In the manuscript the blow up phenomena for classical nonlinear equations of parabolic type with double nonlinearities and for hyperbolic type equations with positive energy are studied. Moreover the systems of similar equations are also considered. In the investigation the modified energetic method, developed earlier by M.O.Korpusov and A.G.Sveshnikov, is used. By this method sufficient conditions of a finite time blow-up for considered problems are obtained.

In this monograph, we consider problems on the local solvability
and the blow-up in a finite time of solutions to classical
nonlinear equations and systems of nonlinear equations of
mathematical physics. In particular, we consider equations and
systems of equations of parabolic type with double nonlinearities,
wave equations with positive energy, and systems of equations of
composite type.
These equations and systems of equations appear in nonlinear mathematical
physics. We present several examples of equations considered. We start from
parabolic equations with double nonlinearities.
Propagation of heat in a undeformable, immovable, homogeneous,
isotropic medium is described by the equation

Maksim Olegovich Korpusov received a mathematical physics degree and a Ph.D. degree in mathematical physics from the Physics Department of Moscow State University, Russia, in 1998 and 2005, respectively. Since 2008, he has been a Full Professor at the Mathematical Division of Moscow State University, where he is heading a research team working in the area of nonlinear functional analysis and its applications. He received the 1998, 2004, 2006 Moscow State University Award for young mathematicians and the President Grant for Young Mathematicians (2005, 2007, 2009) (Russia). He is currently a Member of the Editorial Board of Differential Equations and Applications (DEA, Croatia).

Maksim Olegovich Korpusov received a mathematical physics degree and a Ph.D. degree in mathematical physics from the Physics Department of Moscow State University, Russia, in 1998 and 2005, respectively. Since 2008, he has been a Full Professor at the Mathematical Division of Moscow State University, where he is heading a research team working in the area of nonlinear functional analysis and its applications. He received the 1998, 2004, 2006 Moscow State University Award for young mathematicians and the President Grant for Young Mathematicians (2005, 2007, 2009) (Russia). He is currently a Member of the Editorial Board of Differential Equations and Applications (DEA, Croatia).

Alexey Vitalievich OVCHINNIKOV is a well-known expert in the field of nonlinear analysis.