 So first of all I would like to thank the organizers for giving me the opportunity to talk in this nice conference. Today I will talk about a very recent work done jointly with Franck Merle. So let me first start with the equation. We have the semilinear wave equation in two space dimensions. So here the nonlinearity is u to the p where p is subconformal. And the subconform, the conform exponent is less than the sub exponent. So here p is less than 5 because most of the time I will be in two dimensions. Of course I will talk first about the one dimensional case which is very well understood but then I will focus on the 2D case. As far as blow up is concerned we had many many works before. So many people in this conference room have already worked on this equation. So I said some of them and I apologize for those who I might have forgotten. Now I will not consider global end time solutions but only solutions which exist up to some time here t bar. But then from the finite speed of propagation the solution may stop existing at this minimal existing time and then it continues to exist somewhere else up to some surface which is a graph x gives capital T of x which is a kind of local blow up time. And from the finite speed of propagation this graph is already one lip sheets. Why? Because my domain of definition is simply a union of backward light cones. So a union of backward light cones is either the whole half space t positive or the sub graph of one lip sheets function. All this comes directly from the Kushi theory which we do in H1 times L2 by the way because we are sub left sub critical anyway. Okay so you see as I said if you take any point on this surface the light cone with vertex x t of x is included in the domain of definition. Then we will give a geometric definition. If you can change the slope of your light cone which is in blue here and make a green cone with a slope which is strictly less than one then today in the domain of definition we will say that we have a non-characteristic point. Their set will be called script R. All the other points will be called characteristic points and their set will be called S. S like singular because in fact the solution blows up everywhere you see. Okay so you are either non-characteristic or characteristic points will be the nasty ones where you will have a more complicated behavior. Now I start with the case N equals 1. First thing any blow up solution has a non-characteristic point. Why? Because you always take the minimum, the minimal time, the place where you have the minimal time and then you will see that of course you can have even a flat cone with slope 0 if you want. Of course I take initial data for this in H1 lock uniform times L2 lock uniform meaning that the L2 norm on every wall with radius 1 is uniformly bounded. So you have always your solution which is defined in a strip. At least okay your domain of definition contains a strip so you have a minimal time and there you have a non-characteristic point. But then this is for free if I can say. What is difficult to have is to have a solution with a characteristic point and this was an open problem until some years ago when we solved that with Franck-Mehl with this small example. Take initial data which is odd and then with large plateaus here and here so from the finite speed of propagation it will stay constant and dependent of X in a smaller plateau, in a smaller interval of space. So here you can solve it like the ODE. You know that it will blow up. You can see that it blows up. The ODE is explicit. So this is not a global solution. It's a non-global solution which will blow up. And then because it's initially odd it will stay odd all the time. So U of 0 will be always equal to 0 and we will have a characteristic point at the origin. And by the way having characteristic points is connected to sign changing. In fact, as we will see this in a moment. An important property in one space dimension, this solution is stable meaning that you may perturb it and breaking the symmetry and still have a characteristic point in the middle close to 0 not necessarily at 0 but close to 0. So this is a robust property having a characteristic point with one change of sign is stable with respect to initial data. So this is the picture in one space dimension with the existence of non-characteristic points which is free and then having a characteristic point this is something more difficult. Now I would like to move to the asymptotic behavior near blow up points. So we will have two kind of behaviors near non-characteristic points and near characteristic points. To see in a nice way the behavior it's useful to introduce similarity variables. So here we have a new function, new space variable, new time variable. Let me start with time. Time is simply a slow time. s equals negative log of capital T of x0 minus t going to infinity as t goes to capital T minus x0. y is a zoom near the singularity and this zoom is in the wave style meaning that we have x minus x0 to the power 1 and capital T minus t to the power 1 unlike the heat equation where as you may know we have a square root here because space and time do not play the same role. And then for the new function we simply divide u by the rate of the ODE. So the question