An evolution equation is an equation for a function depending on a time variable and some other variables which can often be thought of as spatial variables. There is also the case where there are no variables , which is the one relevant for ordinary differential equations. We can reinterpret the function as being something of the form , a function of which depends on . I am thinking of the case where takes its values in a Euclidean space. Then should be thought of as taking values in a function space. Different regularity assumptions on the solutions naturally lead to different choices of function space. Suppose, for instance, I consider the one-dimensional heat equation . Then I could choose the function space to be , the space of continuous functions, , the space of twice continuously differentiable functions or . For some choices of function spaces we are forced to consider weak solutions. It is tempting to consider the evolution equation as an ODE in a function space. This can have advantages but also disadvantages. It gives us intuition which can suggest ideas but the analogues of statements about ODE often do not hold for more general evolution equations, in particular due to loss of compactness. (In what follows the function space considered will always be a Banach space.) We can formally write the equation of the example as for a linear operator . If we choose the function space to be then this operator cannot be globally defined, since does not map from to itself. This leads to the consideration of unbounded operators from to . This is a topic which requires care with the use of language and the ideas which we associate to certain formulations. An unbounded operator from to is a linear mapping from a linear subspace , the domain of , to . Just as there may be more than one space which is of interest for a given evolution equation there may be more than one choice of domain which is of interest even after the space has been chosen. To take account of this an unbounded operator is often written as a pair . In the example we could for instance choose to be the space of functions or the Sobolev space .

In the finite-dimensional case we know the solution of the equation with initial datum . It is . It is tempting to keep this formula even when is unbounded, but it must then be supplied with a suitable interpretation. There are general ways of defining nonlinear functions of unbounded linear operators using spectral theory but here I want to pursue another direction, which uses a kind of axiomatic approach to the exponential function . It should have the property that and it should satisfy the semigroup property for all non-negative and . It remains to require some regularity property. One obvious possibility would be to require that is a continuous function from into the space of bounded operators with the topology defined by the operator norm. Unfortunately this is too much. Let us define an operator whenever this limit exists in and to be the linear subspace for which it exists. In this way we get an unbounded operator on a definite domain. The problem with the continuity assumption made above is that it implies that . In other words, if the operator is genuinely unbounded then this definition cannot apply. In particular it cannot apply to our example. It turns out that the right assumption is that for and any . This leads to what is called a strongly continuous one-parameter semigroup. is called the infinitesimal generator of . Its domain is dense and it is a closed operator, which means that its graph is a closed subset (in fact linear subspace) of the product with the topology defined by the product norm. In a case like the example above the problem with continuity is only at . The solution of the heat equation is continuous in in any reasonable topology on any reasonable Banach space for but not for . In fact it is even analytic for , something which is typical for linear parabolic equations.

In this discussion we have said how to start with a semigroup and get its generator but what about the converse? What is a criterion which tells us for a given operator that it is the generator of a semigroup? A fundamental result of this type is the Hille-Yosida theorem. I do not want to go into detail about this and related results here. I will just mention that it has to do with spectral theory. It is possible to define the spectrum of an unbounded operator as a generalization of the eigenvalues of a matrix. The complement of the spectrum is called the resolvent set and the resolvent is , which is a bounded operator. The hypotheses made on the generator of a semigroup concern the position of the resolvent set in the complex plane and estimates on the norm of the resolvent at infinity. In this context the concept of a sectorial operator arises.

My interest in these topics comes from an interest in systems of reaction-diffusion equations of the form . Here is vector-valued, is a diagonal matrix with non-negative elements and the Laplace operator is to be applied to each component of . I have not found it easy to extract the results I would like to have from the literature. Part of the reason for this is that I am interested in examples where not all the diagonal elements of are positive. That situation might be described as a degenerate system of reaction diffusion equations or as a system of reaction-diffusion equations coupled to a system of ODE. In that case a lot of results are not available ‘off the shelf’. Therefore to obtain an understanding it is necessary to penetrate into the underlying theory. One of the best sources I have found is the book ‘Global Solutions of Reaction-Diffusion Systems’ by Franz Rothe.

May 21, 2021 at 9:03 am |

[…] be unbounded. In fact the case I am most interested in here is that where is the generator of a strongly continuous semigroup of linear operators on . Hence may not be globally defined on . It will be assumed that is […]

May 22, 2021 at 1:09 pm |

Note that eventually compact, eventually differentiable, analytic and uniformly continuous semigroups are eventually norm-continuous so that the spectral determined growth condition holds in particular for those semigroups.