Blow-up theory for nonlinear partial differential equations. Critical nonlinearities

Global analysis of nonlinear partial differential equations reveals existence of critical nonlinearities determined by corresponding growth rates.
These critical exponents depend on the structure and order of the differential operator, space dimension, singularity of the coefficients, and on the data of the problem for the corresponding equation.
It turns out that if the nonlinearity belongs to the critical range, then each solution of the corresponding problem necessarily blows up, i.e., a catastrophe occurs in finite time or outside of a certain volume, irrespectively of boundary conditions.
Later it was established that the regularity of the solutions also depends on the critical exponents of the nonlinearities.
The first results in this direction were obtained by:

• for elliptic equations in 1965 (S. I. Pohozaev), 1981 (B. Gidas and J. Spruck),
• for parabolic equations in 1966 (H. Fujita),
• for hyperbolic ones in 1979 (F. John), 1980 (T. Kato).

In this talk we consider a new approach to the study of nonlinearities based on the concept of nonlinear capacity generated by the nonlinear operator.
As applications we demonstrate concrete examples, including those from nonlinear mathematical physics, such as the Kuramoto-Sivashinsky equation.