# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022042
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## Qualitative and quantitative analysis for a nonlocal and nonlinear reaction-diffusion problem with in-homogeneous Neumann boundary conditions

 1 Alexandru Ioan Cuza University - UAIC, Bd. Carol I, No. 11, 700506, Iaşi, Romania 2 Faculty of Electrical Engineering and Computer Science, Stefan cel Mare University of Suceava, Universitatii 13, 720225, Suceava, Romania

Received  October 2021 Revised  January 2022 Early access February 2022

The main goal of this paper is to introduce and analyze a new nonlocal reaction-diffusion model with in-homogeneous Neumann boundary conditions. We prove the existence and uniqueness of a solution in the class $C((0, T], L^\infty(\Omega))$ and the dependence on the data. Proofs are based on the Banach fixed-point theorem. Our results extend the results already proven by other authors. A numerical approximating scheme and a series of numerical experiments are also presented in order to illustrate the effectiveness of the theoretical result. The overall scheme is explicit in time and does not need iterative steps; therefore it is fast.

Citation: Costică Moroşanu, Bianca Satco. Qualitative and quantitative analysis for a nonlocal and nonlinear reaction-diffusion problem with in-homogeneous Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022042
##### References:
 [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, in: Mathematical Surveys and Monographs, 165, American Mathematical Society, 2010. doi: 10.1090/surv/165. [2] T. Barbu, A. Miranville and C. Moroşanu, A qualitative analysis and numerical simulations of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy-Neumann boundary conditions, Applied Mathematics and Computation, 350 (2019), 170-180.  doi: 10.1016/j.amc.2019.01.004. [3] P. W. Bates, S. Brown and J. Han, Numerical analysis for a nonlocal Allen-Cahn equation, Int. J. Numerical Analysis and Modeling, 6 (2009), 33-49. [4] M. Bogoya and C. A. Gómez S., On a nonlocal diffusion model with Neumann boundary conditions, Nonlinear Analysis, 75 (2012), 3198-3209.  doi: 10.1016/j.na.2011.12.019. [5] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to aproximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8. [6] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002. [7] M. Craus and S.-D. Pavăl, An accelerating numerical computation of the diffusion term in a nonlocal reaction-diffusion equation, Mathematics, 8 (2020), 2111.  doi: 10.3390/math8122111. [8] P. C. Fife, Some nonclassical trends in parabolic and paraboli-like evolutions, Trends in Nonlinear analysis, Springer, Berlin, 2003,153–191. doi: 10.1007/978-3-662-05281-5_3. [9] L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013. [10] A. Miranville and C. Moroşanu, A qualitative analysis of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy-Stefan-Boltzmann boundary conditions, Appl. Math. Optim., 84 (2021), 227-244.  doi: 10.1007/s00245-019-09643-5. [11] C. Moroşanu and A.-M. Moşneagu, On the numerical approximation of the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions. Case 1D, ROMAI J., 9 (2013), 91-110. [12] C. Moroşanu, S. Pavăl and C. Trenchea, Analysis of stability and errors of three methods associated to the nonlinear reaction-diffusion equation supplied with homogeneous Neumann boundary conditions, Journal of Applied Analysis and Computation, 7 (2017), 1-19.  doi: 10.11948/2017001. [13] M. Pérez-Llanos and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640.  doi: 10.1016/j.na.2008.02.076. [14] J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM J. Sci. Comput., 31 (2009), 3042-3063.  doi: 10.1137/080738398.

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##### References:
 [1] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, in: Mathematical Surveys and Monographs, 165, American Mathematical Society, 2010. doi: 10.1090/surv/165. [2] T. Barbu, A. Miranville and C. Moroşanu, A qualitative analysis and numerical simulations of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy-Neumann boundary conditions, Applied Mathematics and Computation, 350 (2019), 170-180.  doi: 10.1016/j.amc.2019.01.004. [3] P. W. Bates, S. Brown and J. Han, Numerical analysis for a nonlocal Allen-Cahn equation, Int. J. Numerical Analysis and Modeling, 6 (2009), 33-49. [4] M. Bogoya and C. A. Gómez S., On a nonlocal diffusion model with Neumann boundary conditions, Nonlinear Analysis, 75 (2012), 3198-3209.  doi: 10.1016/j.na.2011.12.019. [5] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to aproximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal., 187 (2008), 137-156.  doi: 10.1007/s00205-007-0062-8. [6] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations, 234 (2007), 360-390.  doi: 10.1016/j.jde.2006.12.002. [7] M. Craus and S.-D. Pavăl, An accelerating numerical computation of the diffusion term in a nonlocal reaction-diffusion equation, Mathematics, 8 (2020), 2111.  doi: 10.3390/math8122111. [8] P. C. Fife, Some nonclassical trends in parabolic and paraboli-like evolutions, Trends in Nonlinear analysis, Springer, Berlin, 2003,153–191. doi: 10.1007/978-3-662-05281-5_3. [9] L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, J. Funct. Anal., 251 (2007), 399-437.  doi: 10.1016/j.jfa.2007.07.013. [10] A. Miranville and C. Moroşanu, A qualitative analysis of a nonlinear second-order anisotropic diffusion problem with non-homogeneous Cauchy-Stefan-Boltzmann boundary conditions, Appl. Math. Optim., 84 (2021), 227-244.  doi: 10.1007/s00245-019-09643-5. [11] C. Moroşanu and A.-M. Moşneagu, On the numerical approximation of the phase-field system with non-homogeneous Cauchy-Neumann boundary conditions. Case 1D, ROMAI J., 9 (2013), 91-110. [12] C. Moroşanu, S. Pavăl and C. Trenchea, Analysis of stability and errors of three methods associated to the nonlinear reaction-diffusion equation supplied with homogeneous Neumann boundary conditions, Journal of Applied Analysis and Computation, 7 (2017), 1-19.  doi: 10.11948/2017001. [13] M. Pérez-Llanos and J. D. Rossi, Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term, Nonlinear Anal., 70 (2009), 1629-1640.  doi: 10.1016/j.na.2008.02.076. [14] J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM J. Sci. Comput., 31 (2009), 3042-3063.  doi: 10.1137/080738398.
Initial values $u^0_{i, k}, \ \ i = 1, 2\ldots , I, \ \ k = 1, 2, \ldots , K$
The approximate solution $u^1_{i, k}, \ \ i = 1, 2\ldots , I, \ \ k = 1, 2, \ldots , K$
The approximate solution $u^5_{i, k}, \ \ i = 1, 2\ldots , I, \ \ k = 1, 2, \ldots , K$
The evolution of the parameters $R_*$ and $t^*$ in Lemma 4.1
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