A pyramid-shaped blow-up set for the 2d semilinear wave equation

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A pyramid-shaped blow-up set for the 2d semilinear wave equation
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We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite-time blow-up solution with a pyramid-shaped blow-up surface and an isolated characteristic blow-up point at the origin. Our solution is symmetric with respect to both axes, and anti-symmetric with respect to both bisectrices. The blow-up surface is differentiable outside the bisecrtices. On the bisectrices, it only has directional derivatives. As for the asymptotic behavior in similariy variables, the solution converges to the classical one-dimensional soliton outside the bisectrices, and to a genuinely two dimensional stationary solution, on the bisectrices, outside the origin. At the origin, it behaves like the sum of 4 solitons localized on the two axes, with opposite signs for neighbors. This is the first example of a blow-up solution with a characteristic point in higher dimensions, showing a really two dimensional behavior. Moreover, the points of the bisectrices outside the origin give us the first example of non characteristic points where the blow-up surface is non differentiable.
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it I I'm trying not relying on and in the end it and so 1st 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 his injury recent work done jointly with the Hong Kong man OK so let me 1st start with the equation we have a 17 year wave equation into space dimensions so here the linearity is due to the pig where pay is Sept conform and the separate from the conform exponent is less than the sum of the exponent so here P is less than 5 because most of the time I will be in 2 dimensions of course I would talk 1st about the 1 dimension of case which is very well understood then I would focus on the 2 d case we had as far as blowup is concerned we have many many works before so many people in this room conference room have already work at on the 3rd equation so highfalutin some of them and I apologize for those who I might have for forgotten case no I read
not consider global anytime solutions but only solutions which are not work which exists up to some time the located but then From the finite speed of propagation solution made up of existence but this minimal existing died and then it continues to exist somewhere else up to some surface which is a grass Banks capital killed OK which is a kind of local go up top and from the finite speed of propagation this graph is already 1 Lipschitz 1 because my domain of definition is simply the union of backward like columns so in Union of backward like courses in either vote have space to positive all the sudden drop of 1 Lipschitz function all this comes directly from the Commission's hearings which we do in H 1 times and 2 by the way because we have suffered a subject "quotation mark anyway OK so he said said if you take any point on the surface the backward like cool with verdicts X the effect is included in the domain of the Finnish the will give the geometric definition if you can change the scope of your light gold which is in Brule here and make mean goal with that slope which is strictly less than 1 and still stay in the domain of definition would say that we have non characters to people in their effect would be Collins skipped off or the other points would be called characteristic points and they said would be called as as like light-fingered because in fact the solution blows up everywhere missing OK so you're even uncharacteristic or characteristic and in fact the characteristic .period will be the nasty ones when we you will have a more complicated no I start with the case and it was 1 1st thing any blowhards who has an uncharacteristic .period why because you always takes a minimum the Dominy was the place where you have the minimum of time and then you have the effect of course you can have even the flat called would still there if you want of course I take the initial data for the H 1 looking uniform times and to lock uniform meaning that the effort to log on every wall with the greatest 1 is uniformly bound so you have always a solution which is defined in a strict but pieced OK your domain of definition contains as the strip so he had a minimal time and there you have an uncharacteristic .period but then this is for free the 2nd said which we will which is difficult to have is to have the solution with characteristic .period and this was an open problems and have some years ago when we saw that with Faulkner with this small example think initial data which is odd OK and then with large petals here and here so from the finite speed of propagation into state constant independent of X in this Asmara plateau In this morning interval of of space so he against the lady Ltd you know that it will blow up you can see that it knows the the oldies is explicit so this is not a global solution it uh a young global solution which will blow up OK and then because it's initially ordered it to a state all the time so you of 0 would be always equal to 0 and we will have a characteristic .period under origin and by the way having characteristic .period is connected to sign changing In fact as you see this in a moment an important property in 1 space dimension this the solution is stable meaning that you may perturbed and breaking the cemetery and still had it known character characteristic .period in the middle close to 0 not necessarily