M. O. Korpusov, A. K. Matveeva, “On critical exponents for weak solutions of the Cauchy problem for a <nobr>$(2+1)$</nobr>-dimensional nonlinear composite-type equation with gradient nonlinearity”, Zh. Vychisl. Mat. Mat. Fiz., 2023, Volume 63, Number 6,Pages <nobr>1006

This article is cited in 4 papers

Partial Differential Equations

On critical exponents for weak solutions of the Cauchy problem for a $(2+1)$-dimensional nonlinear composite-type equation with gradient nonlinearity

M. O. Korpusov^a, A. K. Matveeva^ab

^a Faculty of Physics, Lomonosov Moscow State University, 119992, Moscow, Russia
^b National Engineering Physics Institute "MEPhI", 115409, Moscow, Russia

Abstract: The Cauchy problem for a model nonlinear equation with gradient nonlinearity is considered. We prove the existence of two critical exponents, $q_1=2$ and $q_2=3$, such that this problem has no local-in-time weak (in some sense) solution for $1<q\le q_1$, while such a solution exists for $q>q_1$, but, for $q_1<q\le q_2$, there is no global-in-time weak solution.

Key words: nonlinear Sobolev-type equations, blow-up, local solvability, nonlinear capacity, blow-up time estimates.

UDC: 517.95

Received: 04.05.2022
Revised: 22.12.2022
Accepted: 03.03.2023

DOI: 10.31857/S0044466923060133