we are asking can we compare the growth of the solution of the PDE to the growth of the ODE which is explicit which is known. Okay? Then I'm not writing the equation satisfied by w but of course doing some algebra we can have it and we are working mostly in the w variable but here just to illustrate the blow up behavior. I'm just saying that if you write this PDE you will find a class of trivial solutions. These trivial solutions in the u variable are exactly self-similar blow up solutions. They give you exactly self-similar blow up solutions. When d is equal to 0 you have the ODE solution. When d is non-zero it's just moving the ODE solution with the Lorentz transform. Okay? Now if x0 is non-characteristic then wx0 has a profile and that profile is this guy because this family is the only family the only possibility for having stationary solution in w. We have Lyapunov functional in w. Okay? Decreasing energy so this helps us to have a limit a limit to the set of stationary solutions and this set is completely characterized you have 0 or this family with plus and minus. Now if you have a characteristic point as I have already told you this is the nasty case so here and I'm making connection with the talk of Patrick we have multisolutants. Okay? So look here you have this kind this family that the parameter is moving and we will see that two neighbors have opposite signs. Okay? And this parameter which is moving is explicitly given by this formula so just to give you the result we will see some parameters going to 1 others going to negative 1 and if you have an odd number of solitons the middle solitons soliton will not move. Okay? All these modalities in particular the multisolutants do occur. We were able with Raphael code to construct a solution for any k we have a solution which behaves like that but of course it's better to have a picture and let me here make a picture for you when k equals 4. If you have four solitons it's nice to further change the variables because let me go back a little bit here if x and t is in your light code with vertex x0 t of x0 then y is in the unit bulb. Okay? Now I will make further change of variable y equals hyperbolic tangent of xi so xi is in R y is still in the unit bulb okay? negative 1 1 okay? and I multiply w by this rate and here miracle you see the kdv solitons so your w bar is decomposing into a sum of alternate kdv solitons okay? and you see that two neighbors have opposite signs and the two of the left will go to the left and the two on the right okay? if you have an odd number of solitons then the middle soliton would stay in the middle okay? what else to be said here? of course the middle of the soliton is given completely sharply we have an explicit formula for that it moves like log of s which makes log of log of capital T minus T in fact this is the motion of the center of the soliton and in fact as Patrick said before there are many many connections between the two talks the center of the solitons obey some kind of orthogonality condition and that orthogonality condition gives us the ODE satisfied by the center of the solitons and this ODE as you see here is maybe familiar to some of you so this is not the TODA system because the TODA system comes with double derivative here second derivative for zeta i here we have only one somehow it's maybe easier to handle but not that much okay? and of course we have this kind of small terms which come here simply because our equation is not linear equation so if you have let me go back here this guy is a solution but some of the some of solitons is not a solution and because it's not a solution we have some defect and that defect is proportional to the small terms and this gives us the low then you see that we have an explicit family which is given here just to check that we have this family is completely easy now we have already talked about the behavior of the solution if you are at a non-characteristic point you have one soliton with a parameter that does not move if you are at a characteristic point then you have a sum of two or more solitons now what about the regularity of the blow-up curve they are connected in fact you cannot have one before the other asymptotic behavior and the regularity of the blow-up set has to be somehow advanced side by side but here I'm just giving you the results so R the set of non-characteristic points is open and T is C1 there okay and then if you know that W approaches capital d of x0 then this d of x0 is the derivative of T you see how asymptotic behavior and the regularity are linked now S the set of characteristic points is finite on compact sets and near each A in S your blow-up curve here in dashed line is tangent to the backward light cone so it has a corner with 90 degrees okay and it's not differentiable so we have half a derivative from the left which is 1 and from the right which is negative 1 and as I told you every characteristic point is isolated then we can have a further expansion of the difference between T and the equation of the backward light cone it's given here so it has a log correction and in the log correction you see the number of solitons okay once again the regularity is linked to the asymptotic behavior you can see it okay so then usually it's not