Reaction-diffusion equation

A system of partial differential equations of the form

$$\frac{\partial u}{\partial t}=D\Delta u+f(u),$$

where $u=u(x,t)=(u_1,\ldots,u_n)$, $\Delta$ is the Laplace operator in the spatial variables $x$, $D$ is a non-negative non-zero diagonal matrix, and $f$ is a function from a domain in $\mathbf R^n$ into $\mathbf R^n$. Many generalizations of these equations have also been studied, such as result when $f$ depends also on the first-order $x$-derivatives of $u$, when the operator $\Delta$ is replaced by other, possibly non-linear, operators, or when the matrix $D$ is not diagonal. If extra first-order terms appear in the system as a model for convective transport effects, the system is sometimes termed reaction-advection-diffusion equation.

Such equations arise as models of diverse natural phenomena [a1], but their most natural roots lie in the study of chemical systems: the components of the vector $u$ may then represent concentrations of chemical species which are present, the term $D\Delta u$ represents the diffusive transport of those species, possibly through a chemical solution, and $f(u)$ represents the production or destruction of species resulting from reactions among them (if the rates of all such reactions are known, as functions of $u$, then the explicit form of $f$ can be written down).

The variable $x$ is often confined to a domain $\Omega$ with boundary $\partial\Omega$, and then solutions are sought which satisfy specific boundary conditions on $\partial\Omega$. These are generally of the form

$$a_i\frac{\partial u_i}{\partial\nu}+b_iu_i==h_i,\quad x\in\partial\Omega,\quad i=1,\ldots,n,$$

where $\partial/\partial\nu$ is the derivative normal to $\partial\Omega$, $a_i$ and $b_i$ are not both zero (unless $u_i$ does not "diffuse" ), and $h_i$ is a given function. Again, generalizations, such as to non-linear boundary conditions, abound.

The specific problems of interest are: i) the initial value problem, in which $u(x,0)$ is given and $u(x,t)$ is sought for $t\geq0$; ii) the steady problem, in which solutions independent of $t$ are sought; and iii) the travelling-wave problem, in which $\Omega=\mathbf R$ and solutions are sought of the special form $u(x,t)=U(x-ct)$.

Because of their strong connections with the applied sciences and the limited number of important properties common to all members of this unwieldy class of systems, the research impetus in this field comes more from viewing the systems as models of specific natural phenomena, rather than from interest in them for their own sake. A typical motivation, for example, may be to ask whether a certain system, in which specific natural effects are modelled, will have solutions which reflect some known natural phenomenon of interest whose causes are incompletely known. Then one looks for the existence and the stability of solutions of the system in question which have properties analogous to the phenomenon in question.

Regarding the initial value problem i), the theory of analytic semi-groups, which in this context relies on the operator $D\Delta$ being sectorial, has developed as one of the most commonly used approaches to existence and uniqueness [a2]. The study of steady solutions ii) has used a variety of methods, such as recasting the problem as a fixed-point problem for a mapping in some suitable function space and using topological-degree methods. In the case when $n=1$ or the system has certain monotonicity properties, methods based on upper-and-lower solutions (cf. Upper-and-lower-functions method) provide easier alternatives (see [a1] and [a3], e.g.).

The travelling-wave problem iii) can be viewed as seeking a parameter $c$ for which there exists a connection between two critical points for the system of ordinary differential equations resulting from the substitution $u=U(x-ct)$. One of the principal tools in this connection has been the powerful Conley index [a4], [a3]. Below some other methods which have been devised more recently are mentioned.

The theory of reaction-diffusion systems can be viewed as incorporating all of the theory of autonomous ordinary differential systems $du/dt=f(u)$ (cf. Autonomous system), since when homogeneous Neumann boundary conditions are imposed, solutions of the latter system automatically constitute $x$-independent solutions of the corresponding reaction-diffusion systems. But, of course, solutions with striking spatial characteristics arise as well; and,in fact, the possible spatial structure of solutions is one of the most often investigated aspects.

Some of the best-studied examples of reaction-diffusion systems are the following.

a) The scalar Fisher equation

$$\frac{\partial u}{\partial t}=\Delta u+f(u),$$

[a5], [a6], where $f$ has exactly two zeros. This equation originally arose in connection with population genetics.

b) The scalar bistable diffusion equation, [a7], [a6], [a8], of the same form but where $f$ has exactly three simple zeros and is negative between the first two. This equation also has connections with population genetics, but knowledge of its properties is even more important in connection with the role it plays as a component part of more complicated systems.

c) The FitzHugh–Nagumo system

$$\frac{\partial u}{\partial t}=\Delta u+f(u)-v,$$

$$\frac{\partial v}{\partial t}=au-bv,$$

where $f$ has the properties given in b) (see the references in [a3] and [a9] for this and generalizations). It is a simplification of higher-order systems such as the Hodgkin–Huxley system, which arise as models of signal transmission on nerve axons and in cardiac tissue.

