Three equations with exponential nonlinearities

Transcription

1 Three equations with exponential nonlinearities Juan Luis Vázquez Departamento de Matemáticas Universidad Autónoma de Madrid Three equations with exponential nonlinearities p. 1/7

2 Three equations with exponential nonlinearities p. 2/7

3 Three equations with exponential nonlinearities Three equations with exponential nonlinearities p. 3/7

4 Three equations with exponential nonlinearities Recent T rends in NLPDEs, Three equations with exponential nonlinearities p. 3/7

5 Three equations with exponential nonlinearities Recent T rends in NLPDEs, Conferencia çelebrada en Salamanca del nuestro Reyno, en loor del Prof. Ireneo Peral Three equations with exponential nonlinearities p. 3/7

6 Three equations with exponential nonlinearities Recent T rends in NLPDEs, Conferencia çelebrada en Salamanca del nuestro Reyno, en loor del Prof. Ireneo Peral Mio Çid de las Çiencias Matemáticas Three equations with exponential nonlinearities p. 3/7

7 Three equations with exponential nonlinearities Recent T rends in NLPDEs, Conferencia çelebrada en Salamanca del nuestro Reyno, en loor del Prof. Ireneo Peral Mio Çid de las Çiencias Matemáticas Cavallero Mayor de los algorismos p-laplaçiados e Hardiqueses Three equations with exponential nonlinearities p. 3/7

8 Three equations with exponential nonlinearities Recent T rends in NLPDEs, Conferencia çelebrada en Salamanca del nuestro Reyno, en loor del Prof. Ireneo Peral Mio Çid de las Çiencias Matemáticas Cavallero Mayor de los algorismos p-laplaçiados e Hardiqueses Real desfazedor de malos argumentos y cálculos errados en el honrrado Reyno de Castiella Three equations with exponential nonlinearities p. 3/7

9 Three equations with exponential nonlinearities Recent T rends in NLPDEs, Conferencia çelebrada en Salamanca del nuestro Reyno, en loor del Prof. Ireneo Peral Mio Çid de las Çiencias Matemáticas Cavallero Mayor de los algorismos p-laplaçiados e Hardiqueses Real desfazedor de malos argumentos y cálculos errados en el honrrado Reyno de Castiella Three equations with exponential nonlinearities p. 3/7

10 Three equations with exponential nonlinearities p. 4/7

11 Introduction The basic problems of Theoretical PDEs are existence, uniqueness, stability, and qualitative behaviour of the solutions of relevant classes of equations. Three equations with exponential nonlinearities p. 5/7

12 Introduction The basic problems of Theoretical PDEs are existence, uniqueness, stability, and qualitative behaviour of the solutions of relevant classes of equations. Relevant classes of equations can refer to one equation that is very important itself because of its applications, or to a large class. Three equations with exponential nonlinearities p. 5/7

13 Introduction The basic problems of Theoretical PDEs are existence, uniqueness, stability, and qualitative behaviour of the solutions of relevant classes of equations. Relevant classes of equations can refer to one equation that is very important itself because of its applications, or to a large class. In dealing with large classes it is always difficult to separate real mathematics from the exercise or from the esoteric. Advice for lost children : solve really open problems, the simpler the better. Three equations with exponential nonlinearities p. 5/7

14 Introduction The basic problems of Theoretical PDEs are existence, uniqueness, stability, and qualitative behaviour of the solutions of relevant classes of equations. Relevant classes of equations can refer to one equation that is very important itself because of its applications, or to a large class. In dealing with large classes it is always difficult to separate real mathematics from the exercise or from the esoteric. Advice for lost children : solve really open problems, the simpler the better. Critical cases are the joy of the pure researcher. Some of them are beautiful, intriguing and important. The exponentials are examples of interesting critical cases. Three equations with exponential nonlinearities p. 5/7

15 Introduction The basic problems of Theoretical PDEs are existence, uniqueness, stability, and qualitative behaviour of the solutions of relevant classes of equations. Relevant classes of equations can refer to one equation that is very important itself because of its applications, or to a large class. In dealing with large classes it is always difficult to separate real mathematics from the exercise or from the esoteric. Advice for lost children : solve really open problems, the simpler the better. Critical cases are the joy of the pure researcher. Some of them are beautiful, intriguing and important. The exponentials are examples of interesting critical cases. I have selected three of my favorite exponentials for Ireneo Three equations with exponential nonlinearities p. 5/7

16 Three equations with exponential nonlinearities p. 6/7

17 Nonlinear Elliptic and Parabolic Pblms All these years I have been busy with understanding Diffusion, and Reaction-Diffusion, both in Evolution and in the Steady State. Three equations with exponential nonlinearities p. 7/7

18 Nonlinear Elliptic and Parabolic Pblms All these years I have been busy with understanding Diffusion, and Reaction-Diffusion, both in Evolution and in the Steady State. Populations diffuse, substances (like particles in a solvent) diffuse, heat propagates, electrons and ions diffuse, the momentum of a viscous (Newtonian) fluid diffuses (linearly), there is diffusion in the markets,... A main question is: how much of it can be explained with linear models, how much is essentially nonlinear? Three equations with exponential nonlinearities p. 7/7

19 Nonlinear Elliptic and Parabolic Pblms All these years I have been busy with understanding Diffusion, and Reaction-Diffusion, both in Evolution and in the Steady State. Populations diffuse, substances (like particles in a solvent) diffuse, heat propagates, electrons and ions diffuse, the momentum of a viscous (Newtonian) fluid diffuses (linearly), there is diffusion in the markets,... A main question is: how much of it can be explained with linear models, how much is essentially nonlinear? The stationary states of diffusion belong to an important world, elliptic equations. Three equations with exponential nonlinearities p. 7/7

20 Nonlinear Elliptic and Parabolic Pblms All these years I have been busy with understanding Diffusion, and Reaction-Diffusion, both in Evolution and in the Steady State. Populations diffuse, substances (like particles in a solvent) diffuse, heat propagates, electrons and ions diffuse, the momentum of a viscous (Newtonian) fluid diffuses (linearly), there is diffusion in the markets,... A main question is: how much of it can be explained with linear models, how much is essentially nonlinear? The stationary states of diffusion belong to an important world, elliptic equations. Elliptic equations, linear and nonlinear, have many relatives: diffusion, fluid mechanics, waves of all types, quantum mechanics,... Three equations with exponential nonlinearities p. 7/7