symmetric this is really something strange because usually when we find we will find a profile which is somehow symmetric but here the first part is symmetric okay but the second part is by no means symmetric, why? because here in this solution this zeta i they are invariant with translation zeta i so you can change zeta i make their barycenter different the barycenter is conserved the center of mass is conserved here so if you take the center of mass which is zero then this guy is symmetric if you take the center of mass here which is not zero then this correction is not symmetric with respect to x0 okay I think that I said everything okay so I can move to the next slide okay I also see time running we have some generalizations which well I would say easy generalization because of what will come later but at that time this was not that easy well anyway you can add some lower order terms f of u where f of u is less than u to the power q with q less than p and even some terms involving derivatives in time in space x and t as far as their growth does not exceed this so here we cannot put other power the maximum power is one unfortunately we could not fill the gap to what the scaling would give us okay so here everything is true for this kind of equations second case if you take radial equation okay but outside the origin if you are outside the origin this term will be of lower order okay so you can handle it as we did here and everything is like the 1D case for example if you have but of course we have radial symmetry so if you have a characteristic point the whole sphere will be characteristic okay etc then you can maybe make a mixture of both cases radial with perturbation this is possible but still outside the origin you can also take the complex case and even the this is very recent contribution by my former student as I is you can take u in r to the power m and have generalizations at least for non-characteristic case and even with strong perturbations going up to u to the p divided by log of u to some power a and this was done by amsa and saidi okay then what happens when n is more than 2 she is of course the topic of my talk here so it's high time for me to start talking about n more than 2 the first result is completely general no symmetry no hypothesis it's about the blow-up rate so you define w the similarity variable version as I told you we defined the similarity variable version like that by dividing u by the rate of the ODE solution so we are asking whether w will be bounded which means that we will have ODE rate okay so this is true okay near non-characteristic points and we and of course sharp also from the solution of the equation h1 times L2 okay and the finite speed of propagation we have a lower bound at characteristic points we have the same bound from above but with some weights we remove the weights by considering only balls of radius one half and then after that as far as classification is concerned we don't have a lot of results but let me concentrate on characteristic points because in fact characteristic points is my aim in this talk although it doesn't appear in the title but you will see it in this pyramid soon okay so here we have no classification of characteristic points in 1D we said that every characteristic point is isolated here no result okay and also for construction for examples if you ask me can you construct a solution with characteristic point then I tell you well I can take a 1D solution and then make some truncations and in some place where the solution will be rigorously 1D I will see for example solution which is in 2D having a line of characteristic point because in 1D it's just one point okay or if I am making truncations with a radial solution then I will see a characteristic set which is a sphere for example but this is always rigorously 1D behavior and that's it we know we have no other examples so the questions when we started with Frank to do this is can we find new blow up solutions with characteristic points with a non 1D behavior that was the challenge and let me here stop a little bit and talk about a general question the geometry of the set of the singular points so our dream with Frank 2 years ago a year and a half ago was to find a solution where S is cross shaped for example in 2D you take the axis x1 equals 0 x2 equals 0 can we have a solution where locally near the origin the set of characteristic points is equal to the axis that was our aim at first but I will tell you if this is true or not in a minute so please wait a little bit but more generally the geometry of the singular set is for me at least the problem which is largely open and for the case of the semi-linear heat equation which is probably the easiest case of PDE having blow up there are the question is largely open as you will see but I'm not sure if many people are aware that this question is completely open look here you can construct you have examples where the blow up set and let me just say something that for the heat equation we have only one blow up time capital T after that the solution does not exist so at capital T we have two kinds of points blow up points where the solution goes to infinity and the regular points where the solution is bounded ok so I will concentrate on the