d) The thermal-diffusion model in chemical reactor theory and combustion [a10], [a11]. In this model, $u=(u_0,\ldots,u_n)$, $u_0$ represents temperature, the other components of $u$ represent concentrations of chemical species, and the components of $f$ are given by

$$f_j(u)=\sum_la_{jl}b_l(u_0)m_l(u),$$

the summation being over all reactions occurring in the material (these reactions are indexed by $l$). Here, $m_l$ is a (mass action) monomial in $u_1,\ldots,u_n$ appropriate to the $l$-th reaction, $b_l$ is the "reaction constant" for that reaction, and the numbers $a_{jl}$ are "stoichiometric parameters" , specifying the amount of species $j$ (or heat, in case $j=0$) produced or consumed in reaction $l$.

In all of the above examples, the existence and the stability of travelling-wave solutions is of paramount importance; and in case d), other spatially or temporally ordered solutions are important as well.

In many applications, solutions with frontal or interfacial properties arise [a9]. For example, a moving surface in $3$-space may exist near which some components of $u$ experience dramatic changes. These changes form an interior layer at the surface in question. They have been studied in the context of phase-field equations (a reaction-diffusion system with non-diagonal $D$), in which they represent phase interfaces, of the bistable equation, and of the FitzHugh–Nagumo equations and their generalizations, in which they may represent phase changes, changes in the electrochemical properties of neural or cardiac tissue, or changes in the chemical state of a medium.

The stability of waves in one space dimension for systems with $n>1$ is a considerably more difficult field of investigation than is their existence. Much of the work here has been done for the FitzHugh–Nagumo equations [a12]. Recently a new technique, the stability index of J. Alexander, R. Gardner and C. Jones [a13], was developed and applied to a number of travelling-wave problems.

For travelling-wave and stationary solutions with interfaces (see above), a technique called the SLEP method has been developed to study stability questions (see [a4] and the references therein).

For FitzHugh–Nagumo and related systems, the most important patterned solutions in two space dimensions are rotating spirals, which are extremely prevalent and apparently extremely stable structures for this and many other models for excitable media. Despite the great interest in these rotating solutions and the large number of papers the concept has generated (see the references in [a9]), their mathematical foundation is still rudimentary. Analogous phenomena exist in three dimensions: structures which rotate about curves in space, called filaments, which themselves migrate according to certain approximate laws. An important challenge for the future is to better understand (and provide a firm mathematical foundation for) the connections between such dynamic spatial patterns and their laws of motion, on the one hand, and the underlying partial differential equations on the other.

References

[a1]	P.C. Fife, "Mathematical aspects of reacting and diffusing systems" , Lect. notes in biomath. , 28 , Springer (1979)
[a2]	D. Henry, "Geometric theory of semilinear parabolic equations" , Lect. notes in math. , 840 , Springer (1981)
[a3]	J. Smoller, "Shock waves and reaction-diffusion equations" , Springer (1983)
[a4]	C. Conley, "Isolated invariant sets and the Morse index" , Amer. Math. Soc. (1978)
[a5]	A.N. Kolmogorov, I.G. Petrovskii, N.S. Piskunov, "A study of the diffusion equation with increase in the quantity of matter, and its application to a biological problem" Byull. Moskov. Gos. Univ. , 17 (1937) pp. 1–72 (In Russian)
[a6]	D.G. Aronson, H.F. Weinberger, "Multidimensional nonlinear diffusion arising in population genetics" Adv. in Math. , 30 (1978) pp. 33–76
[a7]	Ya.I. Kanel', "On the stability of solutions of the Cauchy problem for the equations arising in combustion theory" Mat. Sb. , 59 (1962) pp. 245–288 (In Russian)
[a8]	P.C. Fife, J.B. McLeod, "The approach of solutions of nonlinear diffusion equations to travelling front solutions" Arch. Rational Mech. Anal. , 65 (1977) pp. 335–361
[a9]	P.C. Fife, "Dynamics of internal layers and diffusive interfaces" , CBMS-NSF Reg. Conf. Ser. Appl. Math. , 53 , SIAM (1988)
[a10]	R. Aris, "The mathematical theory of diffusion and reaction in permeable catalysts" , 1–2 , Clarendon Press (1975)
[a11]	J. Buckmaster, G.S.S. Ludford, "Theory of laminar flow" , Cambridge Univ. Press (1982)
[a12]	C. Jones, "Stability of the travelling wave solution of the FitzHugh–Nagumo system" Trans. Amer. Math. Soc. , 286 (1984) pp. 431–469
[a13]	J. Alexander, R. Gardner, C. Jones, "A topological invariant arising in the stability analysis of travelling waves" J. Reine Angew. Math. (To appear)
[a14]	Y. Nishiura, M. Mimura, H. Ikeda, H. Fujii, "Singular limit analysis of stability of travelling wave solutions in bistable reaction-diffusion systems" SIAM J. Math. Anal. , 21 (1990) pp. 85–122