 First of all, I would like to thank the organizers of this conference for inviting me to meet this very special occasion, very special conference in honor of Frank Mel on the occasion of his more or less 60th birthday. And as everybody knows, Frank is an excellent mathematician, everybody says that of course, and with exceptional talent. And this I have seen many times while I was collaborating with him. The first time when I met Frank was, I think it was in Paris, either at the end of the eighties or the beginning of the nineties, maybe by the beginning of the nineties, a little more than 30 years ago. And he came to tell me about his recent result. He works on many problems, but among other things, he also works on blowing up in non-linear heat equations. And he told me, explained, tried to explain, maybe Frank was at that time in the late 20s or below 30, I suppose. And he tried to explain his result, which shows that you can find an initial data such that it blows up at many different points at the same time. And it was very interesting result. And later I started real collaboration with him in 1999, so maybe 24 years ago. And so I invited him and we also collaborated, but there was France, Japan, two-country exchange program of JSPS. And on the Japanese side, it was Mimura and myself, and probably one more. And on the French side, it was Daniel Hillhost, Elizabeth Logak, and Frank. And so we started this. But later, Frank came as invited researcher of JSPS for two months in 2002 or three or three probably. But anyway, Frank is extremely smart. But sometimes you need some effort to understand what he says. But I agree that he is smart, not only in mathematics, but in many other occasions, he has a very quick and correct judgment. Very extremely smart. Of course, he sometimes makes mistakes. You shouldn't just rely on what he says. But nonetheless, often he has a very deep thinking of things even outside of mathematics. And at the beginning, okay, nowadays Frank gives a very good lecture with very neatly made slides. But I suppose that it was made by his collaborators, not by himself. That helps him a lot. Before, maybe in Overlapur Facha or some other places, he gave a talk on the blackboard. And to some people, it could look like a disaster. However, if you are a trained person, it is by no means a disaster. I once talked with Hatem Zag, who is over there. And he told me when he became a student of Frank, it took him six months or more to understand what he wants to say. But after he acquired a sort of ability to interpret his language into a normal language, Hatem realized that Frank was saying very deep things. The depths of his comments Hatem appreciated. You told, say that. Yes. On another occasion, I asked Ivan Martel, who is a frequent collaborator of Frank. And often Frank gives a talk about the joint work. Maybe you forgot it. I asked him, probably after Frank was giving a talk of a joint work with you, what do you think about his talk? And Ivan's answer was surprising. Okay. Sometimes even for a joint work in which I am involved, Frank's talk gives a new aspect to my own joint work. So I can understand more deeply my own work. This is what you said, I think you agree. So it's Frank is not just producing theorems, good theorems, but also have a very deep insight behind the theorems. But also Frank is a person who has a very strong love of his family. He cares a lot of his children. I can say many things, but when he came with a very small baby, Yasha, to Japan and he went to a crab restaurant, restaurant serving crab, nice restaurant. And there were a lot of live crabs in a little tank. And we had to pass next to it to reach the table. And Frank was trying to protect his baby from the crab. And people were saying that no, crabs don't jump. He said just in case. That tells how strong love he has to family. And so this is his eldest son, Yasha, which was taken in 2002. And by the way, today is the birthday of Yasha Mel, right? It's a coincidence that I gave a talk on his birthday. And this was in the cafe in Paris. And it was a little earlier, just shortly after the birth of Yasha. And it was his home. And here is Nuremberg and Rebecca Frank's wife and my wife and myself and Yasha. And then nine years later, he's more grown up. And there's another boy, Maxim, and they were visiting Nara. And often Frank came in summertime to spend summer vacation in Japan. I told him that in Kyoto or Tokyo, in Japan summer is very, very hot and more humid. He said that no, I'm from the south and I don't care. But probably his children cared a lot. But anyway, this is a little later, 11 years ago, when Frank, Daniel and other people are kindly organized a conference on the occasion of my birthday. And this is in this building, an Atem Zag. And this is Adrian Blanchet from Toulouse. Okay. So now here is the outline of my talk. First, I will explain what is a bi-domain problem. Some people may know it, but there may be people who may not know it. And then I will talk about some earlier result and more recent result. And this is joint work with mainly with Yoichiro Mori, who is at the University of Pennsylvania. When I started collaboration with him, he was at the University