set of blow up solution blow up points in fact I don't have characteristic or non-character only blow up points we can have solutions and the singular set is only one point or a finite number of points you can choose them ok or a sphere or a finite number of concentric spheres and you can choose them and that's it for example in 2D we don't have we don't know if we have a solution which may blow up on an ellipse open problem can we have a cross in 2D open problem we have a segment open problem etc can imagine any other geometry and we don't have an answer and the same question of course somehow exists for the similar wave equation similar wave equation but for the similar wave equation we have to change a little bit the definition of singular set because here all points are blow up points but at later times so here the good notion for the singular is the notion of characteristic points because near non-characteristic points the situation is at least in one space dimension completely easy we have a profile where T is C1 etc so here the corresponding question would be can we have for example a solution where the set of characteristic point is a cross that was our initial thought and this is the theorem well we could not answer the question but at least we had we came out with a new type of a new solution so we have a solution in 2D which blows up on a one-lipschitz graph which is pyramid shaped and let me show you here the picture thank you Franck okay so you can see yes the picture is here okay so Tx is T0 minus maximum of X1 minus X2 maximum X1 minus X2 is like the pyramid of Egypt okay you see so something like that so if you see in the direction of the axis X1 okay if you restrict yourself only to to this direction X1 you take X2 0 you will see the corner with 90 degrees like in one space dimension same thing with X2 okay when X1 is equal to 0 but along the bisectress you will see another angle okay which well you will have a slope of 1 over square root of 2 in fact because this is the pyramid okay we have exactly the geometry of the of the pyramid this is local near 0 and let me make some remarks first of all Ux of T is of course non-radial because we have a pyramid pyramid is not radial okay second thing my solution we construct that as being symmetric with respect to the axis and anti-symmetric with respect to bisectrices which means that you will remain always rigorously 0 on bisectrices okay and the surprising result is that only 0 is the characteristic point all other points are non-characteristic and let me tell you about something which is counter-intuitive in some sense because in one space dimension we we have noticed that the change of sign or being 0 is linked to having a characteristic point so here on the bisectrices our solution is always 0 but we have a non-characteristic point so this is something which is completely different from the 1D situation so only the origin is a characteristic point so here in fact this result we proved only in locally okay because we were somehow lazy but we are able to say that all the other points even far in space are non-characteristic this is possible though somehow technical now let me talk about the regularity of the blow-up graph I will simply say okay that the regularity is the same as the regularity of the pyramid okay so the pyramid you see here sorry what happens okay I will no no problem I can do it okay so you see here your pyramid is regular everywhere except at the origin and on bisectrices same for my blow-up graph we are we are C1 everywhere except on the bisectrices okay so if we are outside the bisectrices from symmetries it's enough to consider the case where zero is less than x2 strictly less than x1 here look at your derivatives your derivative is like the derivative of the pyramid negative one but with logarithmic correction like what we had in one space dimension in fact okay and this is almost zero in fact the derivative in the other direction now on the bisectrices outside the origin you have directional derivatives in all directions except in the direction of the bisectrices bisectrics and this is like the pyramid in fact but at the origin you have directional derivatives except once again along the the bisectrices okay and let me insist on something here unlike the 1D we have on the bisectrices the first example of non-characteristic points where T is non-differentiable because in 1D if you are at non-characteristic points then you are differentiable and at characteristic points you have a corner of 90 degrees here we still have a corner at the characteristic point which is the origin but on the on the bisectrices we have non-characteristic points and T which is not differentiable okay okay now what about the behavior so at the origin you remember this change of variable it's always centered at the point where you want to see the behavior okay so when I talk about W0 this means that I'm working in the backward light cone with vertex 0 T of 0 here I will see four solitons two along the X1 direction one going to the right this one and one going to the left they are symmetric okay and two along the X2 direction they are also symmetric and we see the anti-symmetry so this means that if you draw a circle which is the section of your backward