scholarly journals Bounded Solutions to Nonlinear Parabolic Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Jaouad Igbida
Keyword(s):  
Parabolic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Nonlinear Parabolic Equations ◽  
Bounded Solutions ◽  
Test Function ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Compactness Arguments ◽  
Test Function Method

We deal with existence results for nonlinear parabolic equations with general quadratic gradient terms and with absorption term which depend on the solution. We note that no boundedness is assumed on the data of the problem. We prove an existence result of distributional solution via test-function method. A priori estimates and compactness arguments are our main ingredient; the method of sub-supersolution does not apply her.

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.

scholarly journals Keller-Osserman a priori estimates for doubly nonlinear anisotropic parabolic equations with absorption term

2018 ◽  
Vol 32 ◽  
pp. 149-159
Author(s):  
Maria Shan
Keyword(s):  
Parabolic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Quasilinear Parabolic Equation ◽  
Qualitative Theory ◽  
Large Solutions ◽  
Priori Estimates ◽  
Nonnegative Solutions ◽  
Elliptic And Parabolic Equations ◽  
Doubly Nonlinear

We are concerned with divergence type quasilinear parabolic equation with measurable coefficients and lower order terms model of which is a doubly nonlinear anisotropic parabolic equations with absorption term. This class of equations has numerous applications which appear in modeling of electrorheological fluids, image precessing, theory of elasticity, theory of non-Newtonian fluids with viscosity depending on the temperature. But the qualitative theory doesn't construct for these anisotropic equations. So, naturally, that during the last decade there has been growing substantial development in the qualitative theory of second order anisotropic elliptic and parabolic equations. The main purpose is to obtain the pointwise upper estimates in terms of distance to the boundary for nonnegative solutions of such equations. This type of estimates originate from the work of J. B. Keller, R. Osserman, who obtained a simple upper bound for any solution, in any number of variables for Laplace equation. These estimates play a crucial role in the theory of existence or nonexistence of so called large solutions of such equations, in the problems of removable singularities for solutions to elliptic and parabolic equations. Up to our knowledge all the known estimates for large solutions to elliptic and parabolic equations are related with equations for which some comparison properties hold. We refer to I.I. Skrypnik, A.E. Shishkov, M. Marcus , L. Veron, V.D. Radulescu for an account of these results and references therein. Such equations have been the object of very few works because in general such properties do not hold. The main ones concern equations only in the precise choice of absorption term \(f(u)=u^q\). Among the people who published significative results in this direction are I.I. Skrypnik, J. Vetois, F.C. Cirstea, J. Garcia-Melian, J.D. Rossi, J.C. Sabina de Lis. The main result of the paper is a priori estimates of Keller-Osserman type for nonnegative solutions of a doubly nonlinear anisotropic parabolic equations with absorption term that have been proven despite of the lack of comparison principle. To obtain these estimates we exploit the method of energy estimations and De Giorgy iteration techniques.

A nonlinear partial differential system arising in thermoelectricity

2005 ◽  
Vol 16 (6) ◽  
pp. 683-712 ◽  
Author(s):  
A. BERMÚDEZ ◽  
R. MUÑOZ-SOLA ◽  
F. PENA
Keyword(s):  
Source Term ◽  
A Priori Estimates ◽  
A Priori ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Existence Of A Solution ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Partial Differential System ◽  
Temperature Equation

In this paper we prove the existence of a solution for a system of nonlinear parabolic partial differential equations arising from thermoelectric modelling of metallurgical electrodes undergoing a phase change. The model consists of an electromagnetic problem for eddy current computation coupled with a Stefan problem for temperature. The proof uses a regularized problem obtained by truncating the source term in temperature equation. Passing to the limit requires fine a priori estimates leading to compactness.

scholarly journals On a regularity property and a priori estimates for solutions of nonlinear parabolic variational inequalities

2000 ◽  
Vol 160 ◽  
pp. 123-134 ◽  
Author(s):  
Haruo Nagase
Keyword(s):  
Continuous Dependence ◽  
A Priori Estimates ◽  
Lower Semicontinuous ◽  
A Priori ◽  
Regularity Property ◽  
Regularity Properties ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Lower Semicontinuous Functional ◽  
Continuous Dependence Of Solutions

AbstractIn this paper we consider the following nonlinear parabolic variational inequality; u(t) ∈ D(Φ) for all where Δp is the so-called p-Laplace operator and Φ is a proper, lower semicontinuous functional. We have obtained two results concerning to solutions of this problem. Firstly, we prove a few regularity properties of solutions. Secondly, we show the continuous dependence of solutions on given data u0 and f.

DIVERGENCE FORM PARABOLIC EQUATIONS ON TIME-DEPENDENT QUASICONVEX DOMAINS

2012 ◽  
Vol 23 (12) ◽  
pp. 1250128 ◽  
Author(s):  
HUILIAN JIA ◽  
LIHE WANG
Keyword(s):  
Parabolic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Divergence Form ◽  
Time Dependent ◽  
Function Technique ◽  
Parabolic Boundary ◽  
Covering Lemma ◽  
Priori Estimates ◽  
Reifenberg Flat

In this paper, we show the [Formula: see text] regularity of divergence form parabolic equations on time-dependent quasiconvex domains. The objective is to study the optimal parabolic boundary condition for the Lp estimates. The time-dependent quasiconvex domain is a generalization of the time-dependent Reifenberg flat domain, and assesses some properties analog to the convex domain. As to the a priori estimates near the boundary, we will apply the maximal function technique, Vitali covering lemma and the compactness method.

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