of Minnesota. But now he's a chair professor at the University of Pennsylvania, director of the Center of Mathematical Biology and with Mitsunori Nara and the younger person, Koya Sakakibara, who now moved from Okayama University of Science to Kanazawa University. Now here is a formulation of the problem. What is bi-domain model? Bi-domain model is written typically in this form. I'm mainly focusing on bi-domain Alenkan model. So it's an Alenkan or more precisely bi-stable non-linearity. So it's a slightly modified version of reaction diffusion equation. So you have this part and in the reaction diffusion equation you have an aplation of U, but instead you have this equality and this equality. And the non-linearity unknown function is U, Ui minus Ue. And I will explain what Ui and Ue means, but these equalities are for Ui and Ue. So it looks a little strange, but in some sense the unknowns are Ui and Ue, but dynamics is for U, not Ui and Ue. So I called it Alenkan, but to be more precise it is a bi-stable non-linearity, which is quite common in the study of reaction diffusion equations. And the main question is the stability of the planar wave. So where does this bi-domain model come from? The background is electrical conduction in the heart. And the heart has a lot of cardiac cells and the cardiac cells are not, it's a kind of muscle cell, but it can also conduct an electric signal like a neuron. And they are sort of coupled with some what is called gap junction through which a lot of irons pass through and that, so this way they are electrically connected. And then if you look at each cell, which as you saw, it's a bit long, so it's not round shaped. Then inside and outside region, we can call intercellular region and extracellular region, and both inside and outside there is a potential, electric potential. And on the membrane, the the difference between bi and ve is called membrane potential, which is very, very important. So the membrane potential is defined on the boundary of this cell and the difference is the most important things. And then inside and outside it's not necessarily isotropic, so we have conductivity matrix, intracellular and extracellular. And we have... It should be E there, no? One is inside, one is outside? Inside and outside, yes, intra and extra. Then on the other box it's extracellular. Okay, all right. So anyway, inside and outside there are... Okay, sorry, this is E. Quadri-stationary conduction, which means that in this small scale, the electric current does not create much magnetic field, so we don't need to use Maxwell equation. So we have more or less this kind of equation. And the continuity of electric current has to be satisfied along the membrane. And if there are many cells, they form many cells from what is called the tissue, and in this level small cells are invisible in some sense. It is visible, but you can just look at the homogenization limit in some sense. Then this continuity condition will be turned into this. Why is it called the bi-domain model? Because in the homogenization limit, intra-cellular region and extra-cellular region, the difference just disappears. So the homogenized intra-cellular potential and homogenized extra-cellular potential, which we call UI and UE, defined everywhere. So at each point, it is a superposition of intra-cellular and extra-cellular regions. And what is really important is UI minus UE, the difference on the cell membrane. By the way, in the classical model, for the electric, I mean, nervous signal conduction model, the most fundamental one is Hojiki-Hakselen model, which is a system of reaction diffusion type equation plus ODE. And it could be more complicated depending on the situation. And a simplified version is Fitsu Nagumo model, which captures a large part of the future of Hojiki-Hakselen. But it also applies to other physical situations. And it is well known that they received a Nobel Prize for this Hojiki-Hakselen. But in this case, it is 1D Laplacia. And if we consider Hojiki-Hakselen, the paper in 1952, consider the circuit, electric circuit model. And if you just connect this circuit, then in the homogenized limit, you get this diffusion equation. So that's how the Hojiki-Hakselen model was derived. However, if you look at the 2D model with cardiac cells, they also transmit electric signal. However, if it is 1D, the homogenized limit is a diffusion equation. But if it is 2D, it is no longer a diffusion equation. It is a bi-domain model. So that's why it is important to study a bi-domain model to study the electric propagation on the heart. Now, okay, so I'm just comparing the monodomain model. Monodomain means that just classical standard Hojiki-Hakselen office Nagumo model with the bi-domain model, which is here. And the bi-domain model gives more accurate descriptions of the propagation of electric signal on the heart, which is explained in the book of Kina Sunid, for example. And with the diffusion model, you cannot well explain such a phenomenon as cardiac defibrillation. But I'm not so familiar with this. I'm talking more on the mathematical side aspect of this problem. So despite the importance in applications, little was known mathematically about the qualitative