## Abstract

Here, we present a survey concerning parabolic free boundary problems involving a discontinuous hysteresis operator. Such problems describe biological and chemical processes ‘with memory’ in which various substances interact according to hysteresis law. Our main objective is to discuss the structure of the free boundaries and the properties of the so-called ‘strong solutions’ belonging to the anisotropic Sobolev class with sufficiently large *q*. Several open problems in this direction are proposed as well.

## 1. Introduction

Here, parabolic equations containing a discontinuous hysteresis operator on the right-hand side are investigated. For simplicity, we restrict our consideration to the most basic discontinuous hysteresis operator—the so-called non-ideal relay.

If the input function *u* is less than a lower threshold , then the output *h*[*u*] of the non-ideal relay is equal to −1. With increasing *u*, the output remains equal to −1 until the input reaches an upper threshold —at this point the output switches from −1 to 1. Further increases in *u* do not change the output state. Observe that *h*[*u*] switches back to the state −1 when the input *u* decreases again to *α*. This behaviour is illustrated in figure 1. It is evident that the non-ideal relay operator takes the path of a rectangular loop and its next state depends on its past state.

Examples of parabolic equations with non-ideal relay arise in various biological, technological and chemical processes (see, for instance, [1–5] and references therein).

### (a) Statement of the problem

We study solutions of the nonlinear parabolic equation
1.1
satisfying the initial condition
1.2
We also assume that *u* satisfies either the Dirichlet or the Neumann boundary condition on the lateral surface of cylinder *Q*, i.e.
1.3
Here Δ is the Laplace operator, is a domain in and is its boundary, *φ* and *ψ*_{1} (or *ψ*_{2}) are given functions, while *h*[*u*] stands for a non-ideal relay operator acting from to {±1}.

In order to define the operator *h*[*u*], we fix two numbers *α* and *β* (*α*<*β*) and consider a *multi- valued* function
Assuming , we suppose that the values of *h*_{0}(*x*):=*f*(*φ*(*x*)) are prescribed. We set
1.4

After that, for every point *z*=(*x*,*t*)∈*Q* and for the corresponding value of *h*[*u*](*z*) is uniquely defined in the following manner. Let us denote by *E* a set of points
In other words, *E* is a set where *f*(*u*(*z*)) is well defined.

If *z*∈*E* then *h*[*u*](*z*)=*f*(*u*(*z*)). Otherwise, for *z*=(*x*,*t*)∈*Q* such that *α*<*u*(*z*)<*β* we set
where

Observe that for fixed *x* a jump of *h*[*u*](*x*,⋅) can happen only on thresholds {*u*(*x*,⋅)=*α*} and {*u*(*x*,⋅)=*β*}. Moreover, **‘jump down’** (from *h*=1 to −1) is possible on {*u*(*x*,⋅)=*α*} only, whereas **‘jump up’** (from *h*=−1 to 1) is possible on {*u*(*x*,⋅)=*β*} only.

### Remark 1.1

It should be emphasized that the above definition of *h*[*u*] excludes the case *α*=*β* from the consideration.

### Definition 1.2

We say that *u* is a (strong) solution of (1.1)–(1.4) if

(i) ,

*q*>*n*+2, and*h*[*u*] is generated by the function*u*in view of (1.4) as described above,(ii)

*u*satisfies (1.1) a.e. in*Q*,

### Remark 1.3

Recall that is the anisotropic Sobolev space with the norm
where ∥⋅∥_{q,Q} denotes the norm in *L*^{q}(*Q*) with .

Since the right-hand side of equation (1.1) is a discontinuous function depending on *u*, the location of interfaces between regions where *h*[*u*] takes the values +1 and −1 is *a priori* unknown. They can be treated as the free boundaries.

### (b) Historical review

A first attempt to create a mathematical theory of hysteresis was made in [6], where problems with ODEs were studied. We mention also the fundamental books [7–9] in which various hysteretic effects in spatially distributed systems are described.

