## Juan Luis Vazquez

Print publication date: 2006

Print ISBN-13: 9780198569039

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198569039.001.0001

# INTRODUCTION

Chapter:
(p. 1 ) 1 INTRODUCTION
Source:
The Porous Medium Equation
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198569039.003.0001

# Abstract and Keywords

This chapter introduces the mathematical theory of the nonlinear heat equation referred to as the porous medium equation (PME). This equation is one of the simplest examples of a nonlinear evolution equation of parabolic type. The peculiar features of the PME and nonlinear diffusion are discussed. An overview of the chapters in this volume is presented.

# 1.1 The subject

## 1.1.1 The porous medium equation

The aim of the text is to provide a systematic presentation of the mathematical theory of the nonlinear heat equation (PME) $Display mathematics$ usually called the porous medium equation, with due attention paid to its closest relatives. The default settings are: u = u(x, t) is a non-negative scalar function of space x ∈ R d and time t ∈ R, the space dimension is d ≥ 1, and m is a constant larger than 1. Δ = Δ x represents the Laplace operator acting on the space variables. We will refer to the equation by the label PME. The equation can be posed for all x ∈ R d and 0 < t < ∞, and then initial conditions are needed to determine the solutions; but it is quite often posed, especially in practical problems, in a bounded subdomain Ω ⊂ R d for 0 < t < T, and then determination of a unique solution asks for boundary conditions as well as initial conditions.

This equation is one of the simplest examples of a nonlinear evolution equation of parabolic type. It appears in the description of different natural phenomena, and its theory and properties depart strongly from the heat equation, u t = Δu, its most famous relative. Hence the interest of its study, both for the pure mathematician and the applied scientist. We will also discuss in less detail some important variants of the equation.

There are a number of physical applications where this simple model appears in a natural way, mainly to describe processes involving fluid flow, heat transfer or diffusion. Maybe the best known of them is the description of the flow of an isentropic gas through a porous medium, modelled independently by Leibenzon [367] and Muskat [394] around 1930. An earlier application is found in the study of groundwater infiltration by Boussisnesq in 1903 [123]. Another important application refers to heat radiation in plasmas, developed by Zel'dovich and coworkers around 1950 [533]. Indeed, this application was at the base of the rigorous mathematical development of the theory. Other applications have been proposed in mathematical biology, spread of viscous fluids, boundary layer theory, and other fields.

Most physical settings lead to the default restriction u ≥ 0, which is mathematically convenient and currently followed. However, the restriction is not (p. 2 ) essential in developing a mathematical theory on the condition of properly defining the nonlinearity for negative values of u so that the equation is still (formally) parabolic. The most used choice is the antisymmetric extension of the nonlinearity, leading to the so-called signed PME, (sPME) $Display mathematics$ We will also devote much attention to this equation. For brevity, we will often write u m instead of |u| m−1 u even if solutions have negative values in paragraphs where no confusion is to be feared. There is a second important extension, consisting of adding a forcing term in the right-hand side to get the complete form (cPME) $Display mathematics$ where f = f(x, t). The full form is the natural framework of the abstract functional theory for the PME, and has also received much attention when f = f(u) and represents effects of reaction or absorption. The dependence of f on ∇u occurs when convection is taken into account. We will cover the complete form in the text, but the information on the qualitative and quantitative aspects is much less detailed in that generality, and we will not enter into the specific properties of reaction-diffusion models. Specially in the second part of the book, we want to concentrate on the plain equation (PME), hence the simple label for that case. The complete porous medium equation is also referred to as the PME with a source term, or the forced PME.

Equation (PME) for m = 1 is the famous heat equation (HE), that has a well documented theory, cf. Widder [525]. The equation can also be considered for the range of exponents m < 1. Some of the properties in this range are similar to the case m > 1 studied here, but others are quite different, and it is called the fast diffusion equation (FDE). Since it deserves a text of its own, the FDE will only be covered in passing in this book. Note that when m < 0 the FDE has to be written in the ‘modified form’ $Display mathematics$ to keep the parabolic character of the equation. This form of the equation allows us also to include the case m = 0 which reads ∂ t u = div(u −1u) = Δlog(u), and is called logarithmic diffusion.