have 0 but closed to 0 OK so this is a robust property having inked a characteristic .period with 1 change of signing is stable with respect to initial data OK so this is the picture in once they mentioned where the existence of not characteristic .period switches france and then having to the characteristic .period this is something more difficult now I would like to move 2 the asymptotic behavior New Real .period so we have to kind of behaviors neared an uncharacteristic points in your character if it .period to see in a nice way the behavior where it's useful to introduce similarity variables so here we have a new uh function new space variable time variable military stocked with Prime Time is simply a slow time OK as thick with negative love of capital city of active you mind still goes to infinity as goes to Capitol Kidman's x 0 wine is is also near the singularity and this is all that is in the wake
style meaning that we have access my United's Apollo 1 and the capital 2 to the power was unlikely heat equation where as you may know we have a square root here because space and time Monday played the same role and then for the new function with simply divide you'll buy the rates of the ODA so the question we are asking can be compared the growth after the solution of the PDP to the growth of some of the ODA which is explicit which is not OK the I'm not writing the equation satisfied by W. but of course doing some algebra we can we can have it and we working mostly in the W variable but here just to illustrate the behavior I'm just saying that if you write this the PG would find a glass of the trivial solutions this video solutions in the your valuable exactly such similar blow solutions they give you exactly fits ropes of Wilson's Wendy is equal to 0 you have the odious solution Wednesday is non-zero it's just moving the odious solution with Lawrence transfer OK now if I is not characteristic then that Liu Xia has a profile and that profile is this guy because his family is the only founded the only possibility for having stationary solutions in W we have gotten factional in cooking decreasing energy so this helps us to have a limit amid further said that due to the set of the station's solutions and this effect is completely characterized you have 0 or his family with bluster and my now if you have a characteristic .period as I have already told you this is the nest tickets so here and I'm making a connection with the talk of the league we have meant the sentence OK so look here you have this kind of Miss it the family by the parameter is moving and we will see you in the eye of the 2 neighbors have opposite signs OK and this parameter which is moving is explicitly given by this formula so just 2 2 was to give you the desire who is the son Barrett is going to 1 others combed to negative 1 and if you have an odd number of solid the Minnesota times said it and will not move OK on these modalities In particularly conservatives New pocket we were able with African coast to coast to act the solution for any we have a solution which behaves like that but of course it's better to have a picture and letting him make a picture for you when Katie calls for if you have forced on it and it's nice to further change but the variables because let me go back a little bit here if accent she is in your delight cold we've got expects 0 2 x 0 than what is in the unit OK no I would make further change of edible what it was hyperbolic tangent see folks even part why the Indian board but a negative 1 1 OK and I multiplied W by this rate and he miracle you see the KDP solid so UW box it is composing into a sum of Anthony spent KDD soliton attends OK and if that 2 neighbors have opposite signs and the toll of the left go To the left and the toll on the right with God To the right OK if you have a good number of Summit and then the medicine it and would stand in the middle OK Oh what has to be said here Of course the middle of the soliton and is given completely shocked we have an explicit formula for for them it was like love affairs which makes logo from local capital to Munster In fact this is the motion of the the center of the fleet and in fact has but Pique said before with very many many connections between the 2 stocks at the center of the soliton all some kind of possibility that condition and that of the denied a condition uh gives us the OGE satisfied by the center of the and this only as you see here is that may be familiar to some of this is not the told system because the total system comes with double derivatives the 2nd derivative for it by here we have only 1 somehow maybe it is tool to handle but not that much there and of course we have this the kind of small terms which come here simply because of the equation is not linear equations so if you have let me go back here this morning solutions but some of the some of the local sentence is not a solution and because it's not a solution we have some defects and that effect is proportional to the small the terms and this gives us a little than that you have an explicit family which is given here just to 2 check that we have decided is completely isn't no we have
