Entropy solutions of the hyperbolic equations

In this thesis the Cauchy problem for quasi linear partial equation of the first order is investigated. It is known that the solution of this problem has the discontinuity points whose locations unknown beforehand, if the initial profile has both the positive and negative slopes. The solution of this type can be defined by using the concept of the weak solution.However, it is known that too, the uniqueness of the weak solution is spoiled. In this thesis the method for obtaining the points of discontinuity of the solution of the Cauchy problem is suggested. In order to guarantee of uniqueness solution the entropy functions is introduced and the existence of the solution which satisfies of the increase condition of entropy is proved.