 Thank you very much. So it is my great pleasure and honor to have a chance to speak in this very, very exciting workshop or a special program on summer school. And I would like to thank the organizers, Professor Frank Mell, Yvonne Maltel and other organizers for the invitation. And probably I am one of the very few speakers who will talk on parabolic equation in this workshop. However, the result I obtained, this is a joint work with Frank Mell, has a much reminiscent, it's quite reminiscent to what Frank Mell and other people, Kenick and other people are working on nonlinear wave equation. And the results have much similarity despite the difference of the type of equations. And the structure, critical structure, the two equations share the critical structure despite the difference, a lot of difference in the nature, other nature. But this critical structure is so strong that very similar results are obtained for this type of equation. So this is why I wanted to report this result in this workshop, this summer school, I mean. So the goal is to establish a multi-scale resolution of radial solutions. I am just talking about the radially symmetric solutions of the critical heat equation into rescaled ground states, which we may call solitons. In the same spirit of the paper, I'm sorry I cannot pronounce this name well, although I have met him before, but Kenick and Mell of 2013. Their further development in this theory, they worked also on non-radial case, but my talk today is very, very similar in some sense in its spirit to their work. So there is a striking similarity between the two results. So as I said, this is joint work with Frank Mell, and I can call it a parabolic analog of their result, DKW 2013. Short line of the talk is the following. I will make an introduction first just to give a brief review of what is known about nonlinear heat equations rather than wave equations or Schrodinger equations. Although there are many people in the audience who are familiar with these kind of things, but just in case, let me make a quick review, and then I will talk about the main results, and then I will briefly explain the idea of the proof. But the idea of the proof, I will use the blackboard. So let me start with this nonlinear heat equation. This is a power nonlinearity. Ute equals Laplace and U plus U to the power p equals 0, and the solution you can change sign. And I consider a Cauchy program on Rm, and initial data U0 belongs to the energy space. Anyway, the first assumption is that p is the sub-left critical exponent, p equals pS, which is this quantity, and the dimension n is larger than or equal to 3, and I only concentrate on radially symmetric solutions. Still, the analysis of nonlinear solutions is quite difficult, but the result I present here would give some hint for further properties of possibly nonlinear solutions. And then I will assume that the initial data is h1 dot of Rn, so the gradient of U is in L2. But I don't need necessarily, sometimes I may have to make some technical assumption such as U0 being in L2, but in most cases I would like to avoid it. And then, okay, this is, as usual, the norm of h1 dot. And then it is known by the usual parable estimates that if you are given an initial data in h1 dot, then immediately the solution becomes classical in some sense. It belongs to, it's bounded, and it belongs to this space. And also it is known that it is known that if t is the maximum time of existence, if t is finite, then it means that the elevator norm goes to infinity and so does h2 dot norm. But not necessarily h1 dot norm. There are cases where the solution blows up, but h1 dot norm may stay bounded. So hereafter we assume that U0 is bounded, and there is no loss of generality because if the solution will always become bounded immediately. Then I call this finite time blow up. And it is well known, okay, for people here in the audience, most of the people are more experts than myself about this kind of scaling variance. But just in case, let me just make a brief overview of what, of the properties of this critical nonlinear heat equation. So if U is a solution of this equation, then U lambda of xt, which is defined by this function, it's just a rescaling of U, is again a solution. This is just a simple computation, and it doesn't depend on the choice of p. For any p, this is true. And if v is a solution, depending on only on x, then v lambda, which is defined by this, also a solution of this stationary problem. And for the case where p is equal to the sub-left critical exponent, if g lambda, I denote by g lambda this rescaling of g, and then, okay, this 2 over p minus 1 is equal to n minus 2 over 2. And then the h1 norm and the l2 star norm is preserved where 2 star is defined by this. And this is a typical feature of critical nonlinearity. And this identity, in some sense, defines the nature of this problem. And the critical case is distinctly different from the sub-critical and the supercritical problem, as many people in this audience well know very well. Then the energy of this problem, this equation is defined by this for the critical