The problem (1.1)–(1.4) was introduced in [3], where the growth of a colony of bacteria (*Salmonella typhimurium*) on a Petri dish was modelled. The papers [3,4] were devoted to numerical analysis of the problem, however without rigorous justification.

First, existence results for solutions of problems such as (1.1)–(1.4) were proved in [1,2] for modified multi-valued versions of the hysteresis operator. The main reason for these operator extensions was the observation that the discontinuous hysteresis operator *h*[*u*] is not closed with respect to topologies appropriate to its coupling with PDEs.

In [2], problem (1.1)–(1.4) was studied in the one-(space)-dimensional case under the assumption that on the level sets {*u*=*α*} and {*u*=*β*} the right-hand side of equation (1.1) allows us to take any value from the whole interval [−1,1]. The global existence in a specially defined class of weak solutions was established there. In addition, the non-uniqueness and non-stability of such weak solutions were discussed in [2] in several examples.

The multi-(space)-dimensional case with the non-ideal relay replaced by the so-called completed relay was treated in [1], where the global existence result was established for weak solutions. This completed relay operator *h*_{c}[*u*] is the closure of *h*[*u*] in suitable weak topologies. In particular, *h*_{c}[*u*] admits any value from the whole interval [−1,1] on the set {*α*≤*u*≤*β*} (figure 2). For further details concerning the properties of *h*_{c}[*u*], we refer the reader to [10] as well as to ch. IV of [7].

It was also shown in [1] that for *α*=*β* the problem (1.1)–(1.4) with *h*_{c}[*u*] on the right-hand side of equation (1.1) is, in fact, the two-phase parabolic free boundary problem. The properties of solutions of this two-phase problem as well as the behaviour of the corresponding free boundary were completely studied in [11].

### Remark 1.4

Owing to directional restrictions on jumps of the hysteresis on the thresholds *α* and *β*, problem (1.1)–(1.4) with *α*<*β* cannot be reduced to the two-phase parabolic problem even for modified versions of the hysteresis operator.

Another way to overcome the troubles generated by discontinuity of the hysteresis operator has been proposed in [12,13], where a *special class* of strong solutions of (1.1)–(1.4) satisfying the additional *transversality property* was introduced in the one-(space)-dimensional case. This transversality property, roughly speaking, means that the solution *u* has a non-vanishing spatial gradient on the free boundary. More precisely, in the case *n*=1 the transversal solutions are defined as follows.

### Definition 1.5

A function *u* is called a *transversal solution* of (1.1)–(1.4) if *u* is a solution in the sense of definition 1.2 and the following hold:

(i) if and for some and , then in a neighbourhood of ;

(ii) if and for some and , then in a neighbourhood of .

Taking the transversal initial data and assuming that there is only a finite number of points where *φ* takes the values *α* and *β*, Gurevich *et al*. [12] proved the local existence of strong transversal solutions of (1.1)–(1.4) and showed that such solutions depend continuously on initial data. A theorem on the uniqueness of strong transversal solutions was established in [13].

### Remark 1.6

The local well-posedness of problem (1.1)–(1.4) in the multi-(space)-dimensional case is still an open question.

Recently, the approaches developed in the theory of free boundary problems have been applied to the study of strong solutions. Such an activity was initiated in [14], where the local regularity properties of the strong solutions of (1.1)–(1.4), i.e. solutions from the Sobolev space with sufficiently large *q*, were studied in the multi-(space)-dimensional case. Without assuming the transversality property, it was proved in [14] that outside some ‘pathological’ part of the free boundary we have the optimal regularity .

### Remark 1.7

For readers not familiar with free boundary problems, we recommend that they should consult [15], where the main concepts and developed methods are discussed for several model problems.

In the rest of this paper, we describe in more detail the properties of the strong solutions of (1.1)–(1.4) as well as of the free boundaries that are currently known. Sections 2 and 3 are devoted to the general multi-(space)-dimensional case where we do not assume the transversality property. Conversely, §4 deals with the transversal solutions in the one-(space)-dimensional case. The results listed in §§2 and 3 were established in [14], while in §4 we announce the results from [16].