## 1.1.2 The PME as a nonlinear parabolic equation

The PME is an example of nonlinear evolution equation, formally of parabolic type. In a sense, it is the simplest possible nonlinear version of the classical heat equation, which can be considered as the limit m → 1 of the PME. Written in its complete version and in divergence form, (1.1) $Display mathematics$ (p. 3 ) we see that the diffusion coefficient D(u) of the PME equals mu m−1 assuming u ≥ 0, and we have D(u) = m|u| m−1 for signed solutions (D(u) = |u| m−1 in the modified form). It is then clear that the equation is parabolic only at those points where u ≠ 0, while the vanishing of D(u) is recorded as saying that the PME degenerates wherever u = 0. In other words, the PME is a degenerate parabolic equation. The theory of nonlinear parabolic equations in divergence form deals with the class of nonlinear parabolic equations of the form (1.2) $Display mathematics$ where the vector function 𝓐 = (A 1, …, A d ) and the scalar function 𝓑 satisfy suitable structural assumptions and 𝓐 satisfies moreover ellipticity conditions. This topic became a main area of research in PDEs in the second half of the last century, when the tools of functional analysis were ready for it. The theory extends to systems of the same form, in which u = (u 1, …, u k ) is a vector variable, 𝓐 is an (m, d) matrix and 𝓑 is an m-vector. Well-known areas, like reaction-diffusion, are included in this generality. There is a large literature on this topic, cf. e.g. the books [239, 357, 482] that we take as reference works.

The change of character of the PME at the level u = 0 is most clearly demonstrated when we perform the calculation of the Laplacian of the power function in the case m = 2; assuming u ≥ 0 for simplicity, we obtain the form (1.3) $Display mathematics$ It is immediately clear that in the regions where u ≠ 0 the leading term in the right-hand side is the Laplacian modified by the variable coefficient 2u; on the contrary, for u → 0, the equation simplifies into ∂ t u ∼ 2|∇u|2, the eikonal equation (a first-order equation of Hamilton-Jacobi type, that propagates along characteristics). A similar calculation can be done for general m ≠ 1 after introducing the so-called pressure variable, v = cum −1 for some c ≥ 0. We then get (1.4) $Display mathematics$ with a = m/c, b = m/(c(m − 1)). This is a fundamental transformation in the theory of the PME that allows us to get similar conclusions about the behaviour of the equation for u, v ∼ 0 when m ≠ 2. The standard choice for c in the literature is c = m/(m − 1), because it simplifies the formulas (a = m − 1, b = 1) and makes sense for dynamical considerations (to be discussed in Section 2.1), but c = 1 is also used. Mathematically, the choice of constant is not important.

Note that similar considerations apply to the FDE but then (1.5) $Display mathematics$ hence the name of fast diffusion which is well deserved when u ∼ 0. The pressure can be introduced, but being an inverse power of u, its role is different from that in the PME. All this shows the kinship and differences from the start between the two equations.

(p. 4 ) In spite of the simplicity of the equation and of having some important applications, a mathematical theory for the PME has been developed at a slow pace over several decades, due most probably to the fact that it is a nonlinear equation, and also a degenerate one. Though the techniques depart strongly from the linear methods used in treating the heat equation, it is interesting to remark that some of the basic techniques are not very difficult nor need a heavy machinery. What is even more interesting, they can be applied in, or adapted to, the study of many other nonlinear PDEs of parabolic type. The study of the PME can provide the reader with an introduction to, and practice of some interesting concepts and methods of nonlinear science, like the existence of free boundaries, the occurrence of limited regularity, and interesting asymptotic behaviour.

