singularity formation in nonlinear heat and mean curvature flow equations Wenbin Kong

Transcription

1 singularity formation in nonlinear heat and mean curvature flow equations by Wenbin Kong A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright c 21 by Wenbin Kong

2 Abstract singularity formation in nonlinear heat and mean curvature flow equations Wenbin Kong Doctor of Philosophy Graduate Department of Mathematics University of Toronto 21 In this thesis we study singularity formation in two basic nonlinear equations in n dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than n + 1, to the standard 2 n-dimensional sphere, the solution collapses in a finite time t, to a point. We also show that as t t, it looks like a sphere of radius 2n(t t). ii

3 Acknowledgements First, I would like to express my sincere gratitude to my advisor, Professor Israel Michael Sigal, for his continuous support of my graduate study and research with his motivation, enthusiasm, patience and most of all, his immense knowledge and rich experience. He led me into the area of PDE and guided me through the whole period of my Ph.D. study. I also would like to thank my friends Ioannis Anapolitanos, Tim Tzaneteas, Gang Zhou and Xiangqun Zou for many stimulating dicussions. In addition, I am grateful to Professor Almut Burchard, Professor James Colliander, Professor Robert Jerrard, Professor Robert McCann, Professor Mary Pugh, Professor Catherine Sulem for teaching me various courses and preparing me well for my research work. I would like to thank Professor Dan Knopf for very useful remarks on the thesis. Next, I am grateful to our graduate administrator, Ida Bulat, for her support and help through these years. We are so lucky to have her in our department. I really enjoyed the friendship of many students in the department. I would like to thank my fellow students colleagues Jie Chen, Shibing Chen, Xiaolong Cheng, Chao Li, Songhao Li, Xiao Liu, Bin Xu, Chao Yang and Yichao Zhang, who brought me lots of fun and unforgettable memories. I deeply cherish my experience at the math department in last six years. Last but not least, I am deeply grateful to my parents, Fanyun Kong and Zhengchun Wen, and my girlfriend Yun Cao for their endless love, support and concern. iii

4 Contents 1 Introduction 1 2 On blowup in nonlinear heat equations Introduction The Local Well-Posedness of (2.1) Blow-Up Variables and Almost Solutions Reparametrization of Solutions A priori Estimates Proof of Main Theorem Gauge Transform Lyapunov-Schmidt Splitting (Effective Equations) Proof of Estimates (2.46)-(2.48) Rescaling of Fluctuations on a Fixed Time Interval Estimate of the Propagators Estimate of M 1 (τ) (Equation (2.49)) Estimate of M 2 (Equation (2.5)) iv

5 3 Stability of spherical collapse under mean curvature flow Introduction Rescaled equation Differential equation for ρ Collapse center Reparametrization of solutions Lyapunov-Schmidt decomposition Linearized operator Lyapunov functional Proof of Theorem A Feynmann-Kac Formula 77 B Detailed computations and proofs in Chapter 2 84 B.1 Equation (2.25): Computation of A B.2 Derivation of Equation (2.65)-(2.71) B.3 Proof of Lemma B.4 Proof of (2.16) in the case i j = C Proof of Lemma D Proof of (3.43) 92 Bibliography 96 v

6 Chapter 1 Introduction This thesis consists of two parts. The first part deals with the nonlinear heat equation t u = u + u p 1 u (1.1) u(x, ) = u (x) for p > 1. Here u : R n R + R. Its main results are a proof that solutions for certain open set of initial conditions with a single absolute maximum blowup and a characterization of blowup profile. In the second part we study the behavior of mean curvature flow (MCF) of hypersurfaces in R n+1 with initial conditions close to spheres. Given an initial simple, closed hypersurface S in R n+1 the MCF determines a family {S t t } of closed hypersurfaces in R n+1, given by immersions X(, t) : Ω R n+1, satisfying the following evolution equation: X t = H(X)ν(X), (1.2) where Ω R n+1 is a fixed n dimensional hypersurface, ν(x) and H(X) are the outward unit normal vector and mean curvature at X S t, respectively. Here we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than n/2 + 1, to the standard n-dimensional sphere, the solution collapses in a finite time, t, to 1

7 Chapter 1. Introduction 2 a point, z, approaching exponentially fast the spheres of radii 2n(t t), centered at z(t). Equation (1.1) arises in the problem of heat flow, or, more generally, in the problems involving diffusion, and is a model for a large class of nonlinear parabolic equations, which are ubiquitous in mathematics and its applications. The local well-posedness of (1.1) is well known (see, e.g. [8] for the Sobolev spaces H α, α < 2). Moreover for some data u (x), the solutions u(x, t) might blow up in finite time t >. In what follows a solution u(x, t) is said to blow up at time t if it exists in L for [, t ) and sup x u(x, t) as t t. Thus, two key problems about (1.1) are 1. Describe initial conditions for which solutions of Equation (1.1) blow up in finite time; 2. Describe the blow-up profile of such solutions. It is expected (see e.g. [1]) that the blow-up profile is universal it is independent of lower power perturbations of the nonlinearity and of initial conditions within certain spaces. There is rich literature regarding the blow-up problem for Equation (1.1). We review quickly relevant results. Starting with [39], various criteria for blow-up in finite time were derived, see e.g. [39, 8, 16, 31, 65, 66, 74, 76, 82, 33, 37]. For example, if u H 1 L p+1 and E(u ) <, where E(u) is the energy functional for (1.1) given by E(u) := 1 2 u 2 1 p + 1 u p+1, (1.3) then it is proved in [65] that u(t) 2 2 blows up in finite time t. By the observation 1 d 2 dt u(t) 2 2 u(t) p 1 u(t) 2 2

8 Chapter 1. Introduction 3 we have that u(t) blows up in finite time t t also. (In this paper, we denote the norms in the L p spaces by p.) Blow-up at a single point was studied as early as [86] (see also [37]). The first result on asymptotics of the blow-up for arbitrary dimension n 1 was obtained in the pioneering paper [43] where the authors show that under the condition u(x, t) (t t) 1 p 1 is bounded on B1 (, t ), (1.4) where B 1 is the unit ball in R n centred at the origin, and either p n+2 n 2 assuming blow-up takes place at x =, one has lim λ 2 p 1 u(λx, t + λ 2 (t t )) = ± λ ( 1 p 1 ) 1 p 1 (t t) 1 p 1 or. or n 2 and This result was further improved in several papers (see e.g. [45, 44, 49, 34, 69, 84, 35, 36, 37, 1, 71, 68, 72]). A blow-up solution satisfying the bound (1.4) is said to be of type I. This bound was proven under various conditions in [45, 71, 68, 87, 46]. Furthermore, the limits of H 1 -blow-up solutions u(x, t) as t T, outside the blow-up sets were established in [49, 34, 69, 84, 35, 36, 37, 1, 72, 32]. For p > 1, dimension n = 1, Herrero and Velázquez [5] (see also [35]) proved that if the initial condition u is continuous, nonnegative, bounded, even and has only one local maximum at, and if the corresponding solution blows up, then lim (t t) 1 p 1 t t u(y((t t) ln t t ) 1 2, t) = (p 1) 1 p 1 p 1 [1 + 4p y2 ] 1 p 1 (1.5) uniformly on sets y R with R >. Further extensions of this result are achieved in [49, 84, 34, 35]. Later for dimension n = 1 Bricmont and Kupiainen [1] constructed a co-dimension 2 submanifold, of initial conditions such that (1.5) is satisfied on the whole domain. More specifically, given a small function g and a small constant b >, they find constants d and d 1 depending on g and b such that the solution to (1.1) with the datum u (x) = (p 1 + bx 2 ) 1 d p 1 + d 1 x (1 + p 1 + bx ) 1 2 p 1 + g(x) (1.6)

