On the critical exponent “instantaneous blow-up” versus “local solubility” in the Cauchy problem for a model equation of Sobolev type,Izvestiya: Mathematics

当前位置： X-MOL 学术 › Izv. Math. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

On the critical exponent “instantaneous blow-up” versus “local solubility” in the Cauchy problem for a model equation of Sobolev type
Izvestiya: Mathematics ( IF 0.8 ) Pub Date : 2021-02-11 , DOI: 10.1070/im8949
M. O. Korpusov ₁ , A. A. Panin ₁ , A. E. Shishkov ₂

Affiliation

We consider the Cauchy problem for a model partial differential equation of order three with a non-linearity of the form $|\nabla u|^q$ . We prove that when $q\in(1,3/2]$ the Cauchy problem in $\mathbb{R}^3$ has no local-in-time weak solution for a large class of initial functions, while when $q>3/2$ there is a local weak solution.