properties. A lot of the research have been done, as I will mention later, but particularly the qualitative property, like stability or asymptotic behavior. This was not so much studied. And I will focus mostly, in this talk, as a first step to understand this model qualitatively, I will focus on the single equation, which is the bi-domain-alenkan model, to be more precise, bi-domain-by-stable model. So F is a standard alenkan, but it's not balanced. So alenkan is usually balanced by stable non-linearity, but here I just have used the terminology, so it is bi-stable, unbalanced non-linearity. So monodomain is this, and bi-domain is this. But it is not easy to handle this problem if you write it this way. So it will be converted into a closed form of a single equation data, but in that case, the principal term is no longer differential operator, but a fully integral operator. So a typical case which we studied is this case. Both are diagonal matrices, but they have different types of scales in each direction. And the first simple remark is that if AI is a scallop product of AE for some alpha, then it releases the monodomain model, because if this holds, then you can just rewrite this into this form. So it is not a Laplacian, but after linear change of variable, it will become Laplacian. So it's the same as the standard reaction diffusion equation. But this is a very, very special case. In all other cases, the maximum principal doesn't hold. The maximum principal holds only for this special case. And then also it has to be mentioned that if the solution depends only on one direction, and independent of the orthogonal direction, then the equation reduces to a single reaction diffusion equation, one dimensional, which means that a planar wave solution is a special solution which depends only in one directional variable and independent of the orthogonal variable. Therefore, the shape of the planar wave is completely the same as the standard classical Alenka model with some coefficient. But the diffusion coefficient depends on the direction of propagation, as I explained later. But the stability properties are completely different. So here are the known results about the bi-domain models. So it has been studied by many people, particularly in the Italian group, Cody Franzone and his group had many early contributions for the analysis of bi-domain models, and also some French group studied the well-poseness and also former derivation of the sharp interface limit, and also a book of Kin and Sneed on the mathematical physiology explains this bi-domain model. Excuse me. Are the interior and exterior matrices constant? In my talk, yes, to make, oh sorry, to make it simpler. If it is not constant, of course, you can define the same way, but it will make the problem much more complicated. So I should have said that, but constant, which is already quite difficult. So very little was known, as I said, qualitative properties like stability before. The first work in this direction was by Yochiro Mori and myself, which appeared in CPAM 2016, which I will explain later. But today's talk is based on the following papers with the 2016 paper and more recent papers. So first, let me explain about linear stability. So my work with Yochiro Mori was in 2016, was on the linear stability, and it is very, very interesting, which was a surprise to us. So, but let me go quickly, as Elizabeth looked at the watch, so maybe I should be careful about the time. Okay, so the example is again this case, but it is, and also A and B are constant, but it is already interesting enough. So, as I said, if the solution is written, depends only in the direction n, where n is a unit vector, then you satisfy the classical Alen Kahn equation, and which I will show here. C is the direction of propagation, and eta is orthogonal, but I assume that u is independent of eta, then this function v, satisfy this standard reaction diffusion equation, where k is a constant, which depends on theta. So, this is the speed of, wave speed in direction of theta. It is, if f is fixed and we change theta, then the wave speed is proportional to this constant. And this is very, very important, that's a matter of fact, this, how the wave speed changes according to the direction. And, okay, now let's first recall what is known for the classical reaction diffusion equation, which is of this form. And in this case, it is well known that the planar waves are all stable, and the proof is rather easy. Some weak stability can be proved by using just comparison argument, and it gives stability in L infinity on the whole domain, in the whole space, even not necessarily in 2D, but in every dimension. The most naive proof is if this is a traveling wave, which we are looking at, and just look at a special shift of the two traveling, two special shift of the traveling wave, they all travel at the same speed. So, any, if you give perturbation, within these two traveling waves, the solution by the comparison principle remains between the two forever, therefore it is stable in a weak sense. However, if you use a better super and sub solution, which Fife and McLeod did in the late 1970s, you can also give a super solution, which is slightly lifted or lowered, and