already talked about the behavior of the solution if you are not among characteristic .period you have 1 solely with the parameters that does not move if you have your at characteristic .period then you have a son of 2 or more funded now what about the regularity of the role curve then connected in fact you cannot have 1 before the ordinary asymptotic behavior and the of the regularity of the vote said has to be somehow advanced side by side but here I'm just giving you the results so far the set of non characteristic .period is open and seems is the 1 there OK and then if you know that W. opposes capital they kept out of the deal exercise within this new act you is the derivative too you see how asymptotic behavior and regularity partly now the set of characteristics .period finite on complex sets and knew each day in Paris your local hearing dashed line is then danger To the backward like color so it has corner With 90 degrees OK and it's not differential so we have have a derivative from tonight From the left which is 1 and from the right which is a negative 1 and as they told you every the characteristic .period is canceled then again have a further extension of the difference between T and the equation of the backward like going if given here so it has a lot corrections and in the local correction the the number of such OK once again the regularity is linked to the asymptotic Nunavut you can see it OK show the Senate usually if not symmetric misses the Indian is something very strange because usually when you find profiles uh we would find a profile which is somehow the symmetric but here the 1st part medical care but the 2nd part is by no means symmetric y because here in the solution this means that there are there invariant with translations in the time so you can change the tire maker very center different the very center of these concerning the center of mass is concerned so it should take the center of mass which is 0 vendors got
symmetric if you take a center of mass here which is not 0 then this correction is not symmetric with respect to Mexico look I think that they said everything I wanted so I can move to the next flight OK I was with from time
running we have some generalizations we what I would say easy generalization because of what will come later but have had time this was not the music well anyway you can and found no terms therefore fuel where 4 view is less than you to the barbecue with Q resident and even some tournaments involving derivatives in time and space accidently as far as the very rules does not exceed this so here we cannot input all the power the maximum priority is 1 unfortunately we could not fill the gap to what the skating would give us OK so here everything is too for this kind of equations 2nd case we should take range equation OK but outside the origin if you are outside the origins this term would be offer lower order OK so we can handle it as we did here and everything is like the 1 in the case for example if you have a bit of course we have Simmons's so if you have a characteristic .period the policies will be characteristic located then again maybe make a mixture of both cases riddled with perturbations this is possible but still outside the origin you can also take the car of the complex case and even the uh this is very useful contribution by my former student as various uh you can take you in or to the power and and have generalizations about his foot no characteristic case and even with strong perturbations going up to yield to the key divided by a global fuel some politics and this was done by arms confined OK then what happens
when there is more than 2 his cost the topic of my talk here so it's high time for me to start talking about more than 2 the 1st baby that is completely general no synergy no longer behind offices it's about to build operate so you define W similarity of the variable version as it told you we define the similarity very well version like that by dividing you'll buy the rate of the LDP solutions so we are asking whether W will be bound which means that it would have alleviated OK so this is true OK Nienow not characteristic .period and and of course shall also found the solution of the episode in the equation H 1 times and OK and the finite speed of propagation we have around a characteristic once we have the same uh bound From about but with some weights so we really move there at the weights by considering only borders afraid this 1 has and then after that fall as far as gasification is concerned we don't have a lot of results but let me concentrate on characteristic once because in fact characteristic forms is my aim in this talk although he doesn't deal in the type of but with the In this government so OK so here we have no classification of characteristic points in 1 dealer said that as a characteristic .period is isolated he no reason OK and also for construction for example if you ask me Can you construct solution with characteristic .period denied they what I can take a 1 solutions and then make some truncations and in some places where the solution would be rigorously 1 G I will see for example solution which is into a debate over having a line of characteristic .period because in 1 case just 1 point OK all or if I'm making truncations with a regional solution than a see the characteristic set which is this received for example but this has always vigorously 1 the behavior and that's it we know we have no other examples so the question is when we started with folk to do this is gaining and find new goal solutions with characteristic .period willing non 1 behavior that was a challenge and let me he has got a little bit and as an adult about about a general question the geometry of the effect of the singular .period so Oh Jean with Frank 2 years of gold yeah and have a goal was to find a solution where as is crossing for example into takes the axis X-1 1 equals 0 ext week was you can we have the solution