Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions

Bhargav Kumar Kakumani; Suman Kumar Tumuluri

doi:10.3934/dcdsb.2017019

Article Contents

2017, Volume 22, Issue 2: 407-419. Doi: 10.3934/dcdsb.2017019

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Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions

Bhargav Kumar Kakumani^, and
Suman Kumar Tumuluri

School of Mathematics and Statistics, University of Hyderabad, Prof. C.R. Rao road, Gachibowli, Hyderabad, Telangana, India 500046

* Corresponding author: Bhargav Kumar Kakumani
* Corresponding author: Bhargav Kumar Kakumani

Received: February 29, 2016

Revised: June 30, 2016

Published: December 2017

The first author would like to thank CSIR (award number: 09/414(0986)/2011-EMR-I) for providing financial support.

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Abstract

In this paper, we consider a particular type of nonlinear McKend-rick-von Foerster equation with a diffusion term and Robin boundary condition. We prove the existence of a global solution to this equation. The steady state solutions to the equations that we consider have a very important role to play in the study of long time behavior of the solution. Therefore we address the issues pertaining to the existence of solution to the corresponding state equation. Furthermore, we establish that the solution of McKendrick-von Foerster equation with diffusion converges pointwise to the solution of its steady state equations as time tends to infinity.

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Mathematics Subject Classification: Primary:35A05, 35A08, 35K20, 35K55.

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Figure 1. Solutions to (1) and (10) with $d,g,B$ and $u_0$ given in Example 5.1; Left: $u(12,x)$ (continuous line) and $U(x)$ (dotted line) for $0\leq x\leq 10$, Right: The absolute difference between $u(12,x)$ and $U(x)$

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Figure 2. Solutions to (1) and (10) with $d,g,B$ and $u_0$ given in Example 5.2; Left: $u(12,x)$ (continuous line) and $U(x)$ (dotted line) for $0\leq x\leq 5$, Right: The absolute difference between $u(12,x)$ and $U(x)$

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Figure 3. Solutions to (1) and (10) with $d,g,B$ and $u_0$ given in Example 5.3; Left: $u(12,x)$ (continuous line) and $U(x)$ (dotted line) for $0\leq x\leq5$, Right: The absolute difference between $u(12,x)$ and $U(x)$

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Figure 4. Solution to (1) with $d,g,B$ and $u_0$ given in Example 5.3; Left: $u(t,2)$ (continuous line), $u(t,3)$ (dashed line) and $u(t,4)$ (dotted line) for $0\leq t\leq12$, Right: $u|(t,2)-U(2)|$ (continuous line), $|u(t,3)-U(3)|$ (dashed line) and $|u(t,4)-U(4)|$ (dotted line)

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Figure 5. Solutions to (1) and (10) with $d,g,B$ and $u_0$ given in Example 5.4; Left: $u(12,x)$ (continuous line) and $U(x)$ (dotted line) for $0\leq x\leq4$, Right: The absolute difference between $u(12,x)$ and $U(x)$

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Figure 6. Solution to (1) with $d,g,B$ and $u_0$ given in Example 5.4; Left: $u(t,1)$ (continuous line), $u(t,2)$ (dashed line) and $u(t,4)$ (dotted line) for $0\leq t\leq8$, Right: $u|(t,2)-U(2)|$ (continuous line), $|u(t,3)-U(3)|$ (dashed line) and $|u(t,4)-U(4)|$ (dotted line)

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