9 Chapter 1. Introduction 4 has the convergence (1.5) uniformly in y (, + ). The result of [1] was generalized in [7, 32] (see also [41]), where it is shown that there exists a neighborhood U, in the space H := L p+1 H 1, of u, given in (1.6), such that if u U, then the solution u(x, t) blows up in a finite time t and satisfies (1.5) for x R. They conjectured that this asymptotic behavior is generic for any blow-up solution. In 1992, Merle [69] proved that given a finite number of points x 1, x 2,..., x k in I = ( 1, 1) (or any other domain I in R), there is a positive solution to the nonlinear heat equation which blows up up at time T with blow-up points x 1, x 2,..., x k. This theorem can be generalized to allow the sign (+ or ) to be chosen at each blow-up point x i. In [2] precise blowup asymptotics were derived for (1.1) in dimension 1 for even initial conditions. This work developed a different approach based on dynamical rescaling, method of majorants and strong linear estimates. The results of [2] were extended to general initial conditions and consequently to moving blowup point in [6]. Our results and techniques extend those of [2]. The following key properties of equation (1.1) explain important features of the expressions above: (1.1) is invariant with respect to the scaling transformation, u(x, t) λ 2 p 1 u(λx, λ 2 t) (1.7) for any constant λ >, i.e. if u(x, t) is a solution, so is λ 2 p 1 u(λx, λ 2 t). (1.1) has x independent (homogeneous) solutions: u hom = [u p+1 (p 1)t] 1 p 1. (1.8) These solutions blow up in finite time t = ( ) (p 1)u p 1 1 for p > 1.

10 Chapter 1. Introduction 5 The linearization of (1.1) around u hom shows that the solution u hom is unstable. Moreover, it is shown in [43] that if either n 2 or p (n+2)/(n 2), then the equation (1.1) has no other self-similar solutions of the form (T t) 1 p 1 φ ( x/ T t ), φ L, besides u hom. We consider (1.1) with initial conditions which have, modulo a small perturbation, a maximum at the origin, are slowly varying near the origin and are sufficiently small, but not necessarily vanishing, for large x. In particular, the energy E(u) for such initial conditions might be infinite. We show that the solutions of (1.1) for such initial conditions blowup in a finite time t and we characterize asymptotic dynamics of these solutions. As it turns out, the leading term is given by the expression [ λ(t) 2 p 1 c(t) p 1 + λ 2 (t)(x ζ(t))b(t)(x ζ(t)) ] 1 p 1 (1.9) where b(t) a real, symmetric n n-matrix b(t) = (b ij (t)), yby := n i,j=1 y ib ij y j for a n n-matrix b := (b ij ) and the parameters λ(t), b(t), c(t) and ζ(t) obey certain dynamical equations whose solutions give λ(t) = λ (t t) 1 2 (1 + o(1)) b(t) = (p 1)2 4p ln t t (I + O( 1 ln t t 1/2 )) c(t) ζ(t) = 1 = O(1). p 1 2p ln t t (1 + O( 1 ln t t )) (1.1) with λ() = c + 2 p 1 T rb(), c >, b() > depends on the initial datum. Here o(1) is in t t. Moreover, we estimate the remainder, the difference between u(x, t) and (1.9). In our approach, as in [2], we do not fix the time-dependent scale in the self-similarity (blowup) variables but let its behavior, as well as behavior of other parameters (b and c) to be determined by the equation. We will deal, without specifying it, with weak solutions of Equation (1.1) in the sense detailed in section 2.2. The local existence of such solutions is well known and is

11 Chapter 1. Introduction 6 presented for readers convenience there. These solutions can be shown to be classical for t >. In what follows we use the notation x := (1 + x 2 ) 1/2 and f g for two positive functions f and g, satisfying f Cg for some universal constant C. The main result is the following theorem. Theorem 1. Let 1 c 4, b := (b ij ) > be a real, symmetric, positive n n-matrix with b 1. Suppose the initial data u L (R n ) for (1.1) satisfy the conditions ( ( ) ) 1 c p 1 x m u (x) δ m, (1.11) p 1 + xb x with m =, 3, δ 1 and δ 3 = C b 2. Then (1) There exists a time t (, ) such that the solution u(x, t) exists on the interval [, t ) and blows up at t t. (2) When t t, there exist unique, positive, C 1 real valued, n-vector valued and n nmatrix valued functions λ(t) and c(t), ζ(t) and b(t), respectively, with b(t) b(), such that u(x, t) can be decomposed as u(x, t) = λ 2 c(t) p 1 (t)[( p 1 + λ 2 (t)(x ζ(t))b(t)(x ζ(t)) ) 1 p 1 + η(x, t)] with the fluctuation part, η, admitting the estimate λ(t)(x ζ(t)) m η(x, t) δ m (t), m =, 3, with δ (t) = δ 1 and δ 3 (t) = b(t) 2. (3) The functions λ(t), b(t), c(t) and ζ(t) are of the form (1.1). We note that no smoothness of initial conditions is required, also note that (1.1) is an L 2 -gradient system t u = grad E(u), with the energy defined in (1.3). Most of the works mentioned above use this fact in an essential way. It is not used in here and we expect our analysis can be extended to non-gradient systems. Specific forms of initial conditions and nonlinearity enter into only two places: orthogonal decomposition and

12 Chapter 1. Introduction 7 Lyapunov-Schmidt splitting (Sections 2.4 and 2.8). Note that our approach is closest to that of [2] (see also [89]). Our results extend those of [2] in two aspects. First we address the problem of blow-up in arbitrary dimensions. Second we consider more general initial conditions so that the blowup center is not fixed but in general moves. The mean curvature flow is the steepest descent flow for the area functional. It arises in applications, such as models of annealing metals [73] and other problems involving phase separation and moving interfaces ([27, 57, 11]). It has been recently successfully applied by Huisken and Sinestrari to topological classification of surfaces and submanifolds ([54, 55], see also [61]). It is closely related to the Ricci and inverse mean curvature flow. Mean curvature flow was first studied by Brakke [9]. Evans and Spruck [28] constructed a unique weak solution of the nonlinear PDE for certain functions whose zero level set evolves in time according to its mean curvature. Similar results were obtained by Chen, Giga and Goto [15] and by Ambrosio and Soner [3]. The short-time existence in Hölder spaces was proven in [9, 52, 28, 56]. Ecker and Huisken [24] established interior estimates and the short time existence for initial surface which is locally Lipschitz continuous follows from this immediately. Higher codimension mean curvature flows were studied by Mu-Tao Wang [85] (see also [64]). For more results on the existence, uniqueness and regularity of the solution one can see [9, 29, 3, 58, 17]. The question of the long time existence is, as usual, more subtle. Ecker and Huisken [25] showed longtime existence for mean curvature flow in the case of linearly growing graphs. Later they [24] proved that if the initial surface is a locally Lipschitz continuous entire graph over R n then the solution will exist for all times. However, the most interesting aspect of the mean curvature flow is formation of singularities, with two canonical examples being solutions with initial conditions being Eucledian spheres or cylinders. In the latter cases, the solution is again a family of spheres

