Global existence of solutions of parabolic type equation with logarithmic source terms

In the first chapter of this dissertation we indicate various applications of partial differential equations. Also we explain well posedness of a differential equation in the sense of Hadamard. Since differential equations do not always have classical solutions, we mention broaden the solution concept and speak about weak solutions. Weak solutions are solutions which solve an associated integral equation. An evolution equation generally means a partial differential equation with one of the independent variables being time t. Also in this chapter we mention some examples of evolution equations which arise from traffic and quantum mechanics. In the second chapter of this dissertation we give a historical overview of the development of some parabolic equations. We provide a list of researchers work on parabolic equation with logarithmic source term and Petrovsky parabolic equation with logarithmic source term. We summarized each researchers work. In the third chapter of this dissertation we give a brief overview some basic notions and theorems such as, definition of partial differential equation, Aubin-Lions compactness theorem, Hilbert spaces, dual space, weak convergence, lower semi-continuous, Lebesgue spaces, Hölder's inequality, Minkowski inequality, Grönwall's inequality, continuously differentiable functions spaces, weak derivative, Sobolev spaces and some embedding results which are necessary for underground of this dissertation. Also, we mention classification of second-order partial differential equation and give their eliptic, parabolic and hyperbolic equations definitions. Finally, in the fourth chapter of this dissertation we deals with an example of evolution systems of partial differential equations. Systems of differential equations arise naturally in many physical situations that are modeled with more than one equation and involve more than one dependent variable. We give sufficient condition for our equation, which called global existence of solutions of parabolic type equations with logarithmic source terms. In the proof of our partial differential equation we used Galerkin's method, Aubin-Lions compactness theorem, embedding theorems and Poincaré inequality.