## 2. Structure of the free boundary (case *n*≥1)

We denote
The latter means that *Γ*(*u*) is the set where the function *h*[*u*](*z*) has a jump. Note that *Ω*_{±} may consist of several components , respectively. In other words,

We also introduce special notation for some parts of *Γ*(*u*)
and
By definition

Owing to definition 1.2, it is easy to see that the (*n*+1)-dimensional Lebesgue measure of the sets {*u*=*α*} and {*u*=*β*} equals zero.

In addition, the sets {*u*=*α*} and {*u*=*β*} are separated from each other. Moreover, for any *ε*>0 the distance from the level set {*u*=*α*} to the level set {*u*=*β*} in the cylinder with and is estimated from below by a positive constant *δ* depending on , *ε* and *β*−*α* only.

Observe that the level sets {*u*=*α*} and {*u*=*β*} are not always the parts of the free boundary *Γ*(*u*). Indeed, if the level set {*u*=*α*} is locally not a *t*-graph, then a part of {*u*=*α*} may occur inside *Ω*_{−}. In this case, *Γ*(*u*) may contain several components of *Γ*_{α} connected by the parts of cylindrical surfaces with generatrices parallel to the *t*-axis. A similar statement is true for the level set {*u*=*β*}. We will denote by *Γ*_{v} the set of all points *z* lying in such vertical parts of *Γ*(*u*). Some examples of eventual connections of *Γ*_{v} with *Γ*_{α} and *Γ*_{β} are shown in figure 3.

Recall that by definition of *Γ*_{α}(*u*) the function *h*[*u*] has a jump in the *t*-direction from +1 to −1. The latter means that if we cross the free boundary *Γ*_{α}(*u*)∖*Γ*_{v} in the positive *t*-direction, then the corresponding phases change from *Ω*_{+} to *Ω*_{−} (figure 3). A similar statement can be made for the neighbourhood of *Γ*_{β}(*u*), that is, if we cross the free boundary *Γ*_{β}(*u*)∖*Γ*_{v} in the positive *t*-direction then the phases will change from *Ω*_{−} to *Ω*_{+}.

Thus, we have

It should be noted that this *Γ*_{v} is just the ‘pathological’ part of the free boundary mentioned in the Introduction. Indeed, we have no information about the values of *u* on *Γ*_{v}, since *Γ*_{v} is, in general, not the level set {*u*=*α*} as well as not the level set {*u*=*β*}. Furthermore, for any direction functions and are, in general, not sub-caloric near *Γ*_{v}. The latter fact causes serious difficulties in studying the regularity properties of solutions.

We will also distinguish the following parts of *Γ*:
The sets and *Γ**_{β} are defined analogously. In addition, we set

Using the von Mises transformation combined with the parabolic theory and the implicit function theorem, it is possible to show that *Γ**(*u*)∖*Γ*_{v} is locally a *C*^{1}-surface.

## 3. Optimal regularity of *u* beyond *Γ*_{v} (case *n*≥1)

In this section, we discuss the *local* regularity properties of a strong solution *u* of (1.1)–(1.4).

For simplicity of notation, we will denote by *M* a majorant of and will highlight below the dependence of all obtained estimates on *M*.

Recall that the general parabolic theory (e.g. [17]) provides for any *ε*>0 the estimates
where , and . Moreover, the function *u* and its spatial gradient *Du* are Hölder continuous in *Q*, and in the interior of the sets *Ω*_{±}, as well.

We note also that if as well as the values of *u* on the parabolic boundary of *Q* are smooth then the corresponding estimates of *L*^{q}-norm for ∂_{t}*u* and *D*^{2}*u* are true in the whole cylinder *Q*.

