Evolution equation

An equation that can be interpreted as the differential law of the development (evolution) in time of a system. The term does not have an exact definition, and its meaning depends not only on the equation itself, but also on the formulation of the problem for which it is used. Typical of an evolution equation is the possibility of constructing the solution from a prescribed initial condition that can be interpreted as a description of the initial state of the system. The class of evolution equations includes, first of all, ordinary differential equations and systems of the form

$$u'=f(t,u),\quad u''=f(t,u,u'),\label{*}\tag{*}$$

etc., in the case where $u(t)$ can be regarded naturally as the solution of the Cauchy problem; these equations describe the evolution of systems with finitely many degrees of freedom. Allowing for after-effect leads to integro-differential Volterra equations or to differential equations with retarded argument. The description of processes occurring in continuous media reduces to partial differential equations of hyperbolic, parabolic and related types; along with the Cauchy problem, here one can pose a mixed (initial boundary value) problem. If the solution $u(x,t)$ of such an equation is regarded as an element of some space of functions in $x$ that depend on a parameter $t$, then one arrives at abstract differential equations of the form \eqref{*}. All these equations, as well as their corresponding difference equations, usually belong to the class of evolution equations.

Analogues with real processes lead to the formulation of natural problems for an evolution equation (for example, the problem of stability of solutions) and occasionally suggest methods for their study (for example, a technique of establishing mathematical analogues of the laws of conservation or dissipation of the total energy). The evolution character of the equation facilitates its numerical solution, since the values $u(t_k)$ ($t_0<t_1<\ldots$) for a sufficiently small step $\Delta t_k$ can be obtained by means of stepwise reconstruction, starting from the initial condition. Consequently, in numerical calculations many problems concerning the steady state of a medium have come to be regarded as limiting cases of evolution problems as $t\to\infty$. (For example, the solution of the Laplace equation $\Delta u=0$ with prescribed boundary conditions is the limit of solutions of the equation $\partial u/\partial t=\Delta u$ satisfying the same boundary conditions and arbitrary initial conditions; in such cases one speaks of stabilization of the solutions of an evolution equation.)

References

[a1]	J.A. Walker, "Dynamical systems and evolution equations" , Plenum (1980)