# 1.2 Peculiar features of the PME

When considering the linear and quasilinear parabolic theories, the main questions are asked in comparison to what happens for the heat equation, which is the model from which these theories take their inspiration. Thus, the three main questions of existence, uniqueness, and continuous dependence are posed in the literature, as well as the questions of regularity, the validity of maximum principles, the existence of Harnack inequalities, and so on; in some sense, these comparative questions receive positive answers, though the analogy breaks at some points, thus originating novelty and interest.

## 1.2.1 Finite propagation and free boundaries

The same golden rule of comparison with the HE is applied to the theory developed in this book for the PME. The main questions can be posed, but then we see that such questions, though important, do not convey the special flavour of the equation. Indeed, the PME offers a number of very peculiar traits that separate it from the core of the parabolic theory. Mathematically, the difficulties stem from the degenerate character, i.e., the fact that D(u) is not always positive. Explaining the consequences implies changing the way the heat equation theory is developed. We will be led to introducing dynamical concepts to account for the main qualitative difference, which is the property called finite propagation that will be precisely formulated and extensively explored in the text, especially in Chapters 14 and 15. This property is in strong contrast with one of the better known properties of the classical heat equation, the infinite speed of propagation, one of the most contested aspects of the HE on physical grounds. Let us express the contrast in simplest terms:

• HE: ‘A non-negative solution of the heat equation is automatically positive everywhere in its domain of definition’; to be compared with

• PME: ‘Disturbances from the level u = 0 propagate in time with finite speed for solutions of the porous medium equation’.

(p. 5 ) In a sense, the property of finite propagation supports the physical soundness of the PME to model diffusion or heat propagation.

A first consequence of the finite propagation property for the theory of the PME is that the strong maximum principle cannot hold. On the positive side, it means that, whenever the initial data are zero in some open domain of the space, the property of finite propagation implies the appearance of a free boundary that separates the regions where the solution is positive (i.e. where ‘there is gas’, according to the standard interpretation of u as a gas density, see Chapter 2), from the ‘empty region’ where u = 0. Precisely, we define the free boundary as (1.6) $Display mathematics$ where Q is the domain of definition of the solution in space-time, (1.7) $Display mathematics$ is the positivity set, and ∂ denotes boundary. Since Γ moves as time passes, it is also called the moving boundary. In some cases, especially in one space dimension, the name interface is popular.

The theory of free boundaries, or propagation fronts, is an important and difficult subject of the mathematical investigation, covered for instance in the book by A. Friedman [240]. In principle, the free boundary of a nonlinear problem can be a quite complicated closed subset of Q. A main problem of the PME theory consists of proving that it is at least a Hölder continuous (C α) hypersurface in R d+1, and then to investigate how smooth it really is. Let us advance that it is often C smooth, but not always.

Let us illustrate the two main situations that will be encountered. In the first of them, the space domain is R d , the initial data u 0 have compact support, i.e., there exists a bounded closed set S 0 ⊂ R d such that u 0(x) = 0 for all xS 0. In that case, we will prove that the solution u(x, t) vanishes for all positive times t > 0 outside a compact set that changes with time. More precisely, if we define the positivity set at time t as 𝓟 u (t) = {x ∈ R d : u(x, t) > 0}, and the support at time t as 𝓢 u (t) as the closure of 𝓟 u (t), then both families of bounded sets are shown to be expanding in time, or more precisely stated, non-contracting. Note that positivity sets and supports are not defined in the everywhere sense unless solutions are continuous; showing continuity of the solutions is a main issue in the PME theory, and it has been a hot topic in nonlinear elliptic and parabolic equations since the seminal papers of De Giorgi, Nash and Moser.

In the second scenario, the initial configuration ‘has a hole in the support’, i.e., there is a bounded subdomain D 0 ≠ ∅ such that u 0(x) = 0 for every x in the closure of D 0, and u 0(x) > 0 otherwise. Then, the solution has a possibly smaller hole for t > 0. The fact that this hole does disappear in finite time (it is filled up), motivates one of the most beautiful mathematical developments of the PME theory, the so-called focusing problem that we will study in Chapter 19.

