STRONG SOLUTIONS FOR STOCHASTIC NONLINEAR NONLOCAL PARABOLIC EQUATIONS

2010 ◽  
Vol 10 (04) ◽  
pp. 497-508 ◽  
Author(s):  
EDSON A. COAYLA-TERAN
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic ◽  
Multiplicative White Noise ◽  
Nonlocal Parabolic Equations ◽  
Initial Boundary Value

In this article we investigate the existence and uniqueness of strong solutions to the initial-boundary value problem with homogeneous boundary conditions for a stochastic nonlinear parabolic equation of nonlocal type with multiplicative white noise. Moreover, we prove a simple result on the asymptotic behavior for the solution.

A NONLINEAR PARABOLIC SYSTEM ARISING IN THERMOMECHANICS AND IN THERMOMAGNETISM

1992 ◽  
Vol 02 (03) ◽  
pp. 271-281 ◽  
Author(s):  
JOSÉ-FRANCISCO RODRIGUES
Keyword(s):  
Parabolic Equations ◽  
Eddy Currents ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Arbitrary Cross Section ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Nonlinear Parabolic ◽  
Initial Boundary Value ◽  
Dependent Viscosity

We consider a system of two parabolic equations modeling the thermo-convection of a Newtonian fluid, with temperature dependent viscosity of energy dissipation, as well as the thermal effects of the eddy currents, induced by a slowly varying magnetic field, in cylinders with arbitrary cross-section. We show the existence of a weak solution of the corresponding initial-boundary value problem and, under additional assumptions, we consider the question of the uniqueness and regularity of the solution.

XXII.—A Priori Estimates and Nonlinear Parabolic Equations of Arbitrary Order

1972 ◽  
Vol 69 (4) ◽  
pp. 287-293
Author(s):  
D. E. Edmunds ◽  
C. A. Stuart
Keyword(s):  
Parabolic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic Equations ◽  
Linear Parabolic Equation ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Initial Boundary Value

SynopsisIn this paper it is shown that the question of the existence of a classical solution of the first initial-boundary value problem for a non-linear parabolic equation may be reduced to the problem of the derivation of suitable a priori bounds.

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.

scholarly journals A nonlinear parabolic equation modelling surfactant diffusion

2000 ◽  
Vol 11 (4) ◽  
pp. 413-432
Author(s):  
XINFU CHEN ◽  
CHAOCHENG HUANG ◽  
JENNIFER ZHAO
Keyword(s):  
Parabolic Equations ◽  
Single Point ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Parabolic Equations ◽  
Well Posedness ◽  
Initial Value ◽  
Equation Modelling ◽  
Nonlinear Parabolic ◽  
Initial Boundary Value

An initial-boundary value problem for nonlinear parabolic equations modelling surfactant diffusions is investigated. The boundary conditions are of nonlinear adsorptive types, and the initial value has a single point jump. We study the well-posedness of the problem, the convergence of a numerical scheme, and the regularity as well as quantitative behaviour of solutions.

scholarly journals Blowup of Solution for a Class of Doubly Nonlinear Parabolic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Jun Lu ◽  
Qingying Hu ◽  
Hongwei Zhang
Keyword(s):  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Parabolic Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Blowup Of Solutions ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Initial Boundary Value ◽  
Suitable Initial Data

An initial boundary value problem for a class of doubly parabolic equations is studied. We obtain sufficient conditions for the blowup of solutions under suitable initial data using differential inequalities.

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