case. And because of this invariance, the energy remains invariant under this transformation. And it is also well known that the time derivative of the energy is this, and therefore it is non-negative. So it is a Lyapunov function. So in particular, if the energy is known to be bounded from below, you can show that the solution approaches a set of equilibrium, set of steady states. Now, what is known about the stationary solutions for this problem? Again, most of the people in the audience are real experts of this problem, but just in case, let me make a brief quick review. So for the critical case, all the solution of this problem is written, all the radial solution of this problem is written in this form and where q is given by this function explicitly. So this equation, this state, elliptic equation has a solution on the entire space for p critical or supercritical, but for the critical case, it is explicit. And there is also a singular stationary solution, which is given by this, for p larger than certain subcritical value, there is such a solution. And this plays a very important role for the analysis of blow up, for example, but not only for blow up, for the analysis of the soliton resolution, this singular stationary solution plays an important role. And this singular stationary solution decays with this rate, precisely, it's just a power of r. While q of x, since we have r squared here, it decays faster. Again, this is a typical feature of the critical case. The stationary solution for this problem for supercritical p decays with the same order as the singular stationary solution, it is exactly this order. Only for p equals p s, it decays faster. And also, another important thing is that this is what I said. Another important thing is that the intersection number, okay z here denotes the zero number. So it gives the intersection between the two graphs. So for any lambda, the intersection of the graph of q lambda and phi star, which is the singular stationary solution is always two. To check it, all you need to do is to check it for lambda equals one, because all others are the same, because phi star is invariant under this transformation, this rescaling. So if you check it for lambda equals one, it is clear. But this property can also be understood by converting this problem into ODE and by changing the variable, which I may mention later why this is true. It is very, very clear from the phase plane analysis. And also, it is very, very important that for any two values of lambda and the mu, for different values of lambda and the mu, the two stationary solutions intersect only once as a function of r. So this part and this part is symmetric. So let's just consider r positive. And then the intersection number equals one. This is again very important in our analysis. And then I have also, I have to make a brief review of what is known about the blow up of solutions for this nonlinear heat equation. And it is already explained, for example, in the lecture of Professor Kenick, but also other people. Type one blow up is such that the solution blows up at time capital T, where T is a finite number, but this is bounded. This quantity remains bounded. And type two blow up is such that this quantity is unbounded. And it turns out that if it is bounded, then you can show that the solution, this quantity blows up with the same rate as this one. So it converges more or less to some constant. And the sharp estimate of this for the subcritical case, for example, was proved by Neyland-Zag, for example. And but there are many other results even for the supercritical case. But type, okay, this is the same blow up rate as the ODE. U dot equals U to the power P. And also it is the same blow up rate for the self-similar solution, self-similar blow up solution, if such a thing exists. But for the type two blow up, to determine the blow up rate is much harder. It is rather complicated. And it requires much involved analysis in general. And for the critical case, type one implies that H1 dot norm, so the L2 norm of the gradient of U goes to infinity. This you can show. And type two blow up almost implies that this remains bounded. Almost. We could not exclude some pathological cases. There might be some pathological counter example to this, which we could not completely exclude. But in most cases, type two implies this one. So in some sense, one can call this type one blow up, fat blow up profile. It has a fat blow up profile because it goes to infinity. While this profile, local profile, the singularity is slim because this is finite. So this is another way to characterize type one and type two blow up. And about the blow up rate, whether type one exists or not, type two exists or not, type one always exists. For example, ODE solution is of course type one. But the question is whether type two blow up can exist or not for the non-linear heat equation. And it was, there is an early study by Gigant Kohn and later Gigant Matsui Sasayama for the entire space, Gigant Kohn for the star-shaped domain and also for the positive solution on the entire space. And later they studied the sign-changing