21 Nonlinear Elliptic and Parabolic Pblms All these years I have been busy with understanding Diffusion, and Reaction-Diffusion, both in Evolution and in the Steady State. Populations diffuse, substances (like particles in a solvent) diffuse, heat propagates, electrons and ions diffuse, the momentum of a viscous (Newtonian) fluid diffuses (linearly), there is diffusion in the markets,... A main question is: how much of it can be explained with linear models, how much is essentially nonlinear? The stationary states of diffusion belong to an important world, elliptic equations. Elliptic equations, linear and nonlinear, have many relatives: diffusion, fluid mechanics, waves of all types, quantum mechanics,... Three equations with exponential nonlinearities p. 7/7

22 Three equations with exponential nonlinearities p. 8/7

23 I. Exponential Elliptic Equation Working for my doctoral thesis, Profs. Brezis and Bénilan asked me to investigate the possible existence and properties of the solutions of the semilinear perturbation of the Laplace-Poisson equation u + e u = f Three equations with exponential nonlinearities p. 9/7

24 I. Exponential Elliptic Equation Working for my doctoral thesis, Profs. Brezis and Bénilan asked me to investigate the possible existence and properties of the solutions of the semilinear perturbation of the Laplace-Poisson equation u + e u = f The equation is interesting when posed in IR 2, and when f is not a mere integrable function but a Dirac mass, or some other measure not given by a density in L 1. It appeared as a limit case of the Thomas-Fermi theory; it has its own role in vortex theory. Three equations with exponential nonlinearities p. 9/7

25 I. Exponential Elliptic Equation Working for my doctoral thesis, Profs. Brezis and Bénilan asked me to investigate the possible existence and properties of the solutions of the semilinear perturbation of the Laplace-Poisson equation u + e u = f The equation is interesting when posed in IR 2, and when f is not a mere integrable function but a Dirac mass, or some other measure not given by a density in L 1. It appeared as a limit case of the Thomas-Fermi theory; it has its own role in vortex theory. The simplest case is to put f(x) = f 1 (x) + cδ(x), f 1 smooth. This was before I was lucky to find the now well-known critical value c = 4π. There exist solutions of the problem in the plane with f 1 L 1 (IR 2 ), iff c c. Three equations with exponential nonlinearities p. 9/7

26 I. Exponential Elliptic Equation Working for my doctoral thesis, Profs. Brezis and Bénilan asked me to investigate the possible existence and properties of the solutions of the semilinear perturbation of the Laplace-Poisson equation u + e u = f The equation is interesting when posed in IR 2, and when f is not a mere integrable function but a Dirac mass, or some other measure not given by a density in L 1. It appeared as a limit case of the Thomas-Fermi theory; it has its own role in vortex theory. The simplest case is to put f(x) = f 1 (x) + cδ(x), f 1 smooth. This was before I was lucky to find the now well-known critical value c = 4π. There exist solutions of the problem in the plane with f 1 L 1 (IR 2 ), iff c c. Three equations with exponential nonlinearities p. 9/7

27 Three equations with exponential nonlinearities p. 10/7

28 Exponential Elliptic Equation The main result I proved was that the problem can be solved if a certain condition on the measure f is satisfied, but the condition is unilateral: the atomic part has to have each and every mass of positive size equal or less than 4πδ(x x i ). Three equations with exponential nonlinearities p. 11/7

29 Exponential Elliptic Equation The main result I proved was that the problem can be solved if a certain condition on the measure f is satisfied, but the condition is unilateral: the atomic part has to have each and every mass of positive size equal or less than 4πδ(x x i ). If the atomic mass is i c iδ(x x i ) with some c i > 4π, then the natural approximation produces a solution for the reduced problem with right-hand side Π(f) = f E(f), where E(f) = i (c i 4π) + δ(x x i ) Three equations with exponential nonlinearities p. 11/7

30 Exponential Elliptic Equation The main result I proved was that the problem can be solved if a certain condition on the measure f is satisfied, but the condition is unilateral: the atomic part has to have each and every mass of positive size equal or less than 4πδ(x x i ). If the atomic mass is i c iδ(x x i ) with some c i > 4π, then the natural approximation produces a solution for the reduced problem with right-hand side Π(f) = f E(f), where E(f) = i (c i 4π) + δ(x x i ) Recently, H. Brezis, M. Marcus, L. Orsina, A. Ponce and others have discussed a general theory for these elliptic equations with critical values, and the theory has now the spirit of functional analysis, and covers the case where e u is replaced by a monotone g(u). Three equations with exponential nonlinearities p. 11/7

31 Exponential Elliptic Equation The main result I proved was that the problem can be solved if a certain condition on the measure f is satisfied, but the condition is unilateral: the atomic part has to have each and every mass of positive size equal or less than 4πδ(x x i ). If the atomic mass is i c iδ(x x i ) with some c i > 4π, then the natural approximation produces a solution for the reduced problem with right-hand side Π(f) = f E(f), where E(f) = i (c i 4π) + δ(x x i ) Recently, H. Brezis, M. Marcus, L. Orsina, A. Ponce and others have discussed a general theory for these elliptic equations with critical values, and the theory has now the spirit of functional analysis, and covers the case where e u is replaced by a monotone g(u). Three equations with exponential nonlinearities p. 11/7

32 Three equations with exponential nonlinearities p. 12/7

33 Exponential Elliptic Equation In case f is excessive, or as they say now non-admissible, where does the remaining mass go? Here is the magic formula for the approximations u n g(u n ) w 1 + w 2, w 1 L 1, w 1 g(u), w 2 = E(f) Three equations with exponential nonlinearities p. 13/7

34 Exponential Elliptic Equation In case f is excessive, or as they say now non-admissible, where does the remaining mass go? Here is the magic formula for the approximations u n g(u n ) w 1 + w 2, w 1 L 1, w 1 g(u), w 2 = E(f) There are plenty of open problems: a main question is to find cases where the projection is not just cutting some of the atomic mass. There is a very beautiful example worked out by Orsina and collaborators about the admissibility condition for the same equation in IR N, N 3. Of course, it involves measuring the capacities of the sets where the bad measure sits. If it is too small, then diffusion does not see it. Three equations with exponential nonlinearities p. 13/7