light cone with vertex 0 T0 you will see if you go clockwise for example you have positive negative negative okay if you encounter all the axes okay once again we have this kind of sign change in fact like in the 1D situation here the solitons are always the same okay and then the parameter is once again completely explicit and it satisfies the same kind of ODE which is connected to the toda system here because of symmetry so we have only one one center to deal with if we had here D1 D2 D3 D4 we have something involving zeta 1 zeta 2 and zeta 4 okay then well what I would say that D Dbar is negative 1 plus some correction here which is a logarithmic correction because S is negative log of capital T minus T and you see that D goes to negative 1 and negative D goes to 1 which give you the slopes of your pyramid so this is for W sub 0 what happens outside the origin so if X0 is outside the origin as I told you we have non characteristic points and here we have a convergence to a new two stationary solutions okay and we have two cases if we are outside the bisectrices then we will converge like in the 1D situation to this special soliton the same one okay and its slope is given by negative 1 plus this logarithmic correction so we are not rigorously equal to the pyramid but we have a correction to the pyramid which is given like that of course this is given just here but if you change by symmetry you can find the behavior everywhere outside the bisectrices and now if you are on one of the bisectrices then you will find a new blow up a new stationary solution which is not which is not radial okay which is anti-symmetric which is odd well which is anti-symmetric with respect to the bisectrices okay so yes in higher space dimension the kappa d is not the only possibility for the set of stationary solutions we have a new set we are not able to characterize everything but we found a new stationary solution okay then the proof this is maybe the most hard part of the proof I should have started with the proof because at first you are maybe it's easier to follow but at the end it's it's more difficult but of course the results are more important to state anyway we have two major steps first step the construction in the light cone you have finite speed of propagation so it's completely meaningful to start with initial data in the section of a backward light cone and to follow it only in the backward light cone and after that comes step two where we will find the behavior okay outside the origin so here we make a construction with prescribed behavior okay in the light cone and then we find the regularity between the two steps there is a small step which is not very long but well we need to use the finite speed of propagation and extend our initial data outside the light cone in order to have something which is defined hopefully everywhere in space so in fact our initial data is defined in an initial it's compactly supported and defined in a square and the square is suitable to the geometry of the pyramid in fact okay and it is really defined in a set supported in a set which is strictly larger than the section of the light cone okay and as I told already for the 1D case the asymptotic behavior and the regularity of blow up set are completely linked and advanced side by side in the proof okay let me now go to the first step as I told you we are working only in the light cone which means that when I introduce W I will work only in the unit ball which is in the unit ball okay so we are able so this is the goal find a solution in W which obeys this behavior okay so we see our solitons as I stated before in the in the theorem okay so two solitons which are symmetric in the x1 behavior two solitons symmetric in the x2 behavior and we have anti-symmetry matrices okay and I give you all other characterization of the parameter d of s is completely explicit so this is my goal okay the framework is the construction of a solution for PDE with prescribed behavior okay the method what you linearize the equation around the intended behavior and we find three regions in the spectrum negative spectrum and this is controlled thanks to a linearized version of the Lyapunov functional because in W we have a Lyapunov functional for the whole system the nonlinear so even for the linearized equation you have a Lyapunov functional and this helps you to control all the negative part of the spectrum we do not compute any eigenvalue explicitly in the negative range then we have lambda equals zero which is controlled thanks to modulation in the parameter D of the solitons remember that this D parameter is in fact the parameter of the Lorentz transform which operates on the ODE solution in the UXT setting so we change it a little bit to kill the projection on lambda equals zero lambda equals one and not negative one sorry for the mistake okay lambda equals one it's controlled and killed in fact thanks to modulation with respect to this parameter new in this family so this family when new is zero you have your solitons take your soliton kappa of D and Y then go back to UXT and then again to W but with a different time you find this family okay and of course this construction is inspired by the construction we did with Raphael Kot for multi solitons