In contrast, *Δu*−∂_{t}*u* has a jump across the free boundary *Γ*(*u*). Thus, is the best possible regularity of solutions.

The optimal (i.e. ) regularity is not obvious. A crucial point here is the quadratic growth estimate of the type
3.1
with *z*^{0}=(*x*^{0},*t*^{0})∈*Γ*^{0}(*u*)∖*Γ*_{v}. Here *ρ*_{0} denotes the parabolic distance from *z*^{0} to *Γ*_{v}, while *ϵ* stands for the parabolic distance from *z*^{0} to ∂′*Q* and

### Remark 3.1

The parabolic distance dist_{p} from a point *z*^{0}=(*x*^{0},*t*^{0}) to a set is defined as

To show quadratic bound (3.1), we argue by a contradiction and combine this with a local rescaled version of the famous Caffarelli monotonicity formula.

### Remark 3.2

More detailed information about Caffarelli's monotonicity formula and its local rescaled version can be found in [18] and in [14,19], respectively.

Further, it can be verified that the quadratic growth estimate (3.1) implies the corresponding linear bound for |*Du*|
3.2
with *z*^{0}∈*Γ*^{0}(*u*)∖*Γ*_{v}.

The dependence of *N*_{2} on the distance *ρ*_{0} in (3.1) and (3.2) arises due to the monotonicity formula. Unfortunately, near *Γ*_{v} neither the local rescaled version of Caffarelli's monotonicity formula nor its generalization such as the almost monotonicity formula introduced in [20] are applicable to positive and negative parts of the space directional derivatives *D*_{e}*u*.

Besides estimates (3.1) and (3.2), the information about behaviour of ∂_{t}*u* near *Γ**(*u*) plays an important role. As already mentioned above, ∂_{t}*u* may have jumps across the free boundary. Actually, one can show that ∂_{t}*u* is a continuous function in a neighbourhood of *z**∈*Γ**(*u*)∖*Γ*_{v}. In addition, the monotonicity of jumps of *h*[*u*] in the *t*-direction provides one-sided estimates of the time derivative of *u* near *Γ*(*u*). More precisely, the estimate of ∂_{t}*u* from below holds true near *Γ*_{α}(*u*), whereas the estimate of ∂_{t}*u* from above holds true near *Γ*_{β}(*u*). Combining these results with the observation that ∂_{t}*u*≤0 on and ∂_{t}*u*≥0 on leads to an absolute estimate of the time derivative of *u* on the set *Γ**(*u*)∖*Γ*_{v}. Namely,
3.3
where the constant *N*_{3}, contrary to *N*_{2}, depends only on given quantities. In addition, we can show that the mixed second derivatives *D*_{i}(∂_{t}*u*) are -functions in *Q*∖(*Γ*^{0}(*u*)∪*Γ*_{v}).

Now, estimates (3.1)–(3.3) allow us to apply the methods from the theory of free boundary problems and estimate |∂_{t}*u*(*z*)| and |*D*^{2}*u*(*z*)| for any *z* being a point of smoothness for *u*. The corresponding details can be found in [14]. The main result is formulated as follows.

### Theorem 3.3

*Let u be a solution of equation* (*1.1*), *and let z∈Q∖Γ*(*u*). *Then*
3.4
*Here ρ*_{0}*:=*_{p}*{z,Γ*_{v}*} and ϵ:=*_{p}*{z,∂′Q}.*

We emphasize that the constant *C* in (3.4) does not depend on the parabolic distance from *z* to *Γ*^{0}(*u*) as well as to *Γ**(*u*). Unfortunately, we cannot remove the dependence of *C* on *ρ*_{0}, i.e. on the parabolic distance of *z* to *Γ*_{v}.

## 4. Transversal solutions (case *n*=1)

For one-(space)-dimensional transversal solutions of (1.1)–(1.4) the results of §§2 and 3 can be strengthened.

As in [12,13], we will suppose that initially there are only a finite number of different components .