35 Exponential Elliptic Equation In case f is excessive, or as they say now non-admissible, where does the remaining mass go? Here is the magic formula for the approximations u n g(u n ) w 1 + w 2, w 1 L 1, w 1 g(u), w 2 = E(f) There are plenty of open problems: a main question is to find cases where the projection is not just cutting some of the atomic mass. There is a very beautiful example worked out by Orsina and collaborators about the admissibility condition for the same equation in IR N, N 3. Of course, it involves measuring the capacities of the sets where the bad measure sits. If it is too small, then diffusion does not see it. The paper appeared in Proc. Royal Soc. Edinburgh, 1983, and I had forgotten about it. Three equations with exponential nonlinearities p. 13/7

36 Exponential Elliptic Equation In case f is excessive, or as they say now non-admissible, where does the remaining mass go? Here is the magic formula for the approximations u n g(u n ) w 1 + w 2, w 1 L 1, w 1 g(u), w 2 = E(f) There are plenty of open problems: a main question is to find cases where the projection is not just cutting some of the atomic mass. There is a very beautiful example worked out by Orsina and collaborators about the admissibility condition for the same equation in IR N, N 3. Of course, it involves measuring the capacities of the sets where the bad measure sits. If it is too small, then diffusion does not see it. The paper appeared in Proc. Royal Soc. Edinburgh, 1983, and I had forgotten about it. Three equations with exponential nonlinearities p. 13/7

37 Three equations with exponential nonlinearities p. 14/7

38 End of Part I At that time I have taken three main decisions for my future: (i) Getting a position at UAM, where Ireneo Peral with his infinite enthusiasm made life active every day, just as I needed; (ii) Emigrating to the USA for long periods. There, I met Luis Caffarelli, Don Aronson, Mike Crandall, Avner Friedman,..., people who helped change our professional life in years to come; (iii) Going to Italy every year. At the time I was happy to have learned symmetrization from Giorgio Talenti, and I will need in the last part all that I can do. Three equations with exponential nonlinearities p. 15/7

39 End of Part I At that time I have taken three main decisions for my future: (i) Getting a position at UAM, where Ireneo Peral with his infinite enthusiasm made life active every day, just as I needed; (ii) Emigrating to the USA for long periods. There, I met Luis Caffarelli, Don Aronson, Mike Crandall, Avner Friedman,..., people who helped change our professional life in years to come; (iii) Going to Italy every year. At the time I was happy to have learned symmetrization from Giorgio Talenti, and I will need in the last part all that I can do. Three equations with exponential nonlinearities p. 15/7

40 End of Part I At that time I have taken three main decisions for my future: (i) Getting a position at UAM, where Ireneo Peral with his infinite enthusiasm made life active every day, just as I needed; (ii) Emigrating to the USA for long periods. There, I met Luis Caffarelli, Don Aronson, Mike Crandall, Avner Friedman,..., people who helped change our professional life in years to come; (iii) Going to Italy every year. At the time I was happy to have learned symmetrization from Giorgio Talenti, and I will need in the last part all that I can do. Three equations with exponential nonlinearities p. 15/7

41 End of Part I At that time I have taken three main decisions for my future: (i) Getting a position at UAM, where Ireneo Peral with his infinite enthusiasm made life active every day, just as I needed; (ii) Emigrating to the USA for long periods. There, I met Luis Caffarelli, Don Aronson, Mike Crandall, Avner Friedman,..., people who helped change our professional life in years to come; (iii) Going to Italy every year. At the time I was happy to have learned symmetrization from Giorgio Talenti, and I will need in the last part all that I can do. Three equations with exponential nonlinearities p. 15/7

42 End of Part I At that time I have taken three main decisions for my future: (i) Getting a position at UAM, where Ireneo Peral with his infinite enthusiasm made life active every day, just as I needed; (ii) Emigrating to the USA for long periods. There, I met Luis Caffarelli, Don Aronson, Mike Crandall, Avner Friedman,..., people who helped change our professional life in years to come; (iii) Going to Italy every year. At the time I was happy to have learned symmetrization from Giorgio Talenti, and I will need in the last part all that I can do. Three equations with exponential nonlinearities p. 15/7

43 Three equations with exponential nonlinearities p. 16/7

44 II. Nonlinear heat flows In the last 50 years emphasis in PDEs has shifted towards the Nonlinear World. Maths more difficult, more complex and more realistic. My favorite areas are Nonlinear Diffusion and Reaction Diffusion. General formula for Nonlinear Parabolic PDEs u t = i A i (u, u) + B(x, u, u) Three equations with exponential nonlinearities p. 17/7

45 II. Nonlinear heat flows In the last 50 years emphasis in PDEs has shifted towards the Nonlinear World. Maths more difficult, more complex and more realistic. My favorite areas are Nonlinear Diffusion and Reaction Diffusion. General formula for Nonlinear Parabolic PDEs u t = i A i (u, u) + B(x, u, u) Typical nonlinear diffusions: u t = u m, u t = p u Typical reaction diffusion: u t = u + u p Three equations with exponential nonlinearities p. 17/7

46 Three equations with exponential nonlinearities p. 18/7

47 The Nonlinear Diffusion Models The Stefan Problem (Lamé and Clapeyron, 1833; Stefan 1880) u t = k 1 u for u > 0, u = 0, SE : TC : u t = k 2 u for u < 0. v = L(k 1 u 1 k 2 u 2 ). Main feature: the free boundary or moving boundary where u = 0. TC= Transmission conditions at u = 0. Three equations with exponential nonlinearities p. 19/7

48 The Nonlinear Diffusion Models The Stefan Problem (Lamé and Clapeyron, 1833; Stefan 1880) u t = k 1 u for u > 0, u = 0, SE : TC : u t = k 2 u for u < 0. v = L(k 1 u 1 k 2 u 2 ). Main feature: the free boundary or moving boundary where u = 0. TC= Transmission conditions at u = 0. The Hele-Shaw cell (Hele-Shaw, 1898; Saffman-Taylor, 1958) u > 0, u = 0 in Ω(t); u = 0, v = L n u on Ω(t). Three equations with exponential nonlinearities p. 19/7