whether locally nearly regions this is the set of characteristic .period it is equal to the taxes that was held the aim at 1st but I will tell you if this is true or not .period in a minute so pleased with predictable but more generally the geometry of the singular fair is for me at least said the problem which is largely open and for the case of the femininity equation which is probably the easiest cases of PDT having to blow up there the quiz question is largely open as you would see them not too sure if many people are aware that this this question is completely open look here there you can construct you have examples where the blower set and let me just say something that for the heat equation we have only 1 what capital after that the solution does not exist so at Capital to we have 2 kinds of bonds Brill points with a solution goes to infinity and irregular points was solution is mounted OK so I will concentrate on the set of develop solutions that will points in fact I don't have characteristic among convicts on the bill .period without it we can have solutions where the singular said is on the 1 . or a finite number of points you can choose that became a refuge for a finite number of concentric a few and you can trust them and that's it for example 20 we don't have we don't know if we have a solution needlepoint and lips open problems can we haven't crossed into the open problem can we have a segment of people program etc. can imagine any other enjoyment here and we had we don't have an answer and the same question of course somehow investors for the similar wave equation said the union was acquitted but for a similar wave equation we have to change a little bit the definition of Cingular said because here on points of law points but at later times so he the wood notion for the singular is fixed notion of characteristic points because new known characteristic points the situation is at least in 1 space dimension completely easy provided where please see wine etc. so he knew that correspondents the question would be Kennedy have for example the solution with a set of characteristics .period unit costs that was our initial for in this interference 1 we could not answer the question but at least we had we came up with the new the title of a new renewal solution so we have a solution into the which rose up 1 1 Livshits graft which is pyramid-shaped and let me show you hear the picture and thank you long OK so you can see yes the pictures here OK so the these kids you don't mind the maximum of X 1 minus X 2 maximum X 1 9 6 2 0 is like the benefit of Egypt OK this is so something like that so if you see the truth in the direction of the axis X 1 location direct yourself only to have to put this direction X when you take X 2 equals 0 you will the corridor with 90 degrees like in 1 space the same thing with X OK 1 X 1 is equal to 0 back along the by sectors we will see another of the angle OK which is what when you you'd have a slope off 1 of the square root of 2 In fact because this is the pyramid can we have exactly the geometry of the of the parents so this is local hero near 0 and let me make some remarks 1st you access is of course region because you have an opinion is not rated became 2nd thing my solution we consider that as being symmetric with respect to the accident and and isometric with respect to buy factory which means that you would remain always vigorously 0 on by sector at and the surprising result is that 1 needs you this the characteristic .period or other points are not characteristic and let me tell you about something which is counterintuitive in some sense because in 1 space dimensions we we have noticed that there the change of or being 0 is linked to to having characteristic .period solely owned by sectors of solution is always you but we have no characteristic .period so this is something called which is completely different from the 1 the situation so only the origin is the a collective effort .period so here in front of the things that we put money in locally OK because we were somehow lazy but we we're able to say that all the other points even far in space unknown characters it is possible those somehow technical now let me talk about the regularity of the rule of graft I was simply said OK that the regularity is the same as the regularity of the parents OK so the pyramids this year sorry what happened could find no no problem can do it so you see here your
parent is everywhere except at the origin and 1 by since same for libel we have we are the ones everywhere except on the by sector OK so if we are outside the sectors is from cemeteries it's enough to consider the case where 0 is this the next to strictly X 1 here look at your delivered your dividend is like a derivative of the pyramid negative 1 but with the various collections like what we had in 1 space dimension in fact OK this is almost 0 defended the derivatives in the order of direction no on the sectors is outside the origin you have directional derivatives in all directions except in the direction of the basic stresses that basic tricks and this is like that of the pyramid in fact about at the origin you have directional derivatives except once again the long the the the the basic OK and let me insist on something here unlike the 1 we have on the bisected is the 1st example of characteristic points where he is non differential because wondered if she was at noon characteristic points then you differential and at characteristic points you have a accorded from 90 degrees here we still have a quarter land at the