13 Chapter 1. Introduction 8 or cylinders collapsing to their center or axis, with the radii evolving as 2n(t t) or 2(n 1)(t t). The behaviour of solutions in the last two cases are fairly different. In the former case, the first result here was due to Gage and Hamilton [4], who showed that initial convex plane curves shrink to a round point, i.e. approach asymptotically circles of radii 2(t t). Later, Grayson [48] showed that any embedded plane curves always shrink smoothly until they are convex, and then to points by the evolution theorem of convex curves. For higher dimensions the latter result does not hold. In a seminal work, [52], Huisken showed that under mean curvature flow a convex hypersurface in R n, n 3, shrinks smoothly to a point, getting spherical in the limit. For more results see [18, 19, 62, 63, 88]. Our main result is as follows. Theorem 2. Let Ω be the standard n-dimensional sphere and let a surface S, defined by an immersion x H s (Ω), for some s > n 2 + 1, be close to Ω, in the sense that x 1 H s 1. Then there exist t < and z R n+1, s.t. (1.2) has the unique solution, S t, t < t, and this solution contracts to the point z, as t. Moreover, S t is defined by an immersion x(, t) H s (Ω), with the same s, of the form x(ω, t) = z(t) + R(ω, t)ω, for some z(t) R n+1 and R(, t) H s (Ω), satisfying z(t) = z + O((t t) 1 2a (n n ) ) and n R(ω, t) = λ(t)( + ξ(ω, t)), (1.12) a(t) with λ(t), a(t) and ξ(, t) which satisfy λ(t) = 2a (t t) + O((t t) a (1 1 2n ) ), a(t) = λ(t) λ(t) = a + O((t t) 1 2a (1 1 2n ) ) and ξ(, t) H s (t t) 1 2n. Moreover, z 1.

14 Chapter 1. Introduction 9 Remark 1. If the initial condition x is invariant under the transformation x i x i for any i = 1,, n + 1, then z(t) = and the proof below simplifies considerately. The alternative scenario of formation of singularities under the mean curvature flow is neckpinching (see [2, 4, 1, 7, 23, 28, 15, 22, 53, 81, 8] and [5, 6], for the Ricci flow, and references therein). A remarkable difference between this scenario and the one described above is that, unlike spheres, the cylinders are not stable under the mean curvature flow. For instance, it follows from results of [42] that for an open set of initial conditions arbitrarily close to a cylinder which have an arbitrary shallow waist, the solution to the mean curvature flow forms a neck which pinches in a finite time. The form of expression (1.12) above is a reflection of a large class of symmetries of the mean curvature flow: (1.2) is invariant under rigid motions of the surface, i.e. X RX + a, where R O(n + 1), a R n+1 and X = X(u, t) is a parametrization of S t, is a symmetry of (1.2). (1.2) is invariant under the scaling X λx and t λ 2 t for any λ >. Our approach utilizes these symmetries in an essential way. It uses the rescaling of the equation (1.2) by a parameter λ(t) whose behaviour is determined by the equation itself and a series of differential inequalities for a Lyapunov-type functions. This thesis is broken into two parts, with the first part dealing with NLH and the second with MCF. These parts are independent and contain their own introductions. These introductions repeat the corresponding parts of the present introduction plus description of the organizations of each chapter.

15 Chapter 2 On blowup in nonlinear heat equations 2.1 Introduction In this chapter we study the blow-up problem for the n-dimensional nonlinear heat equation (or the reaction-diffusion equation) t u = u + u p 1 u u(x, ) = u (x) (2.1) with p > 1. Here u : R n R + R. Equation (2.1) arises in the problem of heat flow, or, more generally, in the problems involving diffusion, and is a model for a large class of nonlinear parabolic equations, which are ubiquitous in mathematics and its applications. The local well-posedness of (2.1) is well known (see, e.g. [8] for the Sobolev spaces H α, α < 2). Moreover for some data u (x), the solutions u(x, t) might blow up in finite time t >. In what follows a solution u(x, t) is said to blow up at time t if it exists in L for [, t ) and sup x u(x, t) as t t. Thus, two key problems about (2.1) are 1

16 Chapter 2. On blowup in nonlinear heat equations Describe initial conditions for which solutions of Equation (2.1) blow up in finite time; 2. Describe the blow-up profile of such solutions. It is expected (see e.g. [1]) that the blow-up profile is universal it is independent of lower power perturbations of the nonlinearity and of initial conditions within certain spaces. There is rich literature regarding the blow-up problem for Equation (2.1). We review quickly relevant results. Starting with [39], various criteria for blow-up in finite time were derived, see e.g. [39, 8, 16, 31, 65, 66, 74, 76, 82, 33, 37]. For example, if u H 1 L p+1 and E(u ) <, where E(u) is the energy functional for (2.1) given by E(u) := 1 2 u 2 1 p + 1 u p+1, (2.2) then it is proved in [65] that u(t) 2 2 blows up in finite time t. By the observation 1 d 2 dt u(t) 2 2 u(t) p 1 u(t) 2 2 we have that u(t) blows up in finite time t t also. (In this paper, we denote the norms in the L p spaces by p.) Blow-up at a single point was studied as early as [86] (see also [37]). The first result on asymptotics of the blow-up for arbitrary dimension n 1 was obtained in the pioneering paper [43] where the authors show that under the condition u(x, t) (t t) 1 p 1 is bounded on B1 (, t ), (2.3) where B 1 is the unit ball in R n centred at the origin, and either p n+2 n 2 assuming blow-up takes place at x =, one has ( ) 1 lim λ 2 p 1 1 u(λx, t + λ 2 p 1 (t t )) = ± (t t) 1 p 1 or. λ p 1 or n 2 and