49 The Nonlinear Diffusion Models The Stefan Problem (Lamé and Clapeyron, 1833; Stefan 1880) u t = k 1 u for u > 0, u = 0, SE : TC : u t = k 2 u for u < 0. v = L(k 1 u 1 k 2 u 2 ). Main feature: the free boundary or moving boundary where u = 0. TC= Transmission conditions at u = 0. The Hele-Shaw cell (Hele-Shaw, 1898; Saffman-Taylor, 1958) u > 0, u = 0 in Ω(t); u = 0, v = L n u on Ω(t). The Porous Medium Equation (hidden free boundary) u t = u m, m > 1. Three equations with exponential nonlinearities p. 19/7

50 The Nonlinear Diffusion Models The Stefan Problem (Lamé and Clapeyron, 1833; Stefan 1880) u t = k 1 u for u > 0, u = 0, SE : TC : u t = k 2 u for u < 0. v = L(k 1 u 1 k 2 u 2 ). Main feature: the free boundary or moving boundary where u = 0. TC= Transmission conditions at u = 0. The Hele-Shaw cell (Hele-Shaw, 1898; Saffman-Taylor, 1958) u > 0, u = 0 in Ω(t); u = 0, v = L n u on Ω(t). The Porous Medium Equation (hidden free boundary) u t = u m, m > 1. The p-laplacian Equation, u t = div( u p 2 u). Three equations with exponential nonlinearities p. 19/7

51 Three equations with exponential nonlinearities p. 20/7

52 The Reaction Diffusion Models The Standard Blow-Up model (Kaplan, 1963; Fujita, 1966) u t = u + u p Main feature: If p > 1 the norm u(, t) of the solutions goes to infinity in finite time. Hint: Integrate u t = u p. Problem: what is the influence of diffusion / migration? Three equations with exponential nonlinearities p. 21/7

53 The Reaction Diffusion Models The Standard Blow-Up model (Kaplan, 1963; Fujita, 1966) u t = u + u p Main feature: If p > 1 the norm u(, t) of the solutions goes to infinity in finite time. Hint: Integrate u t = u p. Problem: what is the influence of diffusion / migration? General scalar model u t = A(u) + f(u) Three equations with exponential nonlinearities p. 21/7

54 The Reaction Diffusion Models The Standard Blow-Up model (Kaplan, 1963; Fujita, 1966) u t = u + u p Main feature: If p > 1 the norm u(, t) of the solutions goes to infinity in finite time. Hint: Integrate u t = u p. Problem: what is the influence of diffusion / migration? General scalar model u t = A(u) + f(u) The system model: u = (u 1,, u m ) chemotaxis. Three equations with exponential nonlinearities p. 21/7

55 The Reaction Diffusion Models The Standard Blow-Up model (Kaplan, 1963; Fujita, 1966) u t = u + u p Main feature: If p > 1 the norm u(, t) of the solutions goes to infinity in finite time. Hint: Integrate u t = u p. Problem: what is the influence of diffusion / migration? General scalar model u t = A(u) + f(u) The system model: u = (u 1,, u m ) chemotaxis. The fluid flow models: Navier-Stokes or Euler equation systems for incompressible flow. Any singularities? Three equations with exponential nonlinearities p. 21/7

56 The Reaction Diffusion Models The Standard Blow-Up model (Kaplan, 1963; Fujita, 1966) u t = u + u p Main feature: If p > 1 the norm u(, t) of the solutions goes to infinity in finite time. Hint: Integrate u t = u p. Problem: what is the influence of diffusion / migration? General scalar model u t = A(u) + f(u) The system model: u = (u 1,, u m ) chemotaxis. The fluid flow models: Navier-Stokes or Euler equation systems for incompressible flow. Any singularities? The geometrical models: the Ricci flow: t g ij = R ij. Three equations with exponential nonlinearities p. 21/7

57 Three equations with exponential nonlinearities p. 22/7

58 An opinion of John Nash, 1958: The open problems in the area of nonlinear p.d.e. are very relevant to applied mathematics and science as a whole, perhaps more so that the open problems in any other area of mathematics, and the field seems poised for rapid development. It seems clear, however, that fresh methods must be employed... Little is known about the existence, uniqueness and smoothness of solutions of the general equations of flow for a viscous, compressible, and heat conducting fluid... Continuity of solutions of elliptic and parabolic equations, paper published in Amer. J. Math, 80, no 4 (1958), Three equations with exponential nonlinearities p. 23/7

59 Three equations with exponential nonlinearities p. 24/7

60 II. Fujita-Gelfand equation In 1995 Ireneo and myself published a paper on the famous model known as Gelfand equation in combustion. u t u = λf(u), f(u) = e u. Three equations with exponential nonlinearities p. 25/7

61 II. Fujita-Gelfand equation In 1995 Ireneo and myself published a paper on the famous model known as Gelfand equation in combustion. u t u = λf(u), f(u) = e u. There are several motivations both for the evolution problem and for its stationary version. The main one for me is the application to combustion as model by Frank-Kamenetsky in The problem attracted much attention after the work of Fujita in the 1960 s, who carefully examined the question of blow-up. Three equations with exponential nonlinearities p. 25/7

62 II. Fujita-Gelfand equation In 1995 Ireneo and myself published a paper on the famous model known as Gelfand equation in combustion. u t u = λf(u), f(u) = e u. There are several motivations both for the evolution problem and for its stationary version. The main one for me is the application to combustion as model by Frank-Kamenetsky in The problem attracted much attention after the work of Fujita in the 1960 s, who carefully examined the question of blow-up. Dávila has explained in his talk the Joseph-Lundgren analysis, which separates the types of bifurcation plots in dimensions N = 1,2; 3 N 9, and finally N 10. This is one of the origins of the theories of critical numbers, so popular now. Three equations with exponential nonlinearities p. 25/7

63 II. Fujita-Gelfand equation In 1995 Ireneo and myself published a paper on the famous model known as Gelfand equation in combustion. u t u = λf(u), f(u) = e u. There are several motivations both for the evolution problem and for its stationary version. The main one for me is the application to combustion as model by Frank-Kamenetsky in The problem attracted much attention after the work of Fujita in the 1960 s, who carefully examined the question of blow-up. Dávila has explained in his talk the Joseph-Lundgren analysis, which separates the types of bifurcation plots in dimensions N = 1,2; 3 N 9, and finally N 10. This is one of the origins of the theories of critical numbers, so popular now. Three equations with exponential nonlinearities p. 25/7

