A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities. (Q2777816)

In the presented monograph, the authors expose a new general approach for obtaining a priori estimates for solutions to nonlinear partial differential equations and inequalities. This approach can be applied to a sufficiently wide class of nonlinear problems for which the problem of nonexistence of nontrivial solutions is studied. The method is based on using the concept of nonlinear capacity generated by a nonlinear differential operator. NEWLINENEWLINENEWLINEThe contents of this book are divided into three parts devoted to elliptic, parabolic, and hyperbolic nonlinear problems. The following main topics are described in the first part of the book: stationary inequalities which includes second (and higher)-order semilinear equations (and systems), and inequalities both for the cases of bounded and unbounded coefficients, second-order elliptic problems with local (complete) blow-up of a solution, second-order semilinear differential critical degeneracy, coercive problems, generalization of the Bernstein theorem, theorems on the nonexistence of solutions of quasilinear elliptic equations and weakly coupled systems. NEWLINENEWLINENEWLINEIn the second part, first-order evolution (parabolic) inequalities and systems of inequalities are studied. Special attention is paid to the problem of critical exponents in studying nonexistence of solutions. As an example, it is proven the classical Fujita theorem. NEWLINENEWLINENEWLINEThe third part is devoted to second-order evolution problems wherein nonlinear inequalities and systems including the second-order time derivative are considered in \(\mathbb{R}^N\times (0,\infty)\) and \(\mathbb{R}^N\setminus\{0\})\times (0,\infty)\). The authors focus the main attention to problems with noncompact supports for initial datum. The proposed method makes it possible to present the best possible results and to obtain the assertion about complete and instantaneous blow-up of solutions. The following main topics are reflected in this part: energy method and problems with compact support, method of test functions and problems with noncompact support, degenerate and singular hyperbolic problems in \(\mathbb{R}^{N+1}_+\), variational method: periodic solutions, comparison method for a system of wave equations, methods of test functions: nonexistence of solutions to systems of inequalities. NEWLINENEWLINENEWLINEThe final part of the book includes appendices which contain: semilinear differential inequalities on the Heisenberg groups, nonexistence of solutions to higher-order evolution differential inequalities.