The fractional reaction diffusion equation

The scalar reaction-diffusion equation

For (

Equation (

The purpose of this note is a study of the fractional reaction-diffusion equation:

For

The function

The equation occurs in the heuristic hydrodynamic limit of interacting particle populations in which conversion reactions such as (

There is strong evidence that (

The main results of this note are concerned with the existence and nonexistence of a finite speed of spread and take the form (

To prove these results, comparison arguments are employed which follow from integral representations of solutions of (

In this section, it will be shown that fundamental solutions of fractional diffusion equations (without reaction terms) lead to traveling wave solutions

Consider the “free” equation

There is also the special solution

For the remainder of this section, we suppress the subscripts

In the case

It remains to characterize the functions

Let

The function

As

As

The functions

Property (a) follows since

Property (e) will not be used in what follows. It should be noted that

The construction provides travelling wave solutions for (

If the same construction is attempted for the case

This section contains the main results of this paper. As before, the operator

We first discuss the case where

Let

Let

Let

Let

The result shows that the speed of spread is unbounded (i.e., (

In the case

The main result in the case where

Let

The result shows that negative proportional growth at small densities (

The proofs will be given below. The main tools in the proofs are the comparison arguments given in the next section, together with the following crucial auxiliary result.

Let

Let

Let

Let

Suppose that

Consider first statement (a). Let us write

The proof of part (

To prove part (c), let again

To prove part (d), note first that

The proof uses the same argument for all three parts; so we give details only in part (a). Let

As

In case of part (b), the same argument can be used without changes, appealing to part (b) of Lemma

In case of part (c), we have to restrict

We replace

In this section we summarize some basic theory about (

We work in the Banach space

For fixed

Solutions of (

Let

Let

We now employ a standard induction argument to show that

Consider a mild solution

The proof consists in observing that the constant functions

Also required is a comparison result for solutions whose initial data are step functions. Since such initial data are not in

Let

We know that

In this note, conditions for the speed of spread of solutions of fractional scalar reaction-diffusion equations to be finite or infinite have been derived. If the reaction term is positive for all positive arguments, then this speed is shown to be infinite as soon as the reaction term describes some very weak growth for low densities. This is in contrast to the corresponding problem for standard diffusion, where the speed of spread is always finite for such reaction terms. On the other hand, if the reaction term is negative for small positive arguments, then the speed of spread is finite, just as it is for the case of standard diffusion.