Potential Theory for a Nonlinear Equation of the Benjamin–Bona–Mahoney–Burgers Typeстатья
Информация о цитировании статьи получена из
Web of Science,
Scopus
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 28 февраля 2020 г.
Аннотация:For the linear part of a nonlinear equation related to the well-known Benjamin–Bona–
Mahoney–Burgers (BBMB) equation, a fundamental solution is constructed, which is combined with
the second Green formula to obtain a third Green formula in a bounded domain. Then a third Green
formula in the entire space is derived by passage to the limit in some class of functions. The properties
of the potentials entering the Green formula in the entire space are examined. The Cauchy problem
for a nonlinear BBMB-type equation is considered. It is proved that finding its classical solution is
equivalent to solving a nonlinear integral equation derived from the third Green formula. The unique
local-in-time solvability of this integral equation is proved by applying the contraction mapping principle.
Next, the local-in-time classical solvability of the Cauchy problem is proved using the properties
of potentials. Finally, the nonlinear capacity method is used to obtain a global-in-time a priori estimate
for classical solutions of the Cauchy problem.