characteristic .period with which is the only Asian but on the on the sectors since we have no characteristic points and tea which is not at all the differential OK OK now what about the behavior so at the origin and remember this change variable it's always centered at the point where you want to see the behaviors OK so what when I talk about that 0 this means that I'm working in the backward like holding with verdicts 0 to use you here I with the force of intense 2 will be along the X 1 that is directions 1 going to the right this way and 1 gold to the left their signature OK and to all along the Extel direction there also and we will see the anti-Semitic so this means that if you drove the circles which is section of your backward like corn with verdicts 0 0 you see if you go wise for example you have positive negative or positive and OK if you encounter all the actions once again we have this kind of is sign the change in fact like the 1 the situation here with Huntington's are always the same OK and then the parameter is once again completely explicit and satisfies succinct kind of which is connected to the toward system here because of the the symmetry so we have only won once sentenced to 2 to deal with if we had feared the wandered toward affinity for without something involving Zitella wanted to and that therefore OK but the lives and 1 what I would say that the the book is negative 1 plus some correction here which is a Tibetan correction because this His negative log of capital 2 months the fantasy that did goes to a negative 1 and negative dingoes goes to 1 2 1 which give you the slopes of your Co so this is for the W 7 0 what happens outside the origin so if Agassi outside the region as told you we have no characteristic points and here we have a convergence to a new tools stationary solutions OK and we have to places if we are outside the bisected we were converge likened the 1 the situation to the special summit the same 1 OK and it's slope is given by negative 1 plus this they're looking at me "quotation mark action so we have not rigorously equal to the pyramid but we have a correction to the parent to you like that of course this is given just here but if you change by symmetry you can find the behavior of everywhere outside about fictitious and now if you out on the bias is the only 1 of them by sector since then you would find you the blow up a new stationary solution which is not a which is not original OK which is anti-Semitic which is called in a was which is anti-Semitic with respect to the basic OK so yes sir In the high-speed mentioned that cup this is not the only blow possibility for this set of stationary solutions we have a new said we're not able to characterize everything but we found a new station solution OK then the pool
this is maybe the most hired part of prove you I should have started with the political that adds an actress you maybe it's easier to 2 to follow but at the end of the world it's more difficult but of course there is more important to state anyway we have 2 major steps the 1st step the construction and electrical you have finite speed of competition so it's completely meaningful To start with initial data in this section of the backward like gold and to follow it on backward like coal and after that can step to where we find the behavior looking outside villagers so we make a construction with proscribed behavior located in the light cold and then we would find that very regularity between the 2 steps here is a small step which is not very long but uh while we need to use the finite speed of propagation and extend Our initial data outside this section of the alight called in order to have something which is defined hopefully everywhere in space so affect our initial data is defined in an initial is complex ,comma compactly supported and defined in the square and the squares is suitable to the geometry of the 3rd in fact OK and it is really that defined within the set supported usage which is strictly larger than the section of the medical OK and other already for the one-day case the asymptotic behavior and the regularity of real upset by completing an advanced side by side into the OK let me know the 1st step and told you we are working on the Candlelight Court which means that when I introduced W I would work only in the unit why is in wanted more OK so we are they so this is the goal finding a solution W which conveys this behavior OK so we see our freedoms as I stated before in interference OK so to cytokines which are symmetric in the X-1 behavior to fight significant X 2 behavior and we have unprecedented with respect to bisected is OK and I give you order for the characterization of the parameters the is completely explicit so this is my goal OK the framework is the
construction of a solution for beginning with prescribed medication OK the message well you wouldn't arrive equation around intended behavior and we find the regions in the Spektr negative spectrum and this is controlled thanks to minimize version of the up functions because in their view we have from the functional for the whole system the nonlinear so even for the liberalized equation you have upon functional and this helps you to control all the negative part of the spectrum we do not computes the any again value explicitly in the negative range then we have learned that was 0 which is called for thanks to moderation in the parameters in the of the Sowetans remember that this deep parameter is in fact that the parameter of the Lawrence transform which operates on the old ODD solution in you XT setting so we changed it a bit To kill the projection on Monday was you that equals 1 not negative 1 source for the mistake OK lumber equals 1 week it's controlled and killed in fact thanks to modulation with respect to this but new In this family so this family when new ways 0 you have your Sowetans solid and couples and 1 then go back to you XT and then again to W button with a different time you find this family at and of course this construction is inspired by the construction we did with the nothing could food Muttiah Sowetans in 1 space dimension OK