17 Chapter 2. On blowup in nonlinear heat equations 12 This result was further improved in several papers (see e.g. [45, 44, 49, 34, 69, 84, 35, 36, 37, 1, 71, 68, 72]). A blow-up solution satisfying the bound (2.3) is said to be of type I. This bound was proven under various conditions in [45, 71, 68, 87, 46]. Furthermore, the limits of H 1 -blow-up solutions u(x, t) as t T, outside the blow-up sets were established in [49, 34, 69, 84, 35, 36, 37, 1, 72, 32]. For p > 1, dimension n = 1, Herrero and Velázquez [5] (see also [35]) proved that if the initial condition u is continuous, nonnegative, bounded, even and has only one local maximum at, and if the corresponding solution blows up, then lim (t t) 1 p 1 t t u(y((t t) ln t t ) 1 2, t) = (p 1) 1 p 1 p 1 [1 + 4p y2 ] 1 p 1 (2.4) uniformly on sets y R with R >. Further extensions of this result are achieved in [49, 84, 34, 35]. Later for dimension n = 1 Bricmont and Kupiainen [1] constructed a co-dimension 2 submanifold, of initial conditions such that (2.4) is satisfied on the whole domain. More specifically, given a small function g and a small constant b >, they find constants d and d 1 depending on g and b such that the solution to (2.1) with the datum u (x) = (p 1 + bx 2 ) 1 d p 1 + d 1 x (1 + p 1 + bx ) 1 2 p 1 + g(x) (2.5) has the convergence (2.4) uniformly in y (, + ). The result of [1] was generalized in [7, 32] (see also [41]), where it is shown that there exists a neighborhood U, in the space H := L p+1 H 1, of u, given in (2.5), such that if u U, then the solution u(x, t) blows up in a finite time t and satisfies (2.4) for x R. They conjectured that this asymptotic behavior is generic for any blow-up solution. In 1992, Merle [69] proved that given a finite number of points x 1, x 2,..., x k in I = ( 1, 1) (or any other domain I in R), there is a positive solution to the nonlinear heat equation which blows up up at time T with blow-up points x 1, x 2,..., x k. This theorem can be generalized to allow the sign (+ or ) to be chosen at each blow-up

18 Chapter 2. On blowup in nonlinear heat equations 13 point x i. In [2] precise blowup asymptotics were derived for (2.1) in dimension 1 for even initial conditions. This work developed a different approach based on dynamical rescaling, method of majorants and strong linear estimates. The results of [2] were extended to general initial conditions and consequently to moving blowup point in [6]. Our results and techniques extend those of [2]. The following key properties of equation (2.1) explain important features of the expressions above: (2.1) is invariant with respect to the scaling transformation, u(x, t) λ 2 p 1 u(λx, λ 2 t) (2.6) for any constant λ >, i.e. if u(x, t) is a solution, so is λ 2 p 1 u(λx, λ 2 t). (2.1) has x independent (homogeneous) solutions: u hom = [u p+1 (p 1)t] 1 p 1. (2.7) These solutions blow up in finite time t = ( ) (p 1)u p 1 1 for p > 1. The linearization of (2.1) around u hom shows that the solution u hom is unstable. Moreover, it is shown in [43] that if either n 2 or p (n+2)/(n 2), then the equation (2.1) has no other self-similar solutions of the form (T t) 1 p 1 φ ( x/ T t ), φ L, besides u hom. We consider (2.1) with initial conditions which have, modulo a small perturbation, a maximum at the origin, are slowly varying near the origin and are sufficiently small, but not necessarily vanishing, for large x. In particular, the energy E(u) for such initial conditions might be infinite. We show that the solutions of (2.1) for such initial conditions blowup in a finite time t and we characterize asymptotic dynamics of these solutions.

19 Chapter 2. On blowup in nonlinear heat equations 14 As it turns out, the leading term is given by the expression [ λ(t) 2 c(t) p 1 p 1 + λ 2 (t)(x ζ(t))b(t)(x ζ(t)) ] 1 p 1 where b(t) a real, symmetric n n-matrix b(t) = (b ij (t)), yby := n i,j=1 y ib ij y j (2.8) for a n n-matrix b := (b ij ) and the parameters λ(t), b(t), c(t) and ζ(t) obey certain dynamical equations whose solutions give λ(t) = λ (t t) 1 2 (1 + o(1)) b(t) = (p 1)2 4p ln t t (I + O( 1 ln t t 1/2 )) c(t) = 1 p 1 2p ln t t (1 + O( 1 ln t t )) (2.9) ζ(t) = O(1). with λ() = c + 2 T rb(), c p 1 >, b() > depends on the initial datum. Here o(1) is in t t. Moreover, we estimate the remainder, the difference between u(x, t) and (2.8). In our approach, as in [2], we do not fix the time-dependent scale in the self-similarity (blowup) variables but let its behavior, as well as behavior of other parameters (b and c) to be determined by the equation. We will deal, without specifying it, with weak solutions of Equation (2.1) in the sense detailed in the next section. The local existence of such solutions is well known and is presented for readers convenience in the next section. These solutions can be shown to be classical for t >. In what follows we use the notation x := (1 + x 2 ) 1/2 and f g for two positive functions f and g, satisfying f Cg for some universal constant C. The main result of this chapter is the following theorem. Theorem 3. Let 1 c 4, b := (b ij ) > be a real, symmetric, positive n n-matrix with b 1. Suppose the initial data u L (R n ) for (2.1) satisfy the conditions ( ( ) ) 1 c p 1 x m u (x) δ p 1 + xb x m, (2.1) with m =, 3, δ 1 and δ 3 = C b 2. Then

20 Chapter 2. On blowup in nonlinear heat equations 15 (1) There exists a time t (, ) such that the solution u(x, t) exists on the interval [, t ) and blows up at t t. (2) When t t, there exist unique, positive, C 1 real valued, n-vector valued and n nmatrix valued functions λ(t) and c(t), ζ(t) and b(t), respectively, with b(t) b(), such that u(x, t) can be decomposed as u(x, t) = λ 2 c(t) p 1 (t)[( p 1 + λ 2 (t)(x ζ(t))b(t)(x ζ(t)) ) 1 p 1 + η(x, t)] with the fluctuation part, η, admitting the estimate λ(t)(x ζ(t)) m η(x, t) δ m (t), m =, 3, with δ (t) = δ 1 and δ 3 (t) = b(t) 2. (3) The functions λ(t), b(t), c(t) and ζ(t) are of the form (2.9). We note that no smoothness of initial conditions is required, also note that (2.1) is an L 2 -gradient system t u = grad E(u), with the energy defined in (2.2). Most of the works mentioned above use this fact in an essential way. It is not used here and we expect our analysis can be extended to non-gradient systems. Specific forms of initial conditions and nonlinearity enter into only two places: orthogonal decomposition and Lyapunov-Schmit splitting (Sections 2.4 and 2.8). Note that our approach is closest to that of [2] (see also [89]). Our results extend those of [2] in two aspects. First we address the problem of blow-up in arbitrary dimensions. Second we consider more general initial conditions so that the blowup center is not fixed but in general moves. This chapter is organized as follows. In Section 2.2 we give the local well-posedness of (2.1). In Sections we present some preliminary derivations and some motivations for our analysis. In Section 2.5, we formulate a priori bounds on solutions to (2.1) which are proven in Sections 2.9, 2.12 and We use these bounds in Section 2.6 to prove our main result, Theorem 3. In Sections 2.8, 2.1 and 2.11 we lay the ground work for the proof of the a priori bounds of Section 2.5. In particular, in Section 2.8, using a Lyapunov-Schmidt-type argument we derive equations for the parameters a, b and c and

21 Chapter 2. On blowup in nonlinear heat equations 16 fluctuation η. In Section 2.1 we rescale our equations in a convenient way and in Section 2.11 we estimate the corresponding propagators. As was mentioned above, the results of Sections 2.8, 2.1 and 2.11 are used in Sections 2.9, 2.12 and 2.13 in order to prove the a priori estimates. In Appendix, we prove a convenient form of the Feynmann-Kac-type formula. 2.2 The Local Well-Posedness of (2.1) let f be a locally Lipshitz continuous function, i.e. R > there exists C R > such that for all u, v R with u, v R, f(u) f(v) C R u v. (2.11) We consider the following nonlinear heat equation in R n t u = u + f(u) u(x, ) = u (x). (2.12) Let W s := {u L, ( ) s/2 u L }, a Sobolev space. The next theorem uses the notion of mild solution to (2.12), which is given in the proof. Theorem 4. Let u L, and K := 2 f() + 4C 2 u u, with C R the same as in (2.11). Then there exists t > u K 1 s.t. (2.12) has a unique mild solution in C([, t ), L ); u depends continuously on the initial condition u ; Either t = or t < and u(t) as t t ; If u W s, s >, then t u t max(1 s,).

