Nonlinear Elliptic and Parabolic Problems

This monograph presents results on the existence, nonexistence, and qualitative properties of solutions for different classes of nonlinear stationary and evolution equations and inequalities. The authors deal with global, singular, very singular, weak, entropy, and other types of solutions of the considered problems. The book is targeted to researchers and post-graduate students that specialize in the field of the theory of nonlinear PDE’s.

1. Doctor of Science in Physics and Mathematics, Professor Andrej A. Kon’kov works at Lomonosov Moscow State University. His research interests focus on the blow-up phenomenon and behavior of solutions of nonlinear equations and inequalities near singularities. He also carried out research on the stabilization of solutions of nonlinear evolution problems. Professor Andrej A. Kon’kov took part in three international INTAS research projects. He has repeatedly spoken at international mathematical conferences. His papers were published in the well-known mathematical journals such as Nonlinear Analysis, Journal of Mathematical Analysis, Advanced Nonlinear Studies, Nonlinearity, Annali di Matematica Pura ed Applicata.

2. Doctor of Science in Physics and Mathematics, Professor Alexander A. Kovalevsky works at Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences and holds the position of Leading Researcher in the Department of Equations of Mathematical Physics. He is also a professor in the Department of Mathematical Analysis of the Ural Federal University. His research interests focus on the homogenization theory for nonlinear partial differential equations and variational inequalities, the study of Г-convergence of integral functionals and G-convergence of nonlinear differential operators with variable domain of definition, and the investigation of the existence and properties of solutions for different classes of nonlinear elliptic equations and variational inequalities with L-data. Professor Alexander A. Kovalevsky took part in three international INTAS research projects. He also participated in the International Congress of Mathematicians (Zurich, Switzerland, 1994), the Second European Congress of Mathematics (Budapest, Hungary, 1996), and in numerous international mathematical conferences in Russia, France, and Italy. In recent years, he has published in well-known journals such as NonlinearAnalysis, Annali di Matematica Pura ed Applicata, Nonlinear Differential Equations and Applications, Results in Mathematics, and Ricerche di Matematica.

3. Doctor of Science in Physics and Mathematics, Professor Andrey E. Shishkov works at S.N. Nikol'skii Mathematical Institute of RUDN University, Moscow, and holds the position of director of Center of Nonlinear Problems of Mathematical Physics. His research interests are concentrated on different aspects of qualitative theory of nonlinear PDE: blow-up of solutions, asymptotic properties, thin-film equation, Cahn-Hilliard equation, very singular solutions and large solution of semilinear elliptic and parabolic equations, existence-nonexistence of global solutions. Professor Andrey E. Shishkov took part in four international INTAS research projects. He also participated in the International Congresses of Mathematicians (Zurich, Switzerland, 1994; Seoul, Republic Korea, 2014), in the 3rd International Congress on Industrialand Applied Mathematics (Hamburg, Germany, 1995), in the Interdisciplinary Congress: Free Boundary Problems, Theory and Applications (Zakopane, Poland, 1995), in the International AMS conferences "Dynamics, Differential Equations and Applications" ( Poitiers, France, 2006; Madrid, Spain, 2014), Colloquium EUROMECH 490 "Dynamics and Stability of Thin Liquid Films and Slender jets" (Imperial College London, UK, 2007), and numerous international mathematical conferences and workshops in Germany, France, Israel, Italy, Russia, Hungary, Netherland, Slovakia, Ukraine, and other. In November 2019, A.E. Shishkov organized as Chairman the International Conference "Singular Problems, Blow-up, and Regimes with Peaking in Nonlinear PDE" (RUDN University, Moscow, Russia, 2019). In recent years, A.E. Shishkov published papers in well-known advanced journals such as Calculus of Variations and PDE, Mathematische Nachrichten, Annali della Scuola Norm. Super. Pisa Cl. Sci., Interfaces and Free Boundaries, Indiana University Mathematics Journal, Journal of Differential Equations, Journal of Evolution Equations, Annales de l'Institut H. Poincare. Anal. Non Lineaire, Communications in Contemporary Mathematics, Journal of Mathematical Analysis and Applications, Nonlinearity, Advanced Nonlinear Studies, and other.