Problems for nonlinear parabolic equations and variational inequalities with variable exponents of nonlinearity
The thesis consists of the introduction, seven chapters, conclusions, literature and addendum.
Chapter 1 provides an literature review concerning on the topic of the thesis.
Chapter 2 deals nonlinear parabolic equations with variable exponents of nonlinearity. The integro-differential equations with nonmonotonous main part and the systems of fourth order integro-differential parabolic equations are considered and the existence theorems are proved.
Chapter 3 is devoted to the doubly nonlinear equations with variable exponent of nonlinearity. The elliptic-parabolic differential equations and the elliptic-parabolic integro-differential equations with linear main part are investigated. Corresponding parabolic equations are also considered. We prove the existence theorems.
Chapter 4 deals semilinear parabolic equations with variable exponents of nonlinearity. The second order parabolic equations and the Petrovskii-type parabolic systems are considered. We prove the existence of the mild solutions to the initial-boundary-value problems for the mentioned equations and systems.
Chapter 5 is devoted to the parabolic variational inequalities with variable exponents of nonlinearities. The elliptic-parabolic doubly nonlinear variational inequalities and the nonlinear degenerate parabolic variational inequalities are studied. The existence theorems are proved.
Chapter 6 deals nonlinear parabolic variational inequalities and systems of the variational inequalities with monotonous main part and variable exponents of nonlinearities. The existence, uniqueness, smoothness, and comparison theorems are proved.
Chapter 7 is devoted to the parabolic variational inequalities in unbounded with respect to the either time or space variables domains. The existence and uniqueness theorems for second order parabolic variational inequalities and for higher order parabolic variational inequalities are proved.
Keywords: nonlinear parabolic equation, double nonlinear parabolic equation, nonlinear variational inequality, double nonlinear variational inequality, variable exponent of the nonlinearity, in generalized Lebesgue and Sobolev spaces.