22 Chapter 2. On blowup in nonlinear heat equations 17 Proof. Using Duhamel s principle, Equation (2.12) can be written as the fixed point equation u = H(u), where where U(t) = e t. H(u)(t) := U(t)u + t U(t s)f(u(s)) ds. (2.13) We say that the equation (2.12) has a mild solution u if u C([, t ), L ) and u solves u = H(u). Let R 2 u, K R := 2 f() + 2C R R, T = RK 1 R and B R := {u C([, T ], L ) sup u(t) R}. t T We show that H maps B R into itself. Indeed, by (2.11) with v = and for u B R, f(u) f() + C R R =: 1 2 K R, and by the explicit integral kernel of U(t), U(t)w w. Therefore, for u B R, H(u) C([,T ],L ) = sup H(u)(t) sup ( U(t)u + t T t T Hence H : B R B R. sup ( u + 1 t T 2 K RT ) R. Next we prove that H : B R B R is a strict contraction. We have t H(u) H(v) C([,T ],L ) U(t s)(f(u(s)) f(v(s))) ds t = sup U(t s)(f(u(s)) f(v(s))) ds t T sup t T sup t T t t t U(t s)(f(u) f(v)) ds U(t s)f(u(s)) ds C([,T ],L ) f(u) f(v) ds sup T f(u)(t) f(v)(t) t T T C R u v C([,T ],L ). Since T RK 1 R (2C R) 1, we conclude that H(u) H(v) C([,T ],L ) 1 2 u v C([,T ],L ). Therefore, H is a strict contraction. Hence there is a unique solution to H(u) = u in the ball B R.

23 Chapter 2. On blowup in nonlinear heat equations 18 To prove that the solution depends continuously on the initial condition u. Let u, v be the solutions with initial conditions u and v. We estimate t u v C([,T ],L ) U(t)(u v ) C([,T ],L ) + U(t s)(g(u(s)) g(v(s))) ds C([,T ],L ) u v u v C([,T ],L ). This implies u v C([,T ],L ) 2 u v, completing the proof of continuity. The dichotomy that either t = or t < and u(t) as t t follows in a standard way from the fact the local existence time depends only on u (and properties of the function f). By the explicit integral kernel of U(t), u(, t) C (R n ) t >. Finally, let u W s, s >. Then differentiating H(u)(t) w.r.to t and using that t U(t) = U(t) = t (1 ɛ) ( t ) 1 ɛ U(t)( ) ɛ and that ( t ) α U(t) is bounded in L for any α, we conclude that u C 1 ([, t ), L ). The relation u C([, t ), W s ) is shown in the same way as the first statement of the theorem. 2.3 Blow-Up Variables and Almost Solutions Let z(t) R n, λ(t) > be differentiable functions and let α(t) satisfy the equation with a(t) = λ(t)/λ 3 (t). We introduce the blowup variables and define the new function λ 2 α aα = λ 1 ż, (2.14) y := λ(t)(x z(t)) α(t) and τ := t λ 2 (s)ds v(y, τ) := λ 2 p 1 (t)u(x, t). (2.15)

24 Chapter 2. On blowup in nonlinear heat equations 19 Plugging (2.15) into (2.1) we obtain τ v = ( y ay y 2a ) v + v p 1 v, (2.16) p 1 where, as above, a(t) = λ(t)/λ 3 (t). The initial condition for this equation is obtained from the initial condition for (2.1) as v(y, ) = λ 2p p 1 u (z + y+α ), for some λ, z and α. From the local well-posedness of (2.1) and using rescaling, we can conclude that there exists T > s.t. (2.16) has a unique mild solution in C([, T ), L ) and the solution depends continuously on the initial condition. Moreover, either T = or T < and v(τ) as τ T. The equation (2.16) has the following family of homogeneous, static (i.e. y and τ- independent) solutions: a is a constant and v a := λ ( ) 1 2a p 1. (2.17) p 1 This family of solutions corresponds to the homogeneous solution (2.7) of the nonlinear heat equation with the parabolic scaling λ 2 = 2a(T t), where the blowup time, T := [ u p 1 (p 1) ] 1, is dependent on the initial value, u of the homogeneous solution u hom (t). If the parameter a is τ dependent but a τ is small, then the above solutions are good approximations to the exact solutions. A richer family of approximate solutions is obtained by solving the equation ay y v + 2a p 1 v = vp, obtained from (2.16) by neglecting the τ derivative and second order partial derivative in y. This equation has the general solution ( v bc := c p 1 + yby ) 1 p 1 (2.18) for c = 2a and all b := (b ij ), b ij R, real, symmetric n n-matrices. Here recall yby := n i,j=1 y ib ij y j. In what follows we take b, so that v bc is nonsingular. Note that v,2a = v a.

25 Chapter 2. On blowup in nonlinear heat equations Reparametrization of Solutions In this section we split solutions to (2.16) into the leading term - the almost solution V ab (y) = v bc := ( c ) 1 p 1+yby p 1, with c = a + 1, - and a fluctuation η around it. More precisely, we would like to parametrize a solution by a point on the manifold M as := {V ab a R +, b R n n } of almost solutions and the fluctuation orthogonal to this manifold (large slow moving and small fast moving parts of the solution). For technical reasons, it is more convenient to require the fluctuation to be almost orthogonal to the manifold M as. More precisely, we require ξ to be orthogonal to the vectors φ (ij) a, i, j n, where 2 φ () a (y) := e a 4 y 2, φ (i) a (y) = φ (i) a (y) := ay i e a 4 y 2, φ (ij) a (y) := ay i y j e a 4 y 2, 1 i, j n, which are almost tangent vectors to the above manifold, provided b is sufficiently small. From now on, we fix the relation between the parameters a and c as c = a Denote by M n the space of real, symmetric, n n matrices and by M + n, the positive cone in this space. Let u λ,z (y) := λ 2 p 1 u(x), with x = z + λ 1 (y + α). We define neighborhoods U ɛ := {v L (R n ) e 1 9 y 2 (v V ab ) = o( b ) for some 1/4 a 1, < b ɛ } and Ũ ɛ := {u L (R n ) u λ,z U ɛ }. The following statement will be used to reparametrize the initial conditions. Proposition 5. There exist an ɛ > and a unique C 1 functional g : U ɛ R + M + n R n, such that any function u λ,z U ɛ can be uniquely written in the form with η φ (ij) a u λ,z = V ab + η, (2.19), i, j n, in L 2 (R n, e a y 2 4 dy), (a, b, z) = g(u λ,z ). Moreover, if 1 a 4 1, < b ε and y m (u λ,z V a b ) δ m with m =, 3, δ 3 = O( b 2 )