solution for the entire space in 2004. They proved that in the subcritical case, only type one blow up can occur. And this result doesn't depend on the radial symmetry assumption. It is true for any solution without symmetry. And in the supercritical case, much less is known, but it is somewhat understood that situation changes. There are some other critical exponents above the subcritical exponent. There is one called Joseph Rundgren exponent, this, which is finite only for n larger than or equal to 11. But for at least radial solutions, in this lower range of supercritical p, only type one blow up occurs. And there are more recent studies by some other people also on non-rider cases. So this was proved by Frank and me in 2004 for the rider case. While in the upper supercritical range, where p is bigger than Joseph Rundgren exponent, type two blow up can really occur. And this was first approved by Herod and Velazquez in their unpublished paper, long paper, by using much asymptotic expansion. And Mizouuchi simplified the proof, but basically she followed the argument, but somewhat simplified it. And there are also other methods different from much asymptotic expansion, but these are the early work on the existence of type two blow up in this range. Now for the critical case, critical case remained a little subtler. If u is positive, it is not difficult to show that only type one blow up can occur. And for radial case, it is true. But even for the non-rider case, one can show that for positive solutions, critical non-linearity is not a big problem for positive solutions. It is just an easy extension of the subcritical case in some sense, if you assume that u is positive. Difficulty appears only when u changes sign, then the problem becomes much, much more difficult. So for positive solutions, only type one blow up occurs. But type two blow up can occur for sign-changing solutions. This was proved rigorously by Schveier in 2012 for n equals four. And I heard that, okay, four, in Herero Velázquez's result, the reason it was never published is that the paper was more than 100 pages long. And the journal not refused, but asked them to write a book on that, and they refused it. And then very recently, I heard also that they had also some formal expansion, a sympathetic expansion for the critical case already 20 years ago, but it was 200 pages long. And again, they didn't even care even to circulate the preprint. But Schveier's result is rigorous. It is probably 70 pages long. However, it relies heavily on earlier work of Rafael or Manuel. So if you include all those, it may be 150 pages long or something. Anyway, the critical case is very, very difficult. But he proved it, but for dimension four. However, very recent result, which can be very important, appeared on archive. If the dimension is larger than or equal to seven, and if the initial data is very close to Q, Q is the ground state in each one sense, then there is no type two blow up from such initial data. And this is of extreme importance actually for our work, for my work with Frank, which I may mention briefly later. So this is a really important result. And the assumption is such that initial data is very close to Q. And whether this is true, no type two for any initial data even far from Q or not is not completely known. However, combining our analysis, it may be true for more general initial data as at least for radial case, for dimension larger than or equal to seven. So in that case, if that is the case, then the type two blow up for sign changing solutions for the critical heat equation can occur only for dimensions four, five or six. How about three? Also three, okay. Okay. And then I have to also make a brief review of what people studied for blow up problem using the rescale equation. So it is actually standard, similar rescaling. And then this rescale solution in this new variable satisfies this rescale equation. And it was used heavily in the paper of Giga and Com in 1985, 87, 89. And they managed to using this analysis, they managed to show that for the subcritical case, the blow up rate is always type one. And what they showed is that the solution, okay, type one, the blow up being type one is equivalent to saying that W remains bounded as s tends to infinity. By the way, the blow up time, which is small t equals capital T is converted to s equals infinity. So you have infinitely long time to observe the blow up phenomena with this rescaling. And therefore, for example, dynamical systems argument can also be used because we have infinitely long time with this change of variable. And type two is by definition W is unbounded because we are multiplying this. So it is exactly the definition of type one and type two blow up. So the question is, for example, what Giga and Com showed is that for the subcritical problem W is always bounded. And therefore, and this has a weighted energy. And therefore, it converts to some steady state. And steady states are either just zero or some constant, which is the zero of this function, which I will call kappa. So Giga and Com showed that solution always approach either