This article is cited in 1 paper

Blow-up of the solution to the Cauchy problem for one $(N+1)$-dimensional composite-type equation with gradient nonlinearity

M. O. Korpusov^ab, A. A. Panin^ab, A. K. Matveeva<sup>ac

a</sup> Faculty of Physics, Lomonosov Moscow State University, Moscow, Russia
^b Peoples' Friendship University of Russia, Moscow, Russia
^c National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, Russia

Abstract: We consider the Cauchy problem for a third-order nonlinear evolution equation with nonlinearity $|D_xu|^q$. Two exponents, $q_1=N/(N-1)$ and $q_2=(N+1)/(N-1)$, are found such that for $1<q\leqslant q_1$, there is no weak solution local in time for any $T>0$; for $q_1<q\leqslant q_2$, there is a unique weak solution local in time; however, there is no weak solution global in time, i.e., independently of the “value” of the initial function, the solution to the Cauchy problem blows up in a finite time.

Keywords: nonlinear equations of Sobolev type, blow-up, local solvability, nonlinear capacity, blow-up time estimate.

MSC: 74H35, 35K70

Received: 16.03.2025
Revised: 16.03.2025

DOI: 10.4213/tmf10984

 English version: Theoretical and Mathematical Physics, 2025, 225:1, 1811–1829

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