scholarly journals On critical exponents for weak solutions to the Cauchy problem for one nonlinear equation with gradient non-linearity

2020 ◽  
Author(s):  
Maxim Korpusov ◽  
Alexandra Matveeva
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Weak Solution ◽  
Nonlinear Equation ◽  
Critical Exponents ◽  
Weak Sense ◽  
Order Partial Differential Equation ◽  
Third Order ◽  
The Cauchy Problem

In this paper, we consider the Cauchy problem for one nonclassical, third-order, partial differential equation with gradient non-linearity $|\nabla u(x,t)|^q$. The solution to this problem is understood in a weak sense. We show that for $1“3/2$ the existence of the only local-in-time weak solution of Cauchy’s problem.If $3/2”

scholarly journals A note on the nonlocal boundary value problem for a third order partial differential equation

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 801-808 ◽  
Author(s):  
Kh. Belakroum ◽  
A. Ashyralyev ◽  
A. Guezane-Lakoud
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Order Partial Differential Equation ◽  
Nonlocal Boundary Value Problem ◽  
Third Order ◽  
Partial Differential

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space with a self-adjoint positive definite operator is considered. Applying operator approach, the theorem on stability for solution of this nonlocal boundary value problem is established. In applications, the stability estimates for the solution of three nonlocal boundary value problems for third order partial differential equations are obtained.

scholarly journals Continuity equations and ODE flows with non-smooth velocity

2014 ◽  
Vol 144 (6) ◽  
pp. 1191-1244 ◽  
Author(s):  
Luigi Ambrosio ◽  
Gianluca Crippa
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Bounded Variation ◽  
Velocity Fields ◽  
Well Posedness ◽  
Continuity Equations ◽  
Quantitative Estimates ◽  
The Cauchy Problem

In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

Sign in / Sign up

Export Citation Format

Share Document