then let me give you a history of the construction with proscribed behavior once again unsure that I forget some some people and probably some recent work but it was possible to to use these ideas analysts KTV waterways for Odinga maps Indians darken the Seagull wave he Citigroup laps with many many people whichever side here OK no I I moved to step
2 the behavior of W Europe OK and the regularity of the effects 0 1 6 0 is outside the origin letting suggest the following you did because you OK and then if you want to know the behavior of W x 0 which is a completely equivalent to knowing the behavioral few exited in the backward like like along with of the Atkins you killed 6 0 In this case you just you might win and is small the sections of the goal With verdict sex you chill thanks to the goal With verdicts you 2 or 3 0 at almost the same way you are far from the year the singularity OK so far from the singularity you can't start from W X you go back to you and from you go to W's you and you will find that W X U and W 0 and link with these nice and bright identity OK so this is completely explicit scenes that new 0 has 4 solid tumors W X you far from the singularity would also have forced but with the formation because see we are multiplying the museum with effective and the 4 solid innings for W you In
fact involved this cap to kick up a star this standard OK so we have this for
generalized sentence with the defamation when see the formation because we have this factory which means that
here we have a new which is not year OK and then what do
with some dynamics will
follow before so it ends and you will see that To will disappear because
if they go here when you it's positive because new ways and new EPA the power as His new was positive as goes to infinity focus all this will go to 0 so too will disappear and then I will have only 2 Dayton's and in some cases 21 and we need a stronger nations here and the stronger than in the following if you start with 4 cytokines which are the cops which do not see each other you can follow them for a long time the cake and you see if someone would go to 0 or would go to infinite stay close to come out of this OK there is we have
not the bisected he's said in this range there will be left with only 1 solid OK at some point Estell said that this stuff is like negative X 1 so if X 1 is close 2 0 we will have to wait a long time and to making the seaside towns disappeared and had only 1 issue are far from X ones and whether this would happen rather quickly and then we have the trappings results OK which were 1st pro in 1 space dimensions and kind dimension for some complement exponent if you're grows for this sentence send you would converge 2 1 member of that and of course when you converge the parameter here is again into cities and it is close to this In this Parliament so we have some anything To make our solution Close to 1 attend and once it is close to 1 is trapped it would converge 2 1 so it and then since the gradient is like this parameter in gives us directly the gradient also this .period here I forgot something important this is the only known characteristic points because of dropping result works only known characteristic .period so he II outside the backed by sectors since OK if I'm not characteristic that I know the greedy and I have convergence but then at some point later in the pool I will have to show that all of points outside the by citizens of indeed not characterized and this is difficult for the moment they don't have to look at and this would be an important step note once again the link between the asymptotic behavior of W excellent the regularity this city we cover something where see the green OK now case too if we are on the bisexual this situation the solution is and enticing so you cannot have only once when your waste all that you can do this have only to switch tho you would have to cytokines which will decrease to 0 and you are left with tools it like that 1 long 6 1 and the other along textual and the parameter it's almost the same effect OK and here because we have an uncharacteristic .period OK and this comes from the behavior of the neighbors you Wyoming Besiktas but you have neighbors which have not on the basis and then you know the gradient so from that you can't have degraded although at least directional derivatives outside the directions supply sectors and see that you knew of no characteristic then because we have a couple of friction in similarity variables will converge toss solution which will be closed for this the guys so we have a new kind of stationary solutions which are neither original nor when 1 day OK a new kind of stationing solution no we are left with only 1 thing to show that outside the situations all the points are known characteristic OK so In fact this is what they call the world wanted to quickly and dramatically OK maybe it's suitable for today because it's going to be OK but let me
