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May  2022, 21(5): 1567-1580. doi: 10.3934/cpaa.2022030

## Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace

 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany

Received  May 2021 Revised  September 2021 Published  May 2022 Early access  February 2022

Fund Project: The author was supported by the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja Award endowed by the German Federal Ministry of Education and Research

A generalisation of reaction diffusion systems and their travelling solutions to cases when the productive part of the reaction happens only on a surface in space or on a line on plane but the degradation and the diffusion happen in bulk are important for modelling various biological processes. These include problems of invasive species propagation along boundaries of ecozones, problems of gene spread in such situations, morphogenesis in cavities, intracellular reaction etc. Piecewise linear approximations of reaction terms in reaction-diffusion systems often result in exact solutions of propagation front problems. This article presents an exact travelling solution for a reaction-diffusion system with a piecewise constant production restricted to a codimension-1 subset. The solution is monotone, propagates with the unique constant velocity, and connects the trivial solution to a nontrivial nonhomogeneous stationary solution of the problem. The properties of the solution closely parallel the properties of monotone travelling solutions in classical bistable reaction-diffusion systems.

Citation: Anton S. Zadorin. Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1567-1580. doi: 10.3934/cpaa.2022030
##### References:
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##### References:
 [1] A. A. Andronov, A. A. Vitt and S. E. Khaikin, Oscillation Theory, (Russian), Nauka, Moscow, 1981. [2] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics (ed. J. A. Goldstein, Springer, Berlin, Heidelberg, (1975), 5–49. [3] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5. [4] H. Berestycki, J. M. Roquejoffre and L. Rossi, The influence of a line with fast diffusion on Fisher-KPP propagation, J. Math. Biol., 66 (2013), 743-766.  doi: 10.1007/s00285-012-0604-z. [5] H. Berestycki, J. M. Roquejoffre and L. Rossi, Fisher-KPP propagation in the presence of a line: further effects, Nonlinearity, 26 (2013), 2623.  doi: 10.1088/0951-7715/26/9/2623. [6] H. Berestycki, J. M. Roquejoffre and L. Rossi, The shape of expansion induced by a line with fast diffusion in Fisher-KPP equations, Commun. Math. Phys., 343 (2016), 207-232.  doi: 10.1007/s00220-015-2517-3. [7] J. Fang and X. Q. Zhao, Bistable traveling waves for monotone semiflows with applications, J. Euro. Math. Soc., 17 (2015), 2243-2288.  doi: 10.4171/JEMS/556. [8] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 355-369. [9] Y. I. Kanel', Stabilisation of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sborn., 101 (1962), 245-288. [10] A. N. Kolomogorov, I. G. Petrovsky and N. S. Piskunov, The study of the equation, joint with a growth of quantity of substance, Bulletin MGU, Math. Mech, 1 (1937), 1-26. [11] J. Larcher, Multiplications and convolutions in L. Schwartz' spaces of test functions and distributions and their continuity, Analysis, 33 (2013), 319-332.  doi: 10.1524/anly.2013.1200. [12] H. P. McKean Jr., Nagumo's equation, Adv. Math., 4 (1970), 209-223.  doi: 10.1016/0001-8708(70)90023-X. [13] Y. Nec, V. A. Volpert and A. A. Nepomnyashchy, Front propagation problems with sub-diffusion, Discret. Contin. Dynam. Syst. A, 27 (2010), 827-846.  doi: 10.3934/dcds.2010.27.827. [14] S. V. Petrovskii and B. L. Li, Exactly Solvable Models of Biological Invasion, Chapman and Hall/CRC, 2005. [15] V. S. Vladimirov, Generalized Functions in Mathematical Physics, (Russian), Izdatel Nauka, Moscow, 1976. [16] A. I. Volpert, V. A. Volpert and V. A. Volpert, Travelling Wave Solutions of Parabolic Systems, (Vol. 140), American Mathematical Society, 1994. doi: 10.1090/mmono/140. [17] V. A. Volpert, Y. Nec and A. A. Nepomnyashchy, Exact solutions in front propagation problems with superdiffusion, Phys. D, 239 (2010), 134-144.  doi: 10.1016/j.physd.2009.10.011. [18] B. L. van der Waerden and P. Huber, Science Awakening. Vol. 2: The Birth of Astronomy, Leyden: Noordhoff International Publication, and New York: Oxford University Press, 1974. [19] M. H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Bio., 171 (2001), 83-97.  doi: 10.1016/S0025-5564(01)00048-7.
Propagation of a front supported by a plane/line with a piecewise constant growth rate. A. The general geometry of the model. The $z$-axis can be absent. In this case, the model is considered on a plane and the growth happens on a line. B. Piecewise constant approximation of a sigmoidal growth rate $f$ bound by the maximal growth rate $a$ at infinity
The front $w$ of the travelling solution $u(x,y,t) = w(x- v t,y)$ of (1.2) given by (3.12) for $a = 2\pi$, $k = D = 1$, and $u_c = 0.3$ ($v \approx 6.47$). A. A contour plot of $w$ in the $xy$-plane ($y \geqslant 0$). Note that $w(x,y) = w(x,-y)$. B. The value of $w$ on the $x$-axis. Here the production is active on the $x$-axis with $x \in (-\infty,0]$ and inactive everywhere else
Sketches of travelling solution profiles $w$ of the classical piecewise linear equation in the original units on the phase plane $( w, \partial_x w)$. The trajectory that corresponds to a front is depicted as the thick line. A. The regular travelling front that connects the trivial and the nontrivial steady states for the case $v > 0$. It is the standard heteroclinic trajectory on the phase plane. Only separatrices of the saddles are shown. See [19] for the details. B. The travelling solution with $k = 0$ that connects the trivial steady state and infinity with bound derivative at infinity. The case is degenerated and the whole $w$-axis consists of steady states for $w < u_c$. A generic smooth case would correspond to a saddle-node bifurcation at the origin in this situation. C. The homoclinic stationary solution that connects the trivial steady state with itself in the case when the monotone front travels with $v > 0$
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