scholarly journals Existence and uniqueness of global strong solutions to fully nonlinear second order elliptic systems

2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Nikos Katzourakis
Keyword(s):  
Existence And Uniqueness ◽  
Elliptic Systems ◽  
Strong Solutions ◽  
Second Order ◽  
Fully Nonlinear ◽  
Global Strong Solutions

ON THE FIRST BOUNDARY VALUE PROBLEM FOR THE CLASS OF ELLIPTIC SYSTEMS DEGENERATING AT AN INNER POINT

2001 ◽  
Vol 6 (1) ◽  
pp. 147-155 ◽  
Author(s):  
S. Rutkauskas
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Elliptic Systems ◽  
Second Order ◽  
Type Problem ◽  
Dirichlet Type ◽  
Holder Class ◽  
Uniqueness Of The Solution ◽  
Dirichlet Type Problem

The Dirichlet type problem for the weakly related elliptic systems of the second order degenerating at an inner point is discussed. Existence and uniqueness of the solution in the Holder class of the vector‐functions is proved.

scholarly journals Global Solutions for the Kuramoto-Sivashinsky Equation Posed on Unbounded 3D Grooves

2021 ◽  
pp. 293-303
Author(s):  
N.A. Larkin
Keyword(s):  
Mathematical Models ◽  
Existence And Uniqueness ◽  
Three Dimensional ◽  
Global Solutions ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Global Strong Solutions ◽  
Initial Boundary Value ◽  
Sivashinsky Equation

Initial boundary value problems for the three-dimensional Kuramoto-Sivashinsky equation posed on unbounded 3D grooves (that may serve as mathematical models for wildfires) were considered. The existence and uniqueness of global strong solutions as well as their exponential decay have been established.

On a class of fully nonlinear parabolic equations

2016 ◽  
Vol 8 (1) ◽  
pp. 79-100 ◽  
Author(s):  
Stanislav Antontsev ◽  
Sergey Shmarev
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Nonlinear Parabolic Equations ◽  
Fully Nonlinear ◽  
Nonlinear Parabolic ◽  
Homogeneous Dirichlet Problem ◽  
Fully Nonlinear Equation

Abstract We study the homogeneous Dirichlet problem for the fully nonlinear equation u_{t}=|\Delta u|^{m-2}\Delta u-d|u|^{\sigma-2}u+f\quad\text{in ${Q_{T}=\Omega% \times(0,T)}$,} with the parameters {m>1} , {\sigma>1} and {d\geq 0} . At the points where {\Delta u=0} , the equation degenerates if {m>2} , or becomes singular if {m\in(1,2)} . We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as {t\to\infty} . Sufficient conditions for exponential or power decay of {\|\nabla u(t)\|_{2,\Omega}} are derived. It is proved that for certain ranges of the exponents m and σ, every strong solution vanishes in a finite time.

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