zero or plus or minus kappa by using the energy, which is this. This is a weighted energy, where rho is this exponential function. And this analysis turned out to be very, very useful also for the supercritical problem, but also for the critical problem to some extent. And one can show that at least for the radial solution, we have this general point-wise estimate for any solution. Any solution of this satisfies this general point-wise estimate, which I proved with Frank in my paper in 2004. But it was known, at least for positive solutions before. But it is true also for sign-changing solutions. Any solution satisfies this. And even for supercritical nonlinearity or critical nonlinearity. And combining these two, one can show that w, so this is a diapno function. And locally, it is bounded in the y-space locally, away from the origin. So away from the origin, using this diapno function energy, one can show that limit exists, at least for sequences. But for the radial case, one can show that without taking a sequence, the limit really exists. So this limit function, w star of y, which I would call local blow-up profile, characterized the nature of the blow-up very well in some sense. And what is known about the local profile? So the global profile is just a function in the original variable u of xt. And as t tends to capital T, this limit exists because it's a finite time. And away from the blow-up set, the solution is smooth. And the blow-up point doesn't oscillate. One can show. And therefore, outside of the blow-up set, this limit exists. And local profile is what I just mentioned in my previous slide. So this is this function. This is a limit of the solution after this rescaling. And this w star satisfies this equation. And in the subcritical case, w blows up. Okay, w star is either plus or minus kappa, or w star is equal to zero. And Gigant-Corn shows that if it blows up, it is always either plus kappa or minus kappa. And w star being equal to zero implies that there is no blow-up. The solution u is bounded. In the supercritical case, the situation is slightly different because type 2 blow-up can occur. And w star being nonzero and bounded is equivalent to the blow-up being type 1. And we proved in the paper in 2009 that type 2 blow-up, the blow-up is type 2 if and only if the w star is the singular steady state, singular stationary solution. And w star equals zero implies no blow-up. So this part is the same as the subcritical case. But the situation is completely different in the critical case. In the critical case, okay, type w star equals plus minus k, almost implies type 1 blow-up. There might be some pathological cases which we couldn't exclude, but in all cases which we have seen, this implies type 1. The converse is always true. Type 1 blow-up implies w star equals plus minus k, kappa. Then w star equals zero, no longer means no blow-up. W star equals zero in the critical case means either type 2 blow-up or no blow-up. So this is a big difference between both the subcritical and supercritical case. And we are looking at this situation in the critical case. So the main result is the following. Let me, so far, so much for the, okay, I think I have to hurry a little bit, and so much for the normal results. So this is our result. First, about the classification of general behaviors of the solution. So it is a solid-tone resolution for time-global solution. This is theorem one. So assume that solution is global, time-global, and then, okay, we needed some technical assumptions depending on the dimension, but let's not go into the technical details. We wanted to just remove all these technical assumptions, but unfortunately we had to add some technical assumptions. Then this h1.norm remains bounded automatically if it is global. And either of the following alternatives hold. Either this h1.norm goes to zero, so the solution goes to zero in the energy space. Or u converges to more or less approach the sum of such functions. What is this? Q lambda, by the way, is the rescaled ground state. And they are all radially symmetric, but the rescaling factor is so different. Between the neighboring ground state, rescaled ground state, rescaling ratio goes to infinity. Which means that this is one rescaled ground state. Another looks like this. They are all solutions of the steady state of the original problem. And another may look like this, very flat. Because this goes to infinity, the interaction goes to zero. So they are totally independent in some sense. So, okay, they don't travel like a soliton in K-V equation in a real actual space, but they are floating in a parameter space. And moreover, the outermost soliton, this ratio, lambda m is more or less, it shows the position of the soliton in some sense. The ratio between this and square root of T goes to zero, which means that it doesn't expand as fast as the self-similar rate. So it cannot move so fast, even the outermost one. This is also very, very important. And this follows from our lemma, which we call semi-localization lemma, which confines the energy, the energy, initial data energy is like this, then here there