26 Chapter 2. On blowup in nonlinear heat equations 21 and δ small, we have g 1 (u λ,z ) (a, b ) b 2, (2.2) g 2 (u λ,z ) z b, (2.21) y 3 (u λ,z V g(uλ,z ))) b 2, (2.22) u λ,z V g(uλ,z ) δ + b. (2.23) for g(u λ,z ) = (g 1 (u λ,z ), g 2 (u λ,z )), where g 1 (u λ,z ) = (a, b) and g 2 (u λ,z ) = z. Proof. Let V λabz (x) := λ 2 p 1 Vab (y), V µ V λabz with µ = (a, b, z), and ϕ (ij) az (x) := e a y 2 4 φ (ij) a with y := λ(x z) α. The orthogonality conditions on the fluctuation can be written as G(µ, u) =, where µ = (a, b, z) and G : R + M + n R n L (R n ) M n+1 is defined as G(µ, u) := V µ u, ϕ (ij) az. Here and in what follows, all inner products are L 2 (R n ) inner products. Whenever it is convinient we identitify µ with a (n + 1) (n + 1) matrix: µ := a, µ i = µ i = z i, µ ij := b ij, 1 i, j n and let M ++ n+1 := {µ M n+1 a, b, z R n } and M n+1,ɛ := {µ M n+1 a [ 1 4, 1], < b ɛ, z Rn }. Let X := e 1 9 y 2 L (R n ) with the corresponding norm. Using the implicit function (y), theorem we will prove that for any µ := (a, b, z ) M ++ n+1,ɛ there exists a unique C 1 function g : X M n+1, defined in a neighborhood Ũµ X of V µ, such that G( g(u), u) = for all u Ũµ. Let B ε (V µ ) and B δ (µ ) be the balls in X and R n+1 around V µ and µ and of the radii ε and δ, respectively. Note first that the mapping G is C 1 and G(µ, V µ ) = for all µ. We claim that the linear map µ G(µ, V µ ) is invertible. Lemma 6. µ G(µ, u), for u Ũɛ, is invertible.

27 Chapter 2. On blowup in nonlinear heat equations 22 Proof. Let the indices α and β run over the pairs (i, j), i j n. We compute µ G(µ, u) = A 1 + A 2 (2.24) where the (α, β) th entries of A 1 and A 2 are A 1 (α, β) = µα V µ, ϕ (β) az (2.25) and A 2 (α, β) = V µ u, µα ϕ (β) az, respectively. We write A 1 in the block form K 11 K 12 K 13 A 1 = K 21 K 22 K 23 K 31 K 32 K 33, where K 11 = (a,b )V diag µ, ϕ ii az, with i n, K22 = b off-diagv µ, ϕ ij az, with 1 i < j n, K 33 = z V µ, ϕ i az, with 1 i n and similarly for the other entries. For b > and small, we compute using change of variable y = λ(x z) α, that (see Appendix B.1 for more details) 2 K 11 = λ n+ n p 1 (2π) 2 (a + 1) 1 p 1 2 a n 2 (p 1) 1+ 1 p 1 is an (n + 1) (n + 1) matrix, 1 a (p 1)a 1 (p 1)a 1 a (p 1)a 1 (p 1)a 1 (p 1)a 1 a a (p 1)a (p 1)a 3 1 (p 1)a (p 1)a (p 1)a 1 (p 1)a 3 (p 1)a + O( b ) (2.26) K 22 = λ n+ 2 a + 1/2 p 1 ( p 1 ) 1 2 p 1 (p 1) 2 a (2π a )n/2 I n(n 1) + O( b ) (2.27) n(n 1) 2 2 and K 33 = λ n+ 2 p 1 ( a + 1/2 p 1 ) 1 p 1 ( 2π a )n/2 b + o( b ) (2.28)

28 Chapter 2. On blowup in nonlinear heat equations 23 is an n n matrix. Moreover, K ij = o( b ) for 1 i j 3. (2.29) Since K 11, K 22 and K 33 are invertible, the matrix A 1 is also invertible. Furthermore, by the Schwarz inequality A 2 u V a b X = o( b ). (2.3) Therefore there exist ε and ε 1 such that the matrix µ G(µ, u) has an inverse for µ M n+1,ɛ and u µ M n+1,ɛ B ε1 (V µ ). Moreover, from (2.24)-(2.3) we know that µ G can be written as µ G = A 11 A 12 A 21 A 22 + R, where A 11 = O(1) and has an O(1) inverse, A 22 = O( b ) and has an O( b 1 ) inverse, A 12 = o( b ) and A 21 = o( b ). Then we have ( µ G) 1 = B 11 B 12 B 21 B 22, (2.31) where B 11 = (A 11 A 12 A 1 22 A 21 ) 1 = O(1), B 22 = (A 22 A 21 A 1 11 A 12 ) 1 = O( b 1 ), B 12 = A 1 11 A 12 (A 22 A 21 A 1 11 A 12 ) 1 = o(1) and B 21 = A 1 22 A 21 (A 11 A 12 A 1 22 A 21 ) 1 = o(1). Hence by the implicit function theorem, the equation G(µ, u) = has a unique solution µ = g(u) on a neighborhood of every V µ, µ M n+1,ɛ, which is C 1 in u. Our next goal is to determine these neighborhoods. To determine a domain of the function µ = g(u), we examine closely a proof of the implicit function theorem. Proceeding in a standard way, we expand the function G(µ, u) in µ around µ : G(µ, u) = G(µ, u) + µ G(µ, u)(µ µ ) + R(µ, u),

29 Chapter 2. On blowup in nonlinear heat equations 24 where R(µ, u) = O ( µ µ 2 ) uniformly in u X. Here µ 2 = a 2 + b 2 + z 2 for µ = (a, b, z). Inserting this into the equation G(µ, u) = and inverting the matrix µ G(µ, u), we arrive at the fixed point problem α = Φ u (α), where α := µ µ and Φ u (α) := µ G(µ, u) 1 [G(µ, u) + R(µ, u)]. By the above estimates there exists an ε 1 such that the matrix µ G(µ, u) 1 is bounded in u B ε1 (V µ ). Define µ b = a + b + b z for µ = (a, b, z), then from (2.31) we have ( µ G) 1 µ b µ. It follows that Φ u (α) b G(µ, u) + α 2. (2.32) Furthermore, using that α Φ u (α) = µ G(µ, u) 1 [G(µ, u) G(µ, u) + R(µ, u)], we obtain that there exist ε ε 1 and δ such that α Φ u (α) 1 for all u B 2 ε(v µ ) and α B δ (). Pick ε and δ so that ε δ b 1. Then, for all u B ε (V µ ), Φ u is a contraction on the ball B δ () and consequently has a unique fixed point in this ball. This gives a C 1 function µ = g(u) on B ε (V µ ) satisfying µ µ δ. An important point here is that since ε b() we have that b > for all V ab B ε (V µ ). Now, clearly, the balls B ε (V µ ) with µ M n+1,ε cover the neighbourhood Ũε. Hence, the map g is defined on Ũε and is unique, and the same is true for the map g, defined on as g(u λ,z ) = g(u), which implies the first part of the proposition. Now we prove the second part of the proposition. G(µ, u) implies G(µ, u) = λ n+ 2 p 1 ( V a b u λ,z, e a y 2 4 φ ij a (y) The definition of the function ), therefore G(µ, u) e 1 9 y2 (u λ,z V a b ). (2.33) This inequality together with the estimate (2.32) and the fixed point equation α = Φ v (α), where α = µ µ and µ = g(u λ,z ), implies g(u λ,z ) µ b e 1 9 y2 (u λ,z V a b ). (2.34)