let me tell you what they call the ambulance look here in the domain of definition In 1 day but in Much Ado just take the call any cold wave still 1 of the kind of light cold is completely included in the domain of the finish with verdicts X still thinks it but then if you take a number of coaches agreed and starts from the bottom you imagine you're being umbrella is going up OK and then 8 their attached the euro go up and that at some point than that .period this short no characteristic .period OK thank you and then that and that since then every Place the 1st time when you're under the political known characteristic With a slope which is slightly less than 1 that's your graph the point is by definition and no character you see it OK that it's not more complicated than that so he had this is what they will do
for the American connected with the case OK take X
outside the basic for example here thanks to his thinking as an ax 1 and ratio that action character we take an endowment which is rather small between 0 and X 1 square swung in the log the local elections because of slopes negative 1 with another connection so X 1 square is much faster than 1 overload of X 1 OK and you consider family of cold with verdicts sentiment index and hi t 2 years ago and the slope is always 1 got so this is a number of it's coming from the bottom of the Green Umbrella undergo at some point for some value of it will touch the grass at some point next deal flexible immediately ex-lover is not characteristic because of this the slope which is strictly 1 effects of irony equal to X's because maybe the umbrella with attached to the glass at each stop if this happens then we have done thanks is known character if it does not happen if it that as well will try to find a contradiction it's the
imagine that the .period is not the center Of all the schools OK if X but only by sector sees Was this is a little bit complicated and so had talked wanted to mention that at all OK maybe later if you are interested only the paper workers but no 1 thinks about is on the other is not the basics is the argument is in fact is that the imagine that X amount is also but by cemetery here so it has positive coordinates OK so in this place both the gold index about and the grass the financial the code is differential because we are not at the center course but the gravity of differential because we have no characteristic .period thanks bye bye bye construction is no characters imagine 2 surfaces which teach what steps the 2 of them looking at some point so there that they should share the same tangent please OK so their agent has to be the same OK they still have to agree OK so the slope of the cold Is this guy which is less than this 1 and the scope of the graph is this guy and you see the log of X 1 His this than X 1 square because the parameter comes from the X the point where I'm starting and the slope comes from the Dutch Inc . 6 months so this means that X X 1 but is negligible with respect to 1 talking this is on 1 hand on the other hand OK I have a series of 3 but the inequalities which are easy to 20 sample from the effects it is less is more than 2 effects Mandic was still thinks it is because of the cold ended the year the graph in fact OK give X is 1 of them at the top of the cold and the top of the colony's modern too flexible this is 1 of the 2nd thing he is 1 option so TX bodies more than purely humane is excellent but since we are here it's more than X X 1 bar square root of 2 then W X is bounded thanks to our work in 2003 themselves 5 we had the following upper bound which is not shy the effects his effect on white weighs less than 0 0 minus 6 1 6 1 over to now let's put all the the inequalities altogether if you put them this is more than this more than this more than this so you find that X 1 OK Is this happened to ex-wife but but from the previous slide X-1 1 negligible with respect to explain the contradiction looking
not all points outside the bisected is unknown characteristic and they have gradient which is the slope while the betterment negative 1 In this region with In the development collection OK then we went to integrate this estimate between 0 and X will find that the ex-ministers you is that I it X 1 with immigrants make correction and of course by symmetry would find all the present shaped like that thank you for your attention the take question you go out during the search for a long long way to go from 1 rehabilitation work so you did take initiating and it didn't take the rest drinks yesterday for 1st so far I guess OK yes you read that they have that deviate from 1 value was also on the are the yes it is indifferent to the states that solution because on the on the trees we don't have Carrick capital of the don't have the use of solid we have convergence to a new stationary solutions so even if you changed with Laurent transform you will find yes this is good yes We Can we can make that like the like you're saying we find that the new title of no characteristic .period whether profile is not the soliton in winter very good humor this the new prospects an heiress Sita Devi and her look like it was there that the other let me show you it's clear the this this missing and from picture of course if should like that do here really have to 1 of than half the car with you the way was wider than the characteristic I need them in 1 did you either have silk which is legacy 1 although we have something which is differential and here we don't have something which is it's a differential but 2 slopes which which are far far from from Back Of course although OK my