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. BasseB. C. BaguleyE. S. MarshallW. R. JosephB. Van BruntG. 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

show all references

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. BasseB. C. BaguleyE. S. MarshallW. R. JosephB. Van BruntG. 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

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 $U(x)$
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 $U(x)$
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 $U(x)$
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)
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 $U(x)$
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)
[1]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[2]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[3]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[4]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[5]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[6]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[7]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[8]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[9]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[10]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[11]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[12]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[13]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[14]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[15]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[16]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[17]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[18]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[19]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[20]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (59)
  • HTML views (102)
  • Cited by (0)

Other articles
by authors

[Back to Top]