64 II. Fujita-Gelfand equation In 1995 Ireneo and myself published a paper on the famous model known as Gelfand equation in combustion. u t u = λf(u), f(u) = e u. There are several motivations both for the evolution problem and for its stationary version. The main one for me is the application to combustion as model by Frank-Kamenetsky in The problem attracted much attention after the work of Fujita in the 1960 s, who carefully examined the question of blow-up. Dávila has explained in his talk the Joseph-Lundgren analysis, which separates the types of bifurcation plots in dimensions N = 1,2; 3 N 9, and finally N 10. This is one of the origins of the theories of critical numbers, so popular now. Three equations with exponential nonlinearities p. 25/7

65 Three equations with exponential nonlinearities p. 26/7

66 Paper on Fujita-Gelfand equation Our paper (ARMA, 1995) is very specific, we wanted to understand the stability properties of the special singular solution (again, fundamental solution of the Laplacian) U(x) = 2 log( x ) which corresponds to the value λ = 2(N 2) when N 3. Three equations with exponential nonlinearities p. 27/7

67 Paper on Fujita-Gelfand equation Our paper (ARMA, 1995) is very specific, we wanted to understand the stability properties of the special singular solution (again, fundamental solution of the Laplacian) U(x) = 2 log( x ) which corresponds to the value λ = 2(N 2) when N 3. The first thing that we did was to show that there is a very good existence theory for initial data u(x,0) U(x). Moreover, we did a delicate iterative argument to show that for all u 0 U the solution is bounded for all t > 0. Three equations with exponential nonlinearities p. 27/7

68 Paper on Fujita-Gelfand equation Our paper (ARMA, 1995) is very specific, we wanted to understand the stability properties of the special singular solution (again, fundamental solution of the Laplacian) U(x) = 2 log( x ) which corresponds to the value λ = 2(N 2) when N 3. The first thing that we did was to show that there is a very good existence theory for initial data u(x,0) U(x). Moreover, we did a delicate iterative argument to show that for all u 0 U the solution is bounded for all t > 0. Three equations with exponential nonlinearities p. 27/7

69 Paper on Fujita-Gelfand equation Our paper (ARMA, 1995) is very specific, we wanted to understand the stability properties of the special singular solution (again, fundamental solution of the Laplacian) U(x) = 2 log( x ) which corresponds to the value λ = 2(N 2) when N 3. The first thing that we did was to show that there is a very good existence theory for initial data u(x,0) U(x). Moreover, we did a delicate iterative argument to show that for all u 0 U the solution is bounded for all t > 0. Three equations with exponential nonlinearities p. 27/7

70 Paper on Fujita-Gelfand equation Our paper (ARMA, 1995) is very specific, we wanted to understand the stability properties of the special singular solution (again, fundamental solution of the Laplacian) U(x) = 2 log( x ) which corresponds to the value λ = 2(N 2) when N 3. The first thing that we did was to show that there is a very good existence theory for initial data u(x,0) U(x). Moreover, we did a delicate iterative argument to show that for all u 0 U the solution is bounded for all t > 0. Three equations with exponential nonlinearities p. 27/7

71 Three equations with exponential nonlinearities p. 28/7

72 Paper on Fujita-Gelfand equation Years later I proved that there is bounded solution even for u 0 = U. Since obviously U is a solution, this means non-uniqueness. The new minimal solution is selfsimilar and goes down with time. Actually, the result says that you can solve the problem with bounded solutions even when u 0 (x) U(x) + c where c is less than the critical excess". Reference: J. L. V., Domain of existence and blowup for the exponential reaction-diffusion equation. Indiana Univ. Math. Journal, 48, 2 (1999), Three equations with exponential nonlinearities p. 29/7

73 Paper on Fujita-Gelfand equation Years later I proved that there is bounded solution even for u 0 = U. Since obviously U is a solution, this means non-uniqueness. The new minimal solution is selfsimilar and goes down with time. Actually, the result says that you can solve the problem with bounded solutions even when u 0 (x) U(x) + c where c is less than the critical excess". Reference: J. L. V., Domain of existence and blowup for the exponential reaction-diffusion equation. Indiana Univ. Math. Journal, 48, 2 (1999), The next point is to linearize the equation around the singular solution to obtain the equation (λ 0 = 2(N 2)) φ t = L(φ), L(φ) = φ + λ 0 x 2φ Three equations with exponential nonlinearities p. 29/7

74 Paper on Fujita-Gelfand equation Years later I proved that there is bounded solution even for u 0 = U. Since obviously U is a solution, this means non-uniqueness. The new minimal solution is selfsimilar and goes down with time. Actually, the result says that you can solve the problem with bounded solutions even when u 0 (x) U(x) + c where c is less than the critical excess". Reference: J. L. V., Domain of existence and blowup for the exponential reaction-diffusion equation. Indiana Univ. Math. Journal, 48, 2 (1999), The next point is to linearize the equation around the singular solution to obtain the equation (λ 0 = 2(N 2)) φ t = L(φ), L(φ) = φ + λ 0 x 2φ Three equations with exponential nonlinearities p. 29/7

75 Paper on Fujita-Gelfand equation Years later I proved that there is bounded solution even for u 0 = U. Since obviously U is a solution, this means non-uniqueness. The new minimal solution is selfsimilar and goes down with time. Actually, the result says that you can solve the problem with bounded solutions even when u 0 (x) U(x) + c where c is less than the critical excess". Reference: J. L. V., Domain of existence and blowup for the exponential reaction-diffusion equation. Indiana Univ. Math. Journal, 48, 2 (1999), The next point is to linearize the equation around the singular solution to obtain the equation (λ 0 = 2(N 2)) φ t = L(φ), L(φ) = φ + λ 0 x 2φ Three equations with exponential nonlinearities p. 29/7

76 Paper on Fujita-Gelfand equation Years later I proved that there is bounded solution even for u 0 = U. Since obviously U is a solution, this means non-uniqueness. The new minimal solution is selfsimilar and goes down with time. Actually, the result says that you can solve the problem with bounded solutions even when u 0 (x) U(x) + c where c is less than the critical excess". Reference: J. L. V., Domain of existence and blowup for the exponential reaction-diffusion equation. Indiana Univ. Math. Journal, 48, 2 (1999), The next point is to linearize the equation around the singular solution to obtain the equation (λ 0 = 2(N 2)) φ t = L(φ), L(φ) = φ + λ 0 x 2φ Three equations with exponential nonlinearities p. 29/7