in one space dimension okay then let me give you a history of the construction with prescribed behavior well again sorry I'm sure I forget some some people and probably some recent work but it was possible to use these ideas NLS KDV waterways Gensburg Landau Kader Seagal Wave Heat Southern Garoo maps with many many people which I cite here okay now I move to step 2 the behavior of WX0 okay and the regularity of T of X0 when X0 is outside the origin let me suggest the following you take X0 okay and then if you want to know the behavior of WX0 which is completely equivalent to knowing the behavior of UXT in the backward light cone with vertex X0 T of X0 in this case you just remark when X0 is small the sections of the cone with vertex X0 T of X0 and the cone with vertex X0 T of X0 are almost the same when you are far from the singularity okay so far from the singularity you can start from WX0 go back to U and from U go to W0 and you will find that WX0 and W0 are linked with this nice algebraic identity okay so this is completely explicit since W0 has four solitons WX0 far from the singularity will also have four solitons but with deformation because you see we are multiplying W0 with this factor and the four solitons for WX0 in fact involve this kappa tilde kappa star this family okay so we have these four generalized solitons with deformation when I say deformation because we have this factor which means that here we have a nu which is not zero okay and then what we will do with some dynamics we will follow these four solitons and we will see that two will disappear because if I go here when nu is positive because nu is mu e to the power s is mu is positive s going to infinity okay all this will go to zero so two will disappear and then I will have only two solitons and in some cases only one soliton and we need strong analysis here and the strong analysis is the following if you start with four solitons which are decoupled which do not see each other you can follow them for a long time okay and you see if someone would go to zero or would go to infinity or stay close to kappa of d okay then if we are not on the bisectrices say in this range there we will be left with only one soliton okay at some point s star such that s star is like negative log of x1 so if x1 is close to zero we will have to wait a long time until making the three solitons disappear and having only one if you are far from x1 then well this would happen rather quickly okay and then we have a trapping result okay which we first proved in one space dimension and then in any dimension for subconformal exponent if you are close to this family then you would converge to a member of that family and of course when you converge the parameter here is the gradient of t and it is close to this this parameter so we have some analysis to make our solution close to one soliton and once it is close to one soliton it's trapped it would converge to one soliton and then since the gradient is like this parameter it gives us directly the gradient of of t at this point here I forgot something important this is done only at non-characteristic points because our trapping result works only at non-characteristic points so here I am outside the bisectrices okay if I am non-characteristic then I know the gradient and I have convergence but then at some point later in the proof I will have to show that all points outside the bisectrices are indeed non-characteristic and this is difficult for the moment I don't have it okay and this will be an important step note once again the link between the asymptotic behavior of Wx0 and the regularity you see it here we converge to something where we see the gradient okay now case 2 if we are on the bisectrices this situation the solution is anti-symmetric so you cannot have only one soliton you always all that you can do is have only two solitons so you will have two solitons which will decrease to 0 and you are left with two solitons like that one along x1 and the other along x2 and the parameter is almost the same in fact okay and here because we have a non-characteristic point okay and this comes from the behavior of the neighbors you are on a bisectrice but you have neighbors which are not on the bisectrices and there you know the gradient so from that you can have the gradient or at least directional derivatives outside the direction of the bisectrice and see that you are non-characteristic then because we have a Lyapunov functional in similarity variables we will convert two solutions which will be close to this guy so we have a new kind of stationary solution which are neither radial nor when 1D okay a new kind of stationary solution now we are left with only one thing to show that outside the bisectrices all the points are non-characteristic okay so in fact this is what I wanted to call the umbrella technique okay maybe it's suitable for today because it's going to rain okay but let me tell you what I call the umbrella look here in the domain of definition in 1D but in multi-D you just take the cone okay any blue cone with slope 1 okay light cone is completely included in the domain of definition with vertex x t of x here but then if you take an umbrella which is green and you start from the bottom your imagine your green