On critical exponents for weak solutions of the Cauchy problem for a non-linear equation of composite type

We consider the Cauchy problem for a model partial differential equation of third order with non-linearity of the form |u|q, where u = u(x, t) for x ∈ R3 and t ≥ 0. We construct a fundamental solution for the linear part of the equation and use it to obtain analogues of Green’s third formula for elliptic operators, first in a bounded domain and then in unbounded domains. We derive an integral equation for classical solutions of the Cauchy problem. A separate study of this equation yields that it has a unique inextensible-in-time solution in weighted spaces of bounded and continuous functions. We prove that every solution of the integral equation is a local-in-time weak solution of the Cauchy problem provided that q > 3. When q ∈ (1, 3], we use Pokhozhaev’s non-linear capacity method to show that the Cauchy problem has no local-in-time weak solutions for a large class of initial functions. When q ∈ (3, 4], this method enables us to prove that the Cauchy problem has no global-in-time weak solutions for a large class of initial functions. © 2021 Russian Academy of Sciences (DoM) and London Mathematical Society

Authors
Korpusov M.O. 1 , Matveeva A.K.2
Publisher
Institute of Physics Publishing
Number of issue
4
Language
English
Pages
705-711
Status
Published
Volume
85
Year
2021
Organizations
  • 1 Moscow State University, Faculty of Physics, Peoples’ Friendship University of Russia, Moscow, Russian Federation
  • 2 Moscow State University, Faculty of Physics, Russian Federation
Keywords
Blow-up; Bounds for the blow-up time; Local solubility; Non-linear capacity; Non-linear equations of Sobolev type
Date of creation
16.12.2021
Date of change
16.12.2021
Short link
https://repository.rudn.ru/en/records/article/record/76767/
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