# American Institute of Mathematical Sciences

March  2017, 22(2): 407-419. doi: 10.3934/dcdsb.2017019

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

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

* Corresponding author: Bhargav Kumar Kakumani

Received  March 2016 Revised  July 2016 Published  December 2016

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

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.

Citation: Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019
##### References:
 [1] B. Abdellaoui and T. M. Touaoula, Decay solution for the renewal equation with diffusion, Nonlinear Differential Equations and Applications NoDEA, 17 (2010), 271-288.  doi: 10.1007/s00030-009-0053-6.  Google Scholar [2] B. Basse, B. C. Baguley, E. S. Marshall, W. R. Joseph, B. Van Brunt, G. Wake and D. J. N. Wall, A mathematical model for analysis of the cell cycle in cell lines derived from human tumors, Journal of Mathematical Biology, 47 (2003), 295-312.  doi: 10.1007/s00285-003-0203-0.  Google Scholar [3] W. A. Day, Extensions of property of solutions of heat equation to linear thermoelasticity and other theories, Quarterly of Applied Mathematics, 40 (1982), 319-330.   Google Scholar [4] A. Friedman, Partial Differential Equations of Parabolic Type Prentice Hall, Englewood Cliffs, New Jersy, 1964. Google Scholar [5] A. Gladkov and K. I. Kim, Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition, Journal of Mathematical Analysis and Applications, 338 (2008), 264-273.  doi: 10.1016/j.jmaa.2007.05.028.  Google Scholar [6] A. Gladkov and K. I. Kim, Uniqueness and nonuniqueness for reaction-diffusion equation with nonlocal boundary condition, Advances in Mathematical Sciences and Applications, 19 (2009), 39-49.   Google Scholar [7] A. Gladkov and A. Nikitin, A reaction-diffusion system with nonlinear nonlocal boundary conditions, International Journal of Partial Differential Equations, (2014), Article ID 523656, 10 pages. Google Scholar [8] B. K. Kakumani and S. K. Tumuluri, On a nonlinear renewal equation with diffusion, Mathematical Methods in the Applied Sciences, 39 (2016), 697-708.  doi: 10.1002/mma.3511.  Google Scholar [9] P. Michel and T. M. Touaoula, Asymptotic behavior for a class of the renewal nonlinear equation with diffusion, Mathematical Methods in the Applied Sciences, 36 (2013), 323-335.  doi: 10.1002/mma.2591.  Google Scholar [10] M. Iannelli and G. Marinoschi, Approximation of a population dynamics model by parabolic regularization, Mathematical Methods in the Applied Sciences, 36 (2013), 1229-1239.  doi: 10.1002/mma.2675.  Google Scholar [11] C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum, New York, 1992. Google Scholar [12] C. V. Pao, Dynamics of reaction-diffusion equations with nonlocal boundary conditions, Quarterly of Applied Mathematics, 53 (1995), 173-186.   Google Scholar [13] C. V. Pao, Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, Journal of Mathematical Analysis and Applications, 195 (1995), 702-718.  doi: 10.1006/jmaa.1995.1384.  Google Scholar [14] C. V. Pao, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, Journal of Computational and Applied Mathematics, 88 (1998), 225-238.  doi: 10.1016/S0377-0427(97)00215-X.  Google Scholar [15] C. V. Pao, Asymptotic behavior of solutions for finite-difference equations of reaction-diffusion, Journal of Mathematical Analysis and Applications, 144 (1989), 206-225.  doi: 10.1016/0022-247X(89)90369-7.  Google Scholar [16] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary-value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.   Google Scholar

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##### References:
 [1] B. Abdellaoui and T. M. Touaoula, Decay solution for the renewal equation with diffusion, Nonlinear Differential Equations and Applications NoDEA, 17 (2010), 271-288.  doi: 10.1007/s00030-009-0053-6.  Google Scholar [2] B. Basse, B. C. Baguley, E. S. Marshall, W. R. Joseph, B. Van Brunt, G. Wake and D. J. N. Wall, A mathematical model for analysis of the cell cycle in cell lines derived from human tumors, Journal of Mathematical Biology, 47 (2003), 295-312.  doi: 10.1007/s00285-003-0203-0.  Google Scholar [3] W. A. Day, Extensions of property of solutions of heat equation to linear thermoelasticity and other theories, Quarterly of Applied Mathematics, 40 (1982), 319-330.   Google Scholar [4] A. Friedman, Partial Differential Equations of Parabolic Type Prentice Hall, Englewood Cliffs, New Jersy, 1964. Google Scholar [5] A. Gladkov and K. I. Kim, Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition, Journal of Mathematical Analysis and Applications, 338 (2008), 264-273.  doi: 10.1016/j.jmaa.2007.05.028.  Google Scholar [6] A. Gladkov and K. I. Kim, Uniqueness and nonuniqueness for reaction-diffusion equation with nonlocal boundary condition, Advances in Mathematical Sciences and Applications, 19 (2009), 39-49.   Google Scholar [7] A. Gladkov and A. Nikitin, A reaction-diffusion system with nonlinear nonlocal boundary conditions, International Journal of Partial Differential Equations, (2014), Article ID 523656, 10 pages. Google Scholar [8] B. K. Kakumani and S. K. Tumuluri, On a nonlinear renewal equation with diffusion, Mathematical Methods in the Applied Sciences, 39 (2016), 697-708.  doi: 10.1002/mma.3511.  Google Scholar [9] P. Michel and T. M. Touaoula, Asymptotic behavior for a class of the renewal nonlinear equation with diffusion, Mathematical Methods in the Applied Sciences, 36 (2013), 323-335.  doi: 10.1002/mma.2591.  Google Scholar [10] M. Iannelli and G. Marinoschi, Approximation of a population dynamics model by parabolic regularization, Mathematical Methods in the Applied Sciences, 36 (2013), 1229-1239.  doi: 10.1002/mma.2675.  Google Scholar [11] C. V. Pao, Nonlinear Parabolic and Elliptic Equations Plenum, New York, 1992. Google Scholar [12] C. V. Pao, Dynamics of reaction-diffusion equations with nonlocal boundary conditions, Quarterly of Applied Mathematics, 53 (1995), 173-186.   Google Scholar [13] C. V. Pao, Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, Journal of Mathematical Analysis and Applications, 195 (1995), 702-718.  doi: 10.1006/jmaa.1995.1384.  Google Scholar [14] C. V. Pao, Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions, Journal of Computational and Applied Mathematics, 88 (1998), 225-238.  doi: 10.1016/S0377-0427(97)00215-X.  Google Scholar [15] C. V. Pao, Asymptotic behavior of solutions for finite-difference equations of reaction-diffusion, Journal of Mathematical Analysis and Applications, 144 (1989), 206-225.  doi: 10.1016/0022-247X(89)90369-7.  Google Scholar [16] D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary-value problems, Indiana Univ. Math. J., 21 (1972), 979-1000.   Google Scholar
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 $U(x)$
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 $U(x)$
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 $U(x)$
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)
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 $U(x)$
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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