umbrella is going up okay and then it touches the your blow-up graph at some point then that point is for sure non-characteristic point okay take your umbrella and touch in every place the first time when your umbrella or your cone non-characteristic with a slope which is strictly less than one touch your graph that point is by definition a non-characterist you see it okay that's not more complicated than that so here this is what I will do sorry I will get back to the same place okay take x outside the bisectrices for example here x2 is strictly less than x1 and we'll show that x is non-characteristic we take gamma which is rather small here when x1 square smaller than the log the log correction because our slopes are all negative one or one with a log correction so x1 square is much faster than one over log of x1 okay and we consider a family of cones with vertex centered in x and height t t is a parameter and the slope is always one minus gamma so this is an umbrella it's coming from the bottom the green umbrella and going up at some point for some value of t it will touch the graph at some point x bar t of x bar immediately x bar is non-characteristic because of this slope which is strictly less than one if x bar is equal to x because maybe the umbrella will touch the graph at its top if this happens then we are done x is non-characteristic if it does not happen if it touches elsewhere we will try to find a contradiction okay so let's see imagine that the touching point is not the center of all these cones okay if x bar is on the bisectrices while this is a little bit complicated I'm sorry I don't want to to mention that at all okay maybe later if you are interested or in the paper okay but now when x bar is on is not on the bisectrices the argument is in fact easy let's see imagine that x bar is also by symmetry here so it has positive coordinates okay so in this place both the cone in x bar and the graph are differentiable the cone is differentiable because we are not at the center of course but the graph is differentiable because we have a non-characteristic point x bar by construction is non-characteristic imagine two surfaces which teach which touch each other okay at some point so there they should share the same tangent plane okay so their gradient has to be the same okay their slopes have to agree okay so the slope of the cone is this guy which is less than this one and the slope of the graph is this guy and you see log of x1 bar is less than x1 square because the parameter comes from the x I I'm starting and the slope comes from the touching point x bar so all this means that x x1 bar is negligible with respect to x1 okay this is on one hand on the other hand okay I have a series of three inequalities which are easy to understand hopefully t of x is less is more than x bar because t of x is because the cone is under the the graph in fact okay t of x is more than the the top of the cone and the top of the cone is more than t of x bar this is one thing second thing t is one Lipschitz so tx bar is more than t of 0 means x bar but since we are here it's more than x x1 bar square root of 2 then because wx is bounded thanks to our work in 2003-2005 we have the following upper bound which is not sharp t of x is in fact always always less than t of 0 minus x1 x1 over 2 and now let's put all these three inequalities all together if you put them this is more than this more than this more than this more than this so you find that x1 okay is less than 2x1 bar but from the previous slide x1 bar was negligible with respect to x1 a contradiction okay now all points outside the bisectrices are non-characteristic and we have the gradient which is the slope of the pyramid negative 1 in this region with a logarithmic correction okay then we when we integrate this estimate between 0 and x we will find that tx minus t0 is like negative x1 with a logarithmic correction and of course by symmetry we find all the pyramid shape like that okay thank you for your attention so if you take the bisectrics and you Lorentz boost so it's horizontal in what way does it differ from the 1d blow up shape or behavior? sorry can you say it again it takes bisectrics yes and Lorentz boost it so it's horizontal ah yes okay yes then how does that deviate from the 1d behavior which blows up also along the line it's a different profile yes it's a different stationary solution because on the bisectrics we don't have kappa of d we don't have the usual solitons we have convergence to a new stationary solution so even if you change with Lorentz transform you will find yes this is good yes we can make that like you are saying we'll find a new type of non characteristic point the new profile is not the soliton in 1d very good remark yes a new profile and how do I see the deviate I mean where does it look like it makes it look different let me show you it's here it's here you see and Frank can do a picture of course so if you if you see it's like that then it would be with a difference flow one minus but it will be wider wider than the characteristic I mean in 1d you either have slope which is rigorously one or we have something which is differentiable and here we don't have something which is differentiable but two slopes which are far from one