is almost no energy. We were able to show that the energy cannot penetrate through this wall, any wall away from origin, by using some combination, some estimate. And using that, we can prove that. So, lambda j expands slower than the self-similar rate. The next one is the following, theorem to soliton resolution for blow-up solutions. And the assumption is that first, it is a blow-up solution. And moreover, we assume that the local profile is not equal to plus minus kappa. This is the only assumption we make for the blow-up solution. And so, it almost means that the blow-up, it is almost equivalent to saying that the blow-up is type 2. Definitely, this means that the blow-up is type 2. Then W star is actually equal to zero. And this is the global profile minus soliton. So, you consist of solitons, and all this Wm goes to zero very, very fast, because this goes to zero and it goes faster than this one. So, lambda j concentrates faster than the self-similar rate. Oops. Like this. And this never happens in the supercritical case. This is not possible in the supercritical case because of the no-needle lemma, which we proved for supercritical problem in 2009. And in the supercritical case, if the needle becomes thinner and thinner, the energy becomes smaller. And then there is no way for the solution to go farther. But in the critical case, even if it is squeezed, the energy remains the same, and therefore this kind of thin needle can appear. And this is what it says. And then about the existence of multiple solitons is the following. It is an ongoing work, and our claim is the following. If n is larger than or equal to 7, there exists a time-global solution with two solitons. More precisely, there exists a solution, u over xt with time-global solution, such that this minus this goes to zero. For some lambda 1 and lambda 2, satisfying this condition and this condition. And we have more or less proved it, but we are making some final check of all the argument, but basically it is done, but I still call it a claim. But this work, okay, this result heavily relies on an argument, which is a modified version of the result of colloid canic mel. Okay, sorry, sorry, Melnafiel. I'm sorry, Melnafiel. Melnafiel, okay, Melnafiel. And so this is a, but still ongoing work. So I wanted to explain the idea on the blackboard, but I seem to have almost used up my time. I'm sorry. But let me just mention that the proof is based on the combination of certain energy method and the intersection number argument, zero number principle for 1D palavatic equation. And let me just say just very, very quickly how one can prove soliton resolution for the time global solution. For example, the argument for the blow up solution is slightly different, but similar. And it is a function of R. And there is a, we prove the first energy semi-localization. This is done by some energy estimate. We just put some wall here and just calculate how much energy will penetrate through this wall. And if this is quite far away, not much will penetrate. It's very, very small, which means that energy which moves with the square root of T, this moving wall, so it can be arbitrarily small. And therefore, one can show that the energy is confined in a region which is small order of square root of T. And this is one thing. And then what we will show is that to use the intersection number argument we looked at the intersection of the solution with a singular steady state. And we do the rescaling. We normalize it so that it comes to the position at R equals 1, for example. And we do the rescaling limit, take the rescaling limit. In order for the rescaling limit to be possible, we use the fact that EU of T is always positive. Therefore, this is finite. And this quantity, after rescaling, remains the same. And therefore, one can show that this is finite. But it doesn't mean that this is bounded. Energy being positive doesn't mean that the solution is bounded. It's not automatic. So we need further steps. So all one can do first is that any solution satisfies this bound. And therefore, locally away from the origin, one can show that the solution converges to a steady state because of this energy is positive because this is bounded. But all one can say is that the limit function converges to something which is a solution of this in Rn minus the origin. So it may have similarities. So the next task is to show that it doesn't have a similarity at the origin. How do we do that? In order to do that, we look at all the solutions of this problem outside of the origin. It turns out that if you look at this quantity and use z equals log R, the equation, if I call it v, equation is converted into this simple form. And this is a Hamiltonian system. And if you look at its phase plane, it looks like this. These two solutions correspond to phi star, a singular steady state and minus phi star because a singular steady state after multiplying this is a constant. And our q, grand state, is the homoclinic orbit. And all other solutions intersect with phi star infinitely many times because it goes around infinitely many times. But one can show that our