77 Three equations with exponential nonlinearities p. 30/7

78 Paper on Fujita-Gelfand equation Indeed, without linearization we have for φ = U u φ t = φ + λ 0 x 2(1 e φ ) which makes these things subsolutions of the linearized equation. Three equations with exponential nonlinearities p. 31/7

79 Paper on Fujita-Gelfand equation Indeed, without linearization we have for φ = U u φ t = φ + λ 0 x 2(1 e φ ) which makes these things subsolutions of the linearized equation. The analysis of the linearized operator is based on knowing whether L is positive or not (accretive or not). The Hardy inequality plays a crucial role, the constant in the energy formulation is c N = (N 2) 2 /4. The constant that we have in the equation is λ 0 = 2(N 2). Stability holds if N 10. For N < 10 the stationary solution is not attractor and the orbit tends to the minimal solution. Three equations with exponential nonlinearities p. 31/7

80 Paper on Fujita-Gelfand equation Indeed, without linearization we have for φ = U u φ t = φ + λ 0 x 2(1 e φ ) which makes these things subsolutions of the linearized equation. The analysis of the linearized operator is based on knowing whether L is positive or not (accretive or not). The Hardy inequality plays a crucial role, the constant in the energy formulation is c N = (N 2) 2 /4. The constant that we have in the equation is λ 0 = 2(N 2). Stability holds if N 10. For N < 10 the stationary solution is not attractor and the orbit tends to the minimal solution. Three equations with exponential nonlinearities p. 31/7

81 Three equations with exponential nonlinearities p. 32/7

82 Paper on Fujita-Gelfand equation For N 10 the orbits near U tend to U asymptotically. I have calculated the rate with a UK-Russian team by doing matched asymptotics (Dold-Lacey-Galaktionov-JLV). It says u(t) = c 1 t + o(t). Reference: Rate of approach to a singular steady state in quasilinear reaction-diffusion equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 4, Three equations with exponential nonlinearities p. 33/7

83 Paper on Fujita-Gelfand equation For N 10 the orbits near U tend to U asymptotically. I have calculated the rate with a UK-Russian team by doing matched asymptotics (Dold-Lacey-Galaktionov-JLV). It says u(t) = c 1 t + o(t). Reference: Rate of approach to a singular steady state in quasilinear reaction-diffusion equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 4, The most striking result of the paper is the result on instantaneous blow-up If u 0 U and u 0 U and moreover u U, then u(t) + for every t > 0. The proof is based on writing the equation for ψ = u U, ψ t = ψ + λ 0 x 2(eψ 1) L(ψ) Three equations with exponential nonlinearities p. 33/7

84 Paper on Fujita-Gelfand equation For N 10 the orbits near U tend to U asymptotically. I have calculated the rate with a UK-Russian team by doing matched asymptotics (Dold-Lacey-Galaktionov-JLV). It says u(t) = c 1 t + o(t). Reference: Rate of approach to a singular steady state in quasilinear reaction-diffusion equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 4, The most striking result of the paper is the result on instantaneous blow-up If u 0 U and u 0 U and moreover u U, then u(t) + for every t > 0. The proof is based on writing the equation for ψ = u U, ψ t = ψ + λ 0 x 2(eψ 1) L(ψ) Three equations with exponential nonlinearities p. 33/7

85 Three equations with exponential nonlinearities p. 34/7

86 Paper on Fujita-Gelfand equation We compare with the solution ũ with initial data such that φ = U ũ has same initial data as ψ. We get ψ(x, t) φ(x, t) Therefore, u(x, t) 2U(x) ũ(x, t) But φ is bounded, hence the singularity of u is like 2U(x). Three equations with exponential nonlinearities p. 35/7

87 Paper on Fujita-Gelfand equation We compare with the solution ũ with initial data such that φ = U ũ has same initial data as ψ. We get ψ(x, t) φ(x, t) Therefore, u(x, t) 2U(x) ũ(x, t) But φ is bounded, hence the singularity of u is like 2U(x). Iterating the argument, we get u(x, t) cu(x) near x = 0 for every c > 1. Once this implies that e u(t 1) L 1, we conclude that u(t) + for all later times. End of Part II Three equations with exponential nonlinearities p. 35/7

88 Paper on Fujita-Gelfand equation We compare with the solution ũ with initial data such that φ = U ũ has same initial data as ψ. We get ψ(x, t) φ(x, t) Therefore, u(x, t) 2U(x) ũ(x, t) But φ is bounded, hence the singularity of u is like 2U(x). Iterating the argument, we get u(x, t) cu(x) near x = 0 for every c > 1. Once this implies that e u(t 1) L 1, we conclude that u(t) + for all later times. End of Part II Three equations with exponential nonlinearities p. 35/7

89 Three equations with exponential nonlinearities p. 36/7

90 Intermezzo Porous Medium and Fast Diffusion Three equations with exponential nonlinearities p. 37/7

91 Three equations with exponential nonlinearities p. 38/7

92 Porous Medium and Fast Diffusion The equation is written as u t = u m = (c(u) u) density-dependent diffusivity c(u) = mu m 1 [= m u m 1 ] Three equations with exponential nonlinearities p. 39/7

93 Porous Medium and Fast Diffusion The equation is written as u t = u m = (c(u) u) density-dependent diffusivity c(u) = mu m 1 [= m u m 1 ] Equation degenerates at u = 0 if m > 1, Porous Medium Case Three equations with exponential nonlinearities p. 39/7

94 Porous Medium and Fast Diffusion The equation is written as u t = u m = (c(u) u) density-dependent diffusivity c(u) = mu m 1 [= m u m 1 ] Equation degenerates at u = 0 if m > 1, Porous Medium Case Equation is singular at u = 0 if m < 1, Fast Diffusion Case Three equations with exponential nonlinearities p. 39/7