30 Chapter 2. On blowup in nonlinear heat equations 25 From one of the conditions of the proposition, r.h.s. of (2.34) = O( b 2 ) if a [ 1 4, 1]. The last estimate implies (2.2) and (2.21). Using Equation (2.34) we obtain y 3 (u λ,z V g(uλ,z )) y 3 (u λ,z V µ ) + y 3 (V g(uλ,z ) V µ ) y 3 (u λ,z V µ ) + g(u λ,z ) µ y 3 (u λ,z V µ ), which leads to (2.22). Finally, to prove Equation (2.23), we write u λ,z V g(uλ,z ) u λ,z V a,b + V g(uλ,z ) V a,b. A straightforward computation gives V ab V a b a a + b b b. Since by (2.2), a a + b b = O( b 2 ), we have V ab V a b b. This together with the fact u λ,z V a,b δ completes the proof of (2.23). Now we establish a reparametrization of solution u(x, t) on small time intervals. In Section 2.6 we convert this result to a global reparametrization. In the rest of the section it is convenient to work with the original time t, instead of rescaled time τ. We let I t,δ := [t, t + δ] and define for any time t and constant δ > three sets: A t,δ := C 1 (I t,δ, [1/4, 1]), B t,δ,ɛ := C 1 (I t,δ, M + n,ɛ ) and C t,δ := C 1 (I t,δ, [ 1, 1] n ), where, recall the constant ɛ from Proposition 5. Recall u λ,z (y, t) := λ(t) 2 p 1 u(x, t), with x = z(t) + λ 1 (t)(y + α(t)). Suppose u(, t) is a function such that for some λ > sup b 1 (t) y 3 (u λ,z (, t) V a(t),b(t) ) 1 (2.35) t I t,δ for some a A t,δ, b B t,δ,ɛ, z C t,δ, λ(t) satisfying λ(t ) = λ and λ 3 (t) t λ(t) = a(t) and α(t) satisfying α(t ) = α and t α(t) λ 2 (t)a(t)α(t) + λ(t) t z(t) =. We define the set U t,δ,ɛ,λ,α := {u C 1 (I t,δ, y 3 L (R n )) (2.35) holds for some a A t,δ, b B t,δ,ɛ and z C t,δ}.

31 Chapter 2. On blowup in nonlinear heat equations 26 Proposition 7. Suppose u U t,δ,ɛ,λ,α and λ 2 δ 1. Then there exists a unique C 1 map g # : U t,δ,ɛ,λ,α A t,δ B t,δ,ɛ C t,δ, such that for t I t,δ, u(, t) can be uniquely represented in the form u λ (y, t) = V g# (u)(t)(y) + φ(y, t), (2.36) with (a(t), b(t), z(t)) = g # (u)(t) and φ(, t) φ (ij) a(t) in L2 (R n, e a(t) 4 y 2 dy), λ 3 (t) t λ(t) = a(t) and λ(t ) = λ, (2.37) t α(t) λ 2 (t)a(t)α(t) + λ(t) t z(t) = and α(t ) = α. Proof. For any function a A t,δ, we define a function λ(a, t) := (λ 2 2 Let λ(a)(t) := λ(a, t). Next we define a function t t λ α(a, z)(t) := e 2 (s)a(s)ds α t t t a(s)ds) 1 2. t e t s λ2 (γ)a(γ)dγ λ(s) t z(s)ds. Define the C 1 map G # : C 1 (I t,δ, R + ) C 1 (I t,δ, M + n ) C 1 (I t,δ, R n ) C 1 (I t,δ, R (n+2)(n+1) 2 ) as G # (µ, u)(t) := G(µ(t), u λ(a),z (, t)), where t I t,δ, µ = (a, b, z) and G(µ, u) is the same as in the proof of Proposition 5. The orthogonality conditions on the fluctuation can be written as G # (µ, u) =. Using the implicit function theorem we will first prove that for any µ := (a, b, z ) A t,δ B t,δ,ɛ C t,δ there exists a neighborhood U µ of V µ and a unique C 1 map g # : U µ A t,δ B t,δ,ɛ C t,δ such that G # (g # (v), v) = for all v U µ. We claim that µ G # (µ, u) is invertible, provided u λ(a),z is close to V µ. We compute µ G # (µ, u)(t) = µ G(µ(t), u λ(a),z (, t)) = A(t) + B(t), (2.38) where A(t) := µ G(µ, v) v=uλ(a),z, B(t) := v G(µ, v) v=uλ(a),z µ u λ(a),z. (2.39)

32 Chapter 2. On blowup in nonlinear heat equations 27 Note that in (2.39) v G(µ, v) v=uλ(a),z is acting on µ u λ(a),z as an integral with respect to y and let B(t)(y) be the integral kernel of this operator. We have shown in Lemma 6 that the first term on the r.h.s. is invertible, provided u λ(a),z is close to V µ. Now we show that for δ > sufficiently small the second term on the r.h.s. is small. Let v := u λ(a),z. Assuming for the moment that v is differentiable, we compute a v = a (λ 1 )[ 2 p 1 λv (y + α) yv] + λ 1 a α y v. Combining the last two equations together with Equation (2.39) we obtain [B(t)ρ](t) = B(t)(y)[( 2 p 1 λv + (y + α) yv)( a λ 1 ρ) + λ 1 y v( a α)ρ]dy. Integrating by parts the second term in parenthesis gives 2 [B(t)ρ](t) = [( p 1 λv + v y (y + α))( a λ 1 )ρ + λ 1 v y ( a α)ρ]b(t)[y]dy. (2.4) Furthermore, a (λ 1 )ρ = λ(t) t t ρ(s)ds and t t t λ ( a α)ρ = e 2 (s)a(s)ds α [a(s) a λ 2 (s) + λ 2 (s)]ρ(s)ds t t t e t s λ2 (γ)a(γ)dγ t z(s)[λ(s) t s (a(γ) a λ 2 (γ) + λ 2 (γ))ρ(γ)dγ + a λ(s)ρ(s)]ds. Now, using a density argument, we remove the assumption of the differentiability on v and conclude that (2.4) holds without this assumption. Using this expression and the inequality λ(t) 2λ, provided δ (4 sup a) 1 λ 2 1/4λ 2, we estimate B(t)ρ L ([t,t +δ]) δλ 2 v L ρ L ([t,t +δ]). (2.41) So B(t) is small, if δ (λ 2 v L ) 1, as claimed. This shows that µ G # (µ, u) is invertible, provided u λ(a),z is close to V µ. Proceeding as in the proof of Proposition 5 we conclude the proof of Proposition A priori Estimates Let u(x, t), t T be a solution to (2.1) with initial condition u U ɛ and v(y, τ) = λ 2 p 1 (t)u(x, t), where y = λ(x z) α and τ(t) := t λ2 (s)ds. We assume that there