solution, the limit, cannot intersect with phi star infinitely many times because intersection number doesn't increase. I think my time is over but what I wanted to say is that we make several steps based on this kind of intersection number argument. But one can, of course, if you are a very skillful in energy estimate, you can also prove the boundaries of this by using just energy argument. But then the analysis will be much more involved. The combination of this intersection number argument and the energy estimate make the argument much lighter, although it applies only to radial solutions. But at least this kind of argument gives some insight into what's going on, at least for the radial problems. I think I should stop here and ask this in the summary. But open question is are there multiple solutions in this range for this dimension? This is not clear. How fast does the ratio tends to infinity and how about the non-radial case? Okay, I think I should stop here and thank you very much for your attention. I'd just like to clarify something in the statement of your theorem, especially theorem 2. You mentioned this thing called a local profile W star. And you said that if W star is not equals plus minus kappa, then you've got this sort of complexion. So I'm just wondering is when you say W star not equals this, does that mean that it's not a constant function or are you evaluating it as origin or what? Is W star a constant function? So W equation has three constants of steady state. And if and every type one solution converges to plus minus k, because Giga and Korn already proved that even including the critical case, even without a rather symmetry, the only bounded steady state are these solutions. And if it converges to this, definitely h1 dot norm is infinite. And there is no way that you can compose it into finite sum of solitons, because each soliton has finite energy. And therefore, this is excluded. This should be excluded to have our resolution result. But this is the only assumption. If it is not k, automatically it goes to zero and there is a high concentration like this. All the solitons are concentrated. If they are more than one, everything is concentrated. And it is in Y variable. Even if you look at it in a zoomed cell similar coordinate, it will be highly concentrated, every soliton near the blow-up point. And you can also show that there is no soliton between the Y variable and the X variable. In between, there is no possibility. Everything will be concentrated faster than the cell similar scale. Do you know what are the stable regimes for this equation? What are the stable solutions? Stable, as a matter of fact, type 2 solution lies on a threshold in some sense. And usually, if you part of it a little bit, then it will blow up in a type 1 regime. And type 1 regime is usually stable. So it is on the threshold. But of course, to understand the threshold phenomena is also very important, even though it is not stable under arbitrary perturbations. So what we are looking at is, in some sense, a very delicate phenomenon. Otherwise, the solution will either go to zeros and the energy just disappears, or it will blow up to infinity. And is this rigorous? What you just said? To some extent, it is rigorous. To some extent. And the similar threshold phenomena has been studied by many other people. And in some sense, it is rigorous. I would say that. Maybe not 100%, but under some... Okay, 100%. Okay, he says 100%. Maybe I thought that 99%, but he says 100% rigorous. So that's all related to this question. So if you take all the data that leads to all this influence time bubble decouplination, so do you have any idea about this set being open, closed, or how large is this set? And the same question asked about if you have the final time blow up and all the data that leads to this bubble decouplination, how large is this set? Okay, so as I said, it's quite related as a matter of fact. And in some sense, I already answered, but maybe a little bit vaguely. So this is going to zero, global solution, but the energy goes to zero. And this is type one blow up in some sense. And global solution, and even type two blow up solutions lies on this threshold. And so we are looking at the boundary of between two regimes. And type one blow up is stable, and convergence to zero, energy or lost is also stable. Once you are very close to zero, then small perturbation cannot help you to go out. So both are stable. So it's on the peripheral. Oh, here. No, one comment I would say, you say a lot of analogy between the wave and the wind. But of course, there are differences. First, there is no scattering. So the energy goes to zero. This is by dissipation. But there is a big difference. But we prove that the bubble has to be alternated. And for the wave, it seems that they have to be the same size. Oh, this is a big difference. Okay, I forgot to mention clearly, when I showed the theorem, the bubble, neighboring bubbles, adjacent bubbles are opposite sign in the heat equation. But in the wave equation, they have the same sign. So it's a difference. Okay, so this is a big difference. Yes.