It is obvious that for *n*=1 the ‘pathological’ part of the free boundary *Γ*_{v} is a union of vertical segments parallel to the *t*-axis. Further, according to definition 1.5 we conclude that for transversal solutions *u* the inequality
holds true on *Γ*_{α}(*u*)∪*Γ*_{β}(*u*). However, on *Γ*_{v} the function *u*_{x} may vanish even for transversal solutions. In addition, the time derivative ∂_{t}*u* is continuous across *Γ*_{v} except for, eventually, the endpoints of the vertical segments. Thus, for transversal solutions we have *Γ*^{0}(*u*)=∅, and, consequently, *Γ**_{α}(*u*)=*Γ*_{α}(*u*) and *Γ**_{β}(*u*)=*Γ*_{β}(*u*).

The transversality property of a solution *u* implies the monotone behaviour of the free boundary *Γ*(*u*). Indeed, it is possible to show that the set *Ω*^{(k)}_{+} is locally either a subgraph of a monotone curve or a subgraph of a union of two monotone curves with different characters of monotonicity. These monotone curves include the parts of *Γ*_{α}(*u*) and also may contain vertical segments. An example of a possible union of two monotone curves is provided in figure 4.

A similar statement is true for the set near *Γ*_{β}(*u*). So, is locally either a subgraph of a monotone curve or a subgraph of a union of two monotone curves with different characters of monotonicity.

Finally, under the additional assumption that on a time interval (*t*^{1},*t*^{2}) the free boundary part *Γ*_{v} consists of at most a finite number of vertical segments, it is possible to prove the optimal regularity result up to *Γ*_{v}. In other words, one can establish that for any being a point of smoothness for transversal solutions *u* the constant *C* from (3.4) is independent of the parabolic distance from *z* to *Γ*_{v}.

A detailed explanation of all the results presented in this section can be found in [16].

## 5. Conclusion

Below we discuss several open problems and areas of further work.

It should be emphasized that the quadratic growth estimate (3.1) allows us to apply the standard parabolic scaling in points *z*^{0}∈*Γ*^{0}(*u*)∖*Γ*_{v} and to obtain the corresponding blow-up limits of *u*. Using a version of the parabolic Weiss monotonicity formula, it can be shown that all blow-ups are homogeneous functions. Classification of all possible blow-up limits and further study of the regularity of the free boundary *Γ*(*u*)∖*Γ*_{v} are open problems. For *n*=1, some preliminary results of this kind are established in [16].

### Remark 5.1

The original version of the parabolic Weiss monotonicity formula can be found in [21].

Let *u* satisfy on ∂′*Q* the inequalities *α*≤*u*≤*β*. In this case, even for weak solutions, we have the same inequalities *α*≤*u*≤*β* a.e. in *Q*. We suppose that strong solutions of (1.1)–(1.4) do not exist in this case.

Note that numerical examples given in [22] suggest the possible non-existence of the strong solutions with non-transversal initial conditions. Confirmation or rejection of this hypothesis is an open question even for the one-(space)-dimensional case.

Another challenging problem is to define the transversality property for strong multi-(space)-dimensional solutions. Some results in this direction were obtained very recently in [23].

The last (but not least) hypothesis concerns the regularity of the free boundary in the non-transversal case *n*=1. It is proposed to prove or disprove the assertion that the free boundary *Γ*(*u*) is smooth except for the endpoints of the vertical segments provided *Γ*_{v} consists of at most a finite number of such segments.

## Competing interests

We declare we have no competing interests.

## Funding

This work was supported by the Russian Foundation of Basic Research (RFBR) through grant no. 14-01-00534, by the St Petersburg State University grant 6.38.670.2013 and by grant ‘Nauchnye Shkoly’, NSh-1771.2014.

## Acknowledgements

The authors also thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, where part of this work was done during the programme *Free Boundary Problems and Related Topics*.

## Footnotes

One contribution of 15 to a theme issue ‘Free boundary problems and related topics’.

- Accepted January 8, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.