contributes to understanding groundwater movement, pollution transport and petrochemical production processes. Variable exponent equations have also had a significant impact on image processing, enabling more efficient image enhancement techniques. Evolution equations describe how a system changes over time and are widely used in various scientific disciplines such as physics, biology, chemistry and economics. In the first part of the thesis, the historical development of the evolution equations of parabolic type with variable exponent and their importance in applied sciences are discussed. In the second chapter, a literature review of previous works on parabolic equations in Sobolev spaces with variable exponents is presented. In the third chapter, the basic definitions, lemmas, theorems and inequalities that will be used throughout the thesis are detailed. In the fourth chapter, existence and blow-up of solutions of m(x)-biharmonic equations with variable exponent are discussed. In the fifth section, existence and decay of solutions of Kirchhoff equations of parabolic type with variable exponent are analyzed. In the sixth section, existence and blow-up of solutions of a parabolic equation with a nonlinear logarithmic source term with variable exponent are shown. Finally, in the seventh section, the existence and blow-up of solutions of viscoelastic m(x)-biharmonic equations with a logarithmic source term are discussed. In the conclusion, an overview of the analyses is given and the contributions of the findings to the literature and potential areas for future research are discussed.