A Problem with Nonlocal Boundary Conditions for a Quasilinear Parabolic Equation

1999 ◽  
Vol 6 (5) ◽  
pp. 421-428
Author(s):  
T. D. Dzhuraev ◽  
J. O. Takhirov
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Quasilinear Parabolic Equation ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Nonlocal Boundary Value Problem ◽  
Solution Uniqueness ◽  
The Maximum Principle ◽  
Class Of Functions ◽  
Quasilinear Parabolic

Abstract The solvability of the nonlocal boundary value problem 𝑢𝑡 = 𝑎(𝑡, 𝑥, 𝑢, 𝑢𝑥)𝑢𝑥𝑥 + 𝑏(𝑡, 𝑥, 𝑢, 𝑢𝑥), 0 ≤ 𝑡 ≤ 𝑇, |𝑥| ≤ 𝑙, 𝑢(0, 𝑥) = 0, 𝑢(𝑡, –𝑙) = 𝑢(𝑡, 𝑙), 𝑢𝑥(𝑡, –𝑙) = 𝑢𝑥(𝑡, 𝑙) in a class of functions is investigated for a quasilinear parabolic equation. The solution uniqueness follows from the maximum principle.

scholarly journals Optimal control for quasilinear retarded parabolic systems

2001 ◽  
Vol 42 (4) ◽  
pp. 532-551 ◽  
Author(s):  
Liping Pan ◽  
Jiongmin Yong
Keyword(s):  
Optimal Control ◽  
Parabolic Equation ◽  
Maximum Principle ◽  
State Constraints ◽  
Quasilinear Parabolic Equation ◽  
Parabolic Systems ◽  
Terminal State ◽  
Spatial Derivative ◽  
The Cost ◽  
Quasilinear Parabolic

AbstractWe study an optimal control problem for a quasilinear parabolic equation which has delays in the highest order spatial derivative terms. The cost functional is Lagrange type and some terminal state constraints are presented. A Pontryagin-type maximum principle is derived.

scholarly journals Determination of a diffusion coefficient in a quasilinear parabolic equation

2017 ◽  
Vol 15 (1) ◽  
pp. 77-91 ◽  
Author(s):  
Fatma Kanca
Keyword(s):  
Diffusion Coefficient ◽  
Parabolic Equation ◽  
Numerical Experiments ◽  
Input Data ◽  
Quasilinear Parabolic Equation ◽  
Nonlocal Boundary ◽  
Time Dependent ◽  
Integral Overdetermination ◽  
Quasilinear Parabolic

Abstract This paper investigates the inverse problem of finding the time-dependent diffusion coefficient in a quasilinear parabolic equation with the nonlocal boundary and integral overdetermination conditions. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown. Finally, some numerical experiments are presented.

scholarly journals Regularized Asymptotics of the Solution of the Singularly Perturbed First Boundary Value Problem on the Semiaxis for a Parabolic Equation with a Rational “Simple” Turning Point

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 405
Author(s):  
Alexander Yeliseev ◽  
Tatiana Ratnikova ◽  
Daria Shaposhnikova
Keyword(s):  
Parabolic Equation ◽  
Maximum Principle ◽  
Boundary Value Problem ◽  
Regularization Method ◽  
Turning Point ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Asymptotic Convergence ◽  
The Maximum Principle ◽  
Regularized Asymptotics

The aim of this study is to develop a regularization method for boundary value problems for a parabolic equation. A singularly perturbed boundary value problem on the semiaxis is considered in the case of a “simple” rational turning point. To prove the asymptotic convergence of the series, the maximum principle is used.

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

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