On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient nonlinearity

In this paper, we consider the Cauchy problem for one nonclassical, third‐order, partial differential equation with gradient nonlinearity |∇u(x,t)|q$$ {\left|\nabla u\left(x,t\right)\right|}^q $$. The solution to this problem is understood in a weak se...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences 2023-01, Vol.46 (2), p.1574-1630
Main Authors: Korpusov, Maxim O., Matveeva, Alexandra K.
Format: Article
Language:English
Subjects:
blow‐up
blow‐up time estimation
Cauchy problems
critical exponents
local solvability
nonlinear capacity
nonlinear equation of Sobolev‐type
Nonlinear equations
Nonlinearity
Partial differential equations
Smoothness
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Summary:In this paper, we consider the Cauchy problem for one nonclassical, third‐order, partial differential equation with gradient nonlinearity |∇u(x,t)|q$$ {\left|\nabla u\left(x,t\right)\right|}^q $$. The solution to this problem is understood in a weak sense. We show that for 13/2$$ q>3/2 $$, such a solution exists. For 3/2
ISSN:0170-4214
1099-1476
DOI:10.1002/mma.8595