the solution is confined, and this upper solution, the lower solution, will eventually come down to the same level of the original traveling wave, so it is a very nice result. But you can also say more, by using the fact that this traveling wave, u sub x is negative, and using Klein-Rutman theorem, linear stability follows. What does it mean? u sub x, if you differentiate the equation for the traveling wave by x, you will see that u sub x is a eigenfunction for eigenvalue zero, and by Klein-Rutman theorem, if the eigenfunction doesn't change sign, it is the principal eigenvalue, so zero is the principal eigenvalue, and moreover, it is simple, which means that this zero eigenvalue comes from the equivalence with respect to translation, so for the stability, what matters is the rest of the eigenvalues, and the essential spectrum is not really important, because f prime of zero and f prime of one are all negative, and because of that, you will see that all the eigenvalues are negative, and therefore it is stable, which gives linear stability, spectral stability, and therefore if you perturb slightly the traveling wave, it will converge to one of the traveling waves, so it is stable with asymptotic phase. So this is the well-known fact for the classical model, and the maximum principal plays a key role, of course, in this business, and okay, for people working on wave equations and other shredding equations, maximum principal doesn't hold it anyway, so here there are many specialists who don't use maximum principal, but for parabolic people like us, maximum principal is very, very important. However, unfortunately, the by the way model, maximum principal doesn't hold, except for the very, very special case, therefore we have to work on, even though it looks like a reaction diffusion equation, it is similar, but because of lack of maximum principal, many strange things will happen. For example, depending on the direction, planar front, which is slightly perturbed, will develop, as time passes, develop a pattern which looks like sawtooth, and the angle of the sawtooth depends on the direction of the propagation of the planar wave, and at the very beginning, Iochiro Mori and I were not sure what these angles were, and later we came to a conjecture what these angles signify, and we did some numerical simulation to confirm our conjecture, numerically at least, I will explain it later. Now, to study the linear stability, Fourier transform, we do it by Fourier transform. So this equation, equality is this, and we do the Fourier transform, and then everything will become, this differential operator will become something like that, so we can just multiply or divide, and then these matrices, and u i, u can be expressed by u i, and therefore this equation can be written by a single equation, which is this. q k is this homogeneous degree to rational function, q i and q are quadratic polynomials, so this is degree two, and the bi-domain operator, so the equation is equivalent to this equation, just a single closed form, where l is what we call a bi-domain operator, whose Fourier symbol is this q, so in the case of Laplacian, it is quadratic polynomial, and this is also degree two, this is also degree two, but it's a rational function, so that makes things quite different. For example, if we look at the fundamental solution, fundamental solution generated by this bi-domain operator, if a and b are zero, it is just unit matrices, so it's a classical Laplacian, so then the fundamental solution is completely positive. However, as soon as it becomes different a and b are not zero, non-zero, it has a negative part, so therefore the positivity is not preserved, which is of course no surprise to people working on other equations, but for the reaction diffusion people, it is a big problem, because we have to change the approach completely, and we have proved that apart from the very special case where the two matrices are scalar product of each other, in that case it is completely the resulting equation is a standard reaction diffusion equation, but except for this case it has a negative part, and this can be a in physiological viewpoint, it can be interpreted that the current impulse at one location can result in hyperpolarization atmosphere, so you can also call it some sort of lateral inhibition using a biological term, and this property can suggest that a lot of interesting patterns may arise. So let me just go very quickly, skip the details of computations, but anyway I just skip the details, but the stability of the wave depends on the direction of propagation, and therefore it is more natural to consider a coordinate, new coordinate. Xc is the direction of propagation, and eta is orthogonal to that, and the planar wave depends only on c, but its perturbation may also depend on eta, but how does it depend on eta is the problem, so this is the new coordinate, but let me just go relatively quickly, and our analysis was done the following way, so this is a degree two, homogeneous degree two rational function, but we can take out the degree two polynomial out of this, then the remainder is a smaller term, and the degree two polynomial part is the same as Laplacian. Therefore, okay we