95 Porous Medium and Fast Diffusion The equation is written as u t = u m = (c(u) u) density-dependent diffusivity c(u) = mu m 1 [= m u m 1 ] Equation degenerates at u = 0 if m > 1, Porous Medium Case Equation is singular at u = 0 if m < 1, Fast Diffusion Case Situation is inverted as u Three equations with exponential nonlinearities p. 39/7

96 Porous Medium and Fast Diffusion The equation is written as u t = u m = (c(u) u) density-dependent diffusivity c(u) = mu m 1 [= m u m 1 ] Equation degenerates at u = 0 if m > 1, Porous Medium Case Equation is singular at u = 0 if m < 1, Fast Diffusion Case Situation is inverted as u Fast Diffusion can cover in principle the ultrafast range m 0, even if u has changing sign: = c(u) = u m 1. Then u = (c(u) u) = ( u m 1 u/m) Three equations with exponential nonlinearities p. 39/7

97 Three equations with exponential nonlinearities p. 40/7

98 Applied motivation for the PME Flow of gas in a porous medium (Leibenzon, 1930; Muskat 1933) ρ t + div (ρv) = 0, v = k µ p, p = P(ρ). Second line left is the Darcy law for flows in porous media (Darcy, 1856). Porous media flows are potential flows due to averaging of Navier-Stokes on the pore scales. ρ is density, p is pressure. Three equations with exponential nonlinearities p. 41/7

99 Applied motivation for the PME Flow of gas in a porous medium (Leibenzon, 1930; Muskat 1933) ρ t + div (ρv) = 0, v = k µ p, p = P(ρ). Second line left is the Darcy law for flows in porous media (Darcy, 1856). Porous media flows are potential flows due to averaging of Navier-Stokes on the pore scales. ρ is density, p is pressure. If P(ρ) is a power, P = ρ γ 1, we get the PME with m = 1 + γ 2. If not, we get the general Filtration Equation: ρ t = div ( k µ ρ P(ρ)) := Φ(ρ) Three equations with exponential nonlinearities p. 41/7

100 Three equations with exponential nonlinearities p. 42/7

101 Applied motivation II Underground water infiltration (Boussinesq, 1903) m = 2 Three equations with exponential nonlinearities p. 43/7

102 Applied motivation II Underground water infiltration (Boussinesq, 1903) m = 2 Plasma radiation m 4 (Zeldovich-Raizer, < 1950) Three equations with exponential nonlinearities p. 43/7

103 Applied motivation II Underground water infiltration (Boussinesq, 1903) m = 2 Plasma radiation m 4 (Zeldovich-Raizer, < 1950) Spreading of populations (self-avoiding diffusion) m 2. Three equations with exponential nonlinearities p. 43/7

104 Applied motivation II Underground water infiltration (Boussinesq, 1903) m = 2 Plasma radiation m 4 (Zeldovich-Raizer, < 1950) Spreading of populations (self-avoiding diffusion) m 2. Thin films under gravity (no surface tension) m = Three equations with exponential nonlinearities p. 43/7

105 Applied motivation II Underground water infiltration (Boussinesq, 1903) m = 2 Plasma radiation m 4 (Zeldovich-Raizer, < 1950) Spreading of populations (self-avoiding diffusion) m 2. Thin films under gravity (no surface tension) m = Plasma Physics, Okuda-Dawson law: m = 1/2 < 1. Three equations with exponential nonlinearities p. 43/7

106 Applied motivation II Underground water infiltration (Boussinesq, 1903) m = 2 Plasma radiation m 4 (Zeldovich-Raizer, < 1950) Spreading of populations (self-avoiding diffusion) m 2. Thin films under gravity (no surface tension) m = Plasma Physics, Okuda-Dawson law: m = 1/2 < 1. Kinetic limits (Carleman models, McKean, PL Lions and Toscani, Salvarani & JLV, etc.): n = 1, m = 0 (logaritmic diffusion) Three equations with exponential nonlinearities p. 43/7

107 Applied motivation II Underground water infiltration (Boussinesq, 1903) m = 2 Plasma radiation m 4 (Zeldovich-Raizer, < 1950) Spreading of populations (self-avoiding diffusion) m 2. Thin films under gravity (no surface tension) m = Plasma Physics, Okuda-Dawson law: m = 1/2 < 1. Kinetic limits (Carleman models, McKean, PL Lions and Toscani, Salvarani & JLV, etc.): n = 1, m = 0 (logaritmic diffusion) Yamabe Flows in Differential Geometry: n 3, m = (n 2)/(n + 2) Three equations with exponential nonlinearities p. 43/7

108 Applied motivation II Underground water infiltration (Boussinesq, 1903) m = 2 Plasma radiation m 4 (Zeldovich-Raizer, < 1950) Spreading of populations (self-avoiding diffusion) m 2. Thin films under gravity (no surface tension) m = Plasma Physics, Okuda-Dawson law: m = 1/2 < 1. Kinetic limits (Carleman models, McKean, PL Lions and Toscani, Salvarani & JLV, etc.): n = 1, m = 0 (logaritmic diffusion) Yamabe Flows in Differential Geometry: n 3, m = (n 2)/(n + 2) Many more (boundary layers, dopant diffusion, stochastic processes, images,...); you may find m 1 Three equations with exponential nonlinearities p. 43/7

109 Three equations with exponential nonlinearities p. 44/7

110 The basics The equation is re-written for m = 2 as 1 2 u t = u u + u 2 Three equations with exponential nonlinearities p. 45/7

111 The basics The equation is re-written for m = 2 as 1 2 u t = u u + u 2 and you can see that for u 0 it looks like the eikonal equation u t = u 2 This is not parabolic, but hyperbolic (propagation along characteristics). Mixed type, mixed properties. Three equations with exponential nonlinearities p. 45/7

112 The basics The equation is re-written for m = 2 as 1 2 u t = u u + u 2 and you can see that for u 0 it looks like the eikonal equation u t = u 2 This is not parabolic, but hyperbolic (propagation along characteristics). Mixed type, mixed properties. No big problem when m > 1, m 2. The pressure transformation gives: v t = (m 1)v v + v 2 where v = cu m 1 is the pressure; normalization c = m/(m 1). This separates m > 1 PME - from - m < 1 FDE Three equations with exponential nonlinearities p. 45/7

113 Three equations with exponential nonlinearities p. 46/7

114 References About PME J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory", Oxford Univ. Press, 2006 in press. approx. 600 pages About estimates and scaling Three equations with exponential nonlinearities p. 47/7