33 Chapter 2. On blowup in nonlinear heat equations 28 exist C 1 functions a(τ) and b(τ) such that v(y, τ) can be represented as ( v(y, τ) = c(τ) p 1 + yb(τ)y ) 1 p 1 + e a(τ) 4 y 2 ξ(y, τ), (2.42) where ξ(, τ) φ (ij) a(τ) (see (2.19)), λ 3 (t) t λ(t) = a(τ(t)), c = a Since u U ɛ, by Proposition 5, y 3 e a 4 y 2 ξ(y, ) b() 2. (2.43) In this section we formulate a priori bounds on the fluctuation ξ which are proved in later sections. Let the function β(τ), the constant κ be defined as β(τ) := (b() 1 + 4pτ (p 1) 2 I) 1 and κ := min{ 1 2, p 1 }, (2.44) 2 and let β(τ) is the largest eigenvalue of β(τ). For the functions ξ(τ), b(τ) and a(τ) we introduce the following estimating functions (families of semi-norms) M 1 (T ) := max τ T β 2 (τ) y 3 e a 4 y 2 ξ(τ), M 2 (T ) := max τ T e a 4 y 2 ξ(τ), A(T ) := max τ T β 2 (τ) a(τ) 1 2 B(T ) + 2T rb(τ) p 1 := max τ T β (1+κ) (τ) b(τ) β(τ)., (2.45) Proposition 8. Let ξ to be defined in (2.42) and we assume M 1 (), A(), B() 1, M 2 (). Assume there exists an interval [, T ] such that for τ [, T ], M 1 (τ), A(τ), B(τ) β κ/2 (τ). Then in the same time interval the parameters a, b and the function ξ satisfy the following estimates τ b(τ) + 4p (p 1) 2 b2 (τ) β 3 (τ) + β 3 (τ)m 1 (τ)(1 + A(τ)) + β 4 (τ)m1 2 (τ) + β 2p M 2p 1 (τ), and (2.46) B(τ) 1 + M 1 (τ)(1 + A(τ)) + M 2 1 (τ) + M p 1 (τ), (2.47)

34 Chapter 2. On blowup in nonlinear heat equations 29 A(τ) A() β()m 1 (τ)(1 + A(τ)) + β()m 2 1 (τ) + β 2p 2 ()M p 1 (τ), (2.48) M 1 (τ) M 1 () + β κ 2 ()[1 + M 1 (τ)a(τ) + M 2 1 (τ) + M p 1 (τ)] +[M 2 (τ)m 1 (τ) + M 1 (τ)m p 1 2 (τ)], M 2 (τ) M 2 () + β 1/2 ()M 1 () + β 1 3 ()M (T )M (T ) + M 2 2 (τ) + M p 2 (τ) +β κ 2 ()[1 + M 2 (τ) + M 1 (τ)a(τ) + M 2 1 (τ) + M p 1 (τ)]. (2.49) (2.5) Equations (2.46)-(2.48), (2.49) and (2.5) will be proved in Sections 2.9, 2.12 and 2.13 respectively. Corollary 9. Let ξ to be defined in (2.42) and assume M 1 (), A(), B() 1, M 2 () 1. Assume there exists an interval [, T ] such that for τ [, T ], M 1 (τ), A(τ), B(τ) β κ/2 (). Then in the same time interval the parameters a, b and the function ξ satisfy the following estimates M 1 (τ), A(τ), B(τ) 1, M 2 (τ) 1. (2.51) (In fact, M i (τ) M i () + β κ 2 (), i = 1, 2.) Proof. Since β(τ) β() 1, we have, by the continuity (or by Proposition 5), that for a sufficiently small time interval M 1 (τ), B(τ), A(τ) β κ 2 () β κ 2 (τ), (2.52) where, recall, the definitions of β(τ) and κ are given in (2.44). Then Equations ( 2.47)- ( 2.5) imply that for the same time interval M 1 (τ), B(τ), A(τ) 1, M 2 (τ) 1. (2.53) Thus the conditions of the proposition above are satisfied. Since M 1 (τ) β κ 2 (), we can solve (2.48) for A(τ). We substitute the result into Equations (2.49) - (2.5) to obtain inequalities involving only the estimating functions M 1 (τ) and M 2 (τ). Consider

35 Chapter 2. On blowup in nonlinear heat equations 3 the resulting inequality for M 2 (τ). The only terms on the r.h.s., which do not contain β() to a power at least κ/2 as a factor, are M 2 2 (τ) and M p 2 (τ). Hence for M 2 () 1 this inequality implies that M 2 (τ) M 2 () + β κ 2 (). Substituting this result into the inequality for M 1 (τ) we obtain that M 1 (τ) M 1 () + β κ 2 () as well. The last two inequalities together with (2.47) and (2.48) imply the desired estimates on A(τ) and B(τ). 2.6 Proof of Main Theorem 3 We start with an auxiliary statement which eases the induction step. Recall the notation I t,δ := [t, t +δ]. We say that λ(t) is admissible on I t,δ if λ C 1 (I t,δ, R + ) and λ 3 t λ [1/4, 1]. Recall that t is the maximal existence time defined in Section 2.2. Lemma 1. Assume u C 1 ([, t ), x 3 L ), t [, t ) and u λ (, t ) U ɛ /2 for some λ and for ɛ given in Proposition 5. Then there are δ = δ(λ, u) > and λ(t), admissible on I t,δ, s.t. (2.36) and (2.37) hold on I t,δ. Proof. The conditions u C 1 ([, t ), x 3 L ) and u λ (t ) U ɛ /2 imply that there is a δ = δ(λ, u) s.t. u U t,δ,ɛ,λ. By Proposition 7, the latter inclusion implies that there is λ(t), admissible on I t,δ, λ(t ) = λ, s.t. (2.36) and (2.37) hold on I t,δ. Choose b so that C b ɛ with C the same as in (2.1) and with ɛ given in Proposition 5. Let v (y) := λ 2 p 1 u (z + λ 1 y). Then v U 1 2 ɛ, by the condition (2.1) with m = 3, on the initial conditions. Hence Proposition 5 holds for v and we have the splitting (2.19). Denote g(v ) =: (a(), b(), z()). Furthermore, by Lemma 1 there are δ 1 > and λ 1 (t), admissible on [, δ 1 ], s.t. λ 1 () = λ and Equations (2.36) and (2.37) hold on the interval [, δ]. Hence, in particular, the estimating functions M 1 (τ), M 2 (τ), A(τ) and B(τ) of Section 5 are defined on