just skip this anyway, this, okay, okay let me just quickly go, okay, so the eigenvalue can be expanded in L L is the wave number parallel to the front, L, k is a small little k is orthogonal, so to the direction of propagation, and then it turns out that we can project into the function of wave number L, and then consider eigenvalue problem, the eigenvalue, it turns out that eigenvalue is written in this way, and this doesn't affect the stability, so what matters is this, and this alpha zero can be computable, I don't write it down, but it turned out that this alpha zero, okay let me just go quickly, this alpha zero, the sign of this alpha zero is completely the same as the sign of the curvature of the Frank diagram, Frank diagram is the shape which is defined this way, so k theta is the speed of traveling wave in each direction theta, and if you divide it, Frank diagram, this is often for example Christology, how to call it, this Frank diagram and its dual shape, a wolf shape is often studied, and the same appears here, and it turns out that this alpha zero, which is the key quantity to tell you the stability of the front, at least for L small, L small means that wavelength is very large, wavelength is the inverse of the wave number, then here it is, curvature is positive, here it is, curvature is negative, which means that if the planar wave is propagating in the direction where the Frank diagram has positive curvature, then it is stable, at least for the long wavelength or small L, so this is the first result we obtained, and so as far as the long wavelength perturbation are concerned, it is completely, the stability is completely determined by alpha zero, whose sign is the same as the curvature here, so this was the first stability result on Alenkan model, this is a linear analysis but anyway, and moreover the space was L2, so we needed, in the later paper, we needed to convert the result into L infinity space rather than L2, but it is not too difficult, okay, and what is not clear yet is medium wavelength stability, in which L is large, so we were able to expand the eigenvalue in terms of L, for L very small, but when L is large since it is a non-local operator, the stability is not completely clear, but at least we were able to construct some example in which the planar wave is unstable in every direction, so for the long wavelength, it is stable in this direction, but because of the medium wavelength spectrum, it is unstable, for a very, very special left, but in general, for general, we are not yet able to estimate the spectrum, so it is still an open question, but at least for some example we could do it, and that was the linear stability, and then we started to study non-linear stability, however, because of COVID, our progress has been delayed and delayed, and finally we were able to obtain something in 2022, and also in 2021, which I will mention after this, so we wanted to study the stability of a planar wave in an infinite strip of finite waves in direction theta, and non-linear stability is the following, if a planar wave is linearly stable or unstable in the direction theta, which means that, anyway, linearly stable in this domain, I mean, so not in the whole R2, but in this domain, this can be done by just, we also impose periodic boundary conditions, so the spectrum, linear spectrum is a subset of the entire plane, you know, then the theorem 3 is in some sense standard, if you have, if this equation, if L generates an analytic semi-group in some good space, then this comes from a standard analysis, but before we were working on L2, which is not a good space, so we had to rewrite all the results in L infinity, but the spectrum remained the same between L infinity and L2, so this is, but to do that, we use the, we use the fundamental solution, fundamental solution decays with the order of the distance to the power minus 4 or something like that, not as fast as the Laplacian heat channel, but still it was okay, and then non-linear stability, as the second result, is that whatever the direction, if the width of the strip is very narrow, one can also prove that it is stable, regardless of the direction, so this is also not so surprising, many unstable mode is completely eliminated if the width is narrow, and the result is similar to the reaction diffusion equation, and finally we were also studying bifurcation problem to prove the existence of, prove the existence of stable shortest front, and this one is actually not yet published, but anyway, let me just go on to the spreading front, which is mainly numerical simulations on the formal analysis, and in this work, young person Koya Sakakiba also is involved, and what is it? So as I said, this is the speed of propagation for the planar wave, this is the diffusion coefficient actually for the planar wave, it is reduced to one-dimensional reaction diffusion equation for the planar wave, and then we draw Frank diagram, and then between these two spots, only in this direction is a stable direction, so for the moment we don't worry about the medium wavelength stability, but here at least for the long wavelength stability, this is the only possible stable area, and if we consider, okay, this is the wolf shape, which