115 References About PME J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory", Oxford Univ. Press, 2006 in press. approx. 600 pages About estimates and scaling J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Parabolic Equations of Porous Medium Type, Oxford Univ. Press, 2006, 234 pages. About asymptotic behaviour. (Following Lyapunov and Boltzmann) Three equations with exponential nonlinearities p. 47/7

116 References About PME J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory", Oxford Univ. Press, 2006 in press. approx. 600 pages About estimates and scaling J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Parabolic Equations of Porous Medium Type, Oxford Univ. Press, 2006, 234 pages. About asymptotic behaviour. (Following Lyapunov and Boltzmann) J. L. Vázquez. Asymptotic behaviour for the Porous Medium Equation posed in the whole space. Journal of Evolution Equations 3 (2003), Three equations with exponential nonlinearities p. 47/7

117 References About PME J. L. Vázquez, "The Porous Medium Equation. Mathematical Theory", Oxford Univ. Press, 2006 in press. approx. 600 pages About estimates and scaling J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Parabolic Equations of Porous Medium Type, Oxford Univ. Press, 2006, 234 pages. About asymptotic behaviour. (Following Lyapunov and Boltzmann) J. L. Vázquez. Asymptotic behaviour for the Porous Medium Equation posed in the whole space. Journal of Evolution Equations 3 (2003), Three equations with exponential nonlinearities p. 47/7

118 Three equations with exponential nonlinearities p. 48/7

119 Barenblatt profiles (ZKB) These profiles are the alternative to the Gaussian profiles. Three equations with exponential nonlinearities p. 49/7

120 Barenblatt profiles (ZKB) These profiles are the alternative to the Gaussian profiles. They are source solutions. Source means that u(x, t) M δ(x) as t 0. Three equations with exponential nonlinearities p. 49/7

121 Barenblatt profiles (ZKB) These profiles are the alternative to the Gaussian profiles. They are source solutions. Source means that u(x, t) M δ(x) as t 0. Explicit formulas (1950): α = n 2+n(m 1), β = 1 2+n(m 1) < 1/2 B(x, t;m) = t α F(x/t β ), F(ξ) = ( C kξ 2) 1/(m 1) + Three equations with exponential nonlinearities p. 49/7

122 Barenblatt profiles (ZKB) These profiles are the alternative to the Gaussian profiles. They are source solutions. Source means that u(x, t) M δ(x) as t 0. Explicit formulas (1950): α = n 2+n(m 1), β = 1 2+n(m 1) < 1/2 B(x, t;m) = t α F(x/t β ), F(ξ) = ( C kξ 2) 1/(m 1) + u BS x Three equations with exponential nonlinearities p. 49/7

123 Barenblatt profiles (ZKB) These profiles are the alternative to the Gaussian profiles. They are source solutions. Source means that u(x, t) M δ(x) as t 0. Explicit formulas (1950): α = n 2+n(m 1), β = 1 2+n(m 1) < 1/2 B(x, t;m) = t α F(x/t β ), F(ξ) = ( C kξ 2) 1/(m 1) + u BS x Height u = Ct α Free boundary at distance x = ct β Scaling law; anomalous diffusion versus Brownian motion Three equations with exponential nonlinearities p. 49/7

124 Barenblatt profiles (ZKB) These profiles are the alternative to the Gaussian profiles. They are source solutions. Source means that u(x, t) M δ(x) as t 0. Explicit formulas (1950): α = n 2+n(m 1), β = 1 2+n(m 1) < 1/2 B(x, t;m) = t α F(x/t β ), F(ξ) = ( C kξ 2) 1/(m 1) + u BS x Height u = Ct α Free boundary at distance x = ct β Scaling law; anomalous diffusion versus Brownian motion Three equations with exponential nonlinearities p. 49/7

125 Barenblatt profiles (ZKB) These profiles are the alternative to the Gaussian profiles. They are source solutions. Source means that u(x, t) M δ(x) as t 0. Explicit formulas (1950): α = n 2+n(m 1), β = 1 2+n(m 1) < 1/2 B(x, t;m) = t α F(x/t β ), F(ξ) = ( C kξ 2) 1/(m 1) + u BS x Height u = Ct α Free boundary at distance x = ct β Scaling law; anomalous diffusion versus Brownian motion Three equations with exponential nonlinearities p. 49/7

126 Barenblatt profiles (ZKB) These profiles are the alternative to the Gaussian profiles. They are source solutions. Source means that u(x, t) M δ(x) as t 0. Explicit formulas (1950): α = n 2+n(m 1), β = 1 2+n(m 1) < 1/2 B(x, t;m) = t α F(x/t β ), F(ξ) = ( C kξ 2) 1/(m 1) + u BS x Height u = Ct α Free boundary at distance x = ct β Scaling law; anomalous diffusion versus Brownian motion Three equations with exponential nonlinearities p. 49/7

127 Three equations with exponential nonlinearities p. 50/7

128 FDE profiles We again have explicit formulas for 1 > m > (n 2)/n: B(x, t;m) = t α F(x/t β ), F(ξ) = 1 (C + kξ 2 ) 1/(1 m) u(,t) t=1.15 t=1.25 t=1.4 t=1.6 α = β = n ; if n = 2, then α = 1/m 2 n(1 m) 1 2 n(1 m) > 1/2; if n = 2, then β = 1/2m x Solutions for m < 1 with fat tails (polynomial decay; anomalous distributions) Three equations with exponential nonlinearities p. 51/7

129 FDE profiles We again have explicit formulas for 1 > m > (n 2)/n: B(x, t;m) = t α F(x/t β ), F(ξ) = 1 (C + kξ 2 ) 1/(1 m) u(,t) t=1.15 t=1.25 t=1.4 t=1.6 α = β = n ; if n = 2, then α = 1/m 2 n(1 m) 1 2 n(1 m) > 1/2; if n = 2, then β = 1/2m x Solutions for m < 1 with fat tails (polynomial decay; anomalous distributions) Big problem: What happens for m < (n 2)/n? Most active branch of PME/FDE. New asymptotics, extinction, new functional properties, new geometry and physics. Many authors: J. King, geometers,... my book Smoothing. Three equations with exponential nonlinearities p. 51/7