is the dual shape of the Frank diagram, and if it is, okay, if, how do I say, if you have a reaction diffusion equation whose planar wave has this speed in each direction, then if you start from a compactly supported initial data, the front will eventually converge to a wolf shape, this one can sort of expect, and to prove it is not completely trivial, and we have only numerical result, but let me show you one, oops, oh, the movie does not, for example, oops, so the movie, okay, so maybe it's this, it's not incompatibility between Windows movie and Mac, so what happens, what I wanted to say is that if you start from this kind of flat initial data and move in this direction, it will develop, it will develop, okay, unfortunately, okay, media was not somehow, oh, I see, I see, we have to, I have to also copy the video, okay, anyway, so you have, as I showed, this kind of planar front will develop this kind of zigzag pattern, and this kind of front will develop, if it is not 45 degrees, it will be non-symmetric, something like that, and this one moves, so it's a periodic solution, and initially we were not sure what these angles of this sawtooth signify, then we found that, at least numerically, our original conjecture was right in the sense that if this direction is unstable, this unstable direction, they try to deform the front so that it will, oops, it will come to the closest stable direction here, and this angle and this angle is exactly this, and here also the closest stable direction is here and here, and it almost matches, so our conjecture was correct, at least numerically, and to prove it is still not possible yet, but at least the zigzag was enigma for most people, even those people who observed this earlier than we did, no one really knew what this zigzag means, but we have a good interpretation, at least as a conjecture, that the front will choose the closest to stable directions, and our movie, which I couldn't show, really confirms this, and then this is the same as this, there is a movie, but I don't think it will work, eventually if you start from, localize the initial data which is circled, eventually which it will converge to a wolf shape, so it's similar to what we will see in crystallology, crystallography, yes, thank you, and we also did some Fitsu Nagumo model, we are still working theoretically, but Fitsu Nagumo model is in some sense a simpler version of Hodgkin-Huxley model, and many people, and many people using bi-domain model as for numerical simulation, often replace Hodgkin-Huxley or bi-domain Fitsu Nagumo by a standard diffusion model, and we wanted to see if how different they are, and it turned out that the pulse wave, there is a pulse wave for the Fitsu Nagumo model like this, and in the parameter regions in which the front is stable in the case of diffusion, in the bi-domain model in some cases it can collapse, so as time passes it will develop a sawtooth but very thin and eventually collapses, and we want to understand more theoretically how the pulse wave collapses, this is the shape of the wave seen from above, here appears singularity in some sense, but the type of the singularity here is not known, it's completely so far an enigma, but numerically we confirm that things are qualitatively quite different from the diffusion model, okay this is another numerical simulation, in some cases it stays like this zigzag shape and it propagates at a constant speed, but in other cases it's just collapses, so it's still a lot of open problems, and this is the summary, and okay I skip the summary because of the time, so I just wanted to say happy birthday. Thank you. So you said when you go to a machine if you are steeper in some cases? Oh, no, no, no, no, you mean the pattern? The pattern, yeah. No, no, no, no, no, no, no, no, no, no, no, no. Here, no, no. Ah, you mean the Fitsu Nagumo? Yes. Fitsunagumo is still enigmatic, but I don't think it will, it will similar to this case, you know, it will try to choose the closest, closest stable direction so the angle is all, all the time fixed. But what, what this angle signified was not clear before, but now it, using the Frank diagram, it, it is more or less getting somewhat clear, just a second. Okay, yes, this, the angle is eventually fixed, but if you change the theta, the direction propagation, the shape changes, but now we more or less understand what it is, more or less, yes. For some very special choice of wave, for which it was possible to compute the medium length, wavelength spectrum, which means for large L wave number, relatively large L, it is not easy because it's a non-local, you know, involving integral, you know. But for some very special, if we were able to compute it, they turned out that although in the, for the long wavelength small L, in some direction it is always stable because the Frank diagram has a curvature positive direction, but because of the medium wavelength spectrum, thus it cannot be stable. So it is unstable in every direction. There are such very, very strange cases. So in that case, how, we haven't done, because the non-linearity f we chose was very special and numerical, it's very, very difficult to compute how the front collapses. Yes. Any other questions?