## (p. 6 ) 1.2.2 The role of special solutions

Following a standard practice in applied nonlinear analysis and mechanics, before developing a fully fledged theory, the question is posed whether there exist special solutions in explicit or quasi-explicit form that serve as representative examples of the typical or peculiar behaviour. The answer to that question is positive in our case; a reduced number of representative examples have been found and they give both insight and detailed information about the most relevant questions, like existence, finite propagation, optimal continuity, higher smoothness, and so on.

A fundamental example of solution was obtained around 1950 in Moscow by Zel'dovich and Kompaneets [532] and Barenblatt [60], who found and analysed a solution representing heat release from a point source. This solution has the explicit formula (1.8) $Display mathematics$ where (s)+ = max{s, 0}, (1.9) $Display mathematics$ and C > 0 is an arbitrary constant. The solution was subsequently found by Pattle [418] in 1959. The name source-type solution is due to the fact that it takes as initial data a Dirac mass: as t → 0 we have 𝓤(x, t) → Mδ(x), where M is a function of the free constant C (and m and d). We will use the shorter term source solution, and very often the name ZKB solution that looks to us convenient. We recall that the names Barenblatt solution and BarenblattPattle solution are found in the literature.

An analysis of this example shows many of the important features that we have been talking about. Thus, the source solution has compact support in space for every fixed time, since the free boundary is the surface given by the equation (1.10) $Display mathematics$ where c = c(C, m, d). In physical terms, the disturbance propagates with a precise finite speed. This is to be compared with the properties of the Gaussian kernel, $Display mathematics$ which is the source solution for the HE.

There are many other special solutions that have been studied and shed light on different aspects of the theory. Some of the most important will be carefully examined in Chapter 4 and then used in the theory developed in this text. They take the main forms of separate-variables solutions, travelling waves and self-similar solutions. Chapter 16 is entirely devoted to constructing solutions. They (p. 7 ) play a prominent role in Chapters 18 and 19, where the focusing solutions have a key part in settling the regularity issue.

# 1.3 Nonlinear diffusion. Related equations

The PME is but one example of partial differential equation in the realm of what is called nonlinear diffusion. Work in that wide area has frequent overlaps between the different models, both in phenomena to be described, results to be proved and techniques to be used. A quite general form of nonlinear diffusion equation, as it appears in the specialized literature, is (1.12) $Display mathematics$ Suitable conditions should be imposed on the functions H and A i . In particular, ∂ u H(x, t, u) ≥ 0 and the matrix (a ij ) = (∂ uj A i (x, t, u, Du)) should be positive semidefinite. If we want to consider reaction and convection effects, the term 𝓑(x, t, u, Du) is added to the right-hand side. A theory for equations in such a generality has been in the making during the last few decades, but the richness of phenomena that are included in the different examples covered in the general formulation precludes a general theory with detailed enough information.

Progress has been quite remarkable on more specialized topics like ours. Let us mention next four natural extensions of the PME in that direction. Though they have some important traits in common with the PME, they are different territories and we think that the deep study deserves a separate text in each case.

(i) Fast diffusion. Much of the theory can be and has been extended to the simplest generalization of the PME consisting of the same formal equation, but now in the range of exponents m < 1. Since the diffusion coefficient D(u) = |u| m−1 goes now to infinity as u → 0, the equation is called in this new range the fast diffusion equation, FDE. In this terminology, the PME becomes a slow diffusion equation.

There are strong analogies and also marked differences between the PME and the FDE. For instance, the free boundary theory of the PME disappears for the FDE. We will only make small incursions into it. We refer to the monograph [515] and its references as a source of further information.

(ii) Filtration equations. A further extension is the generalized porous medium equation, (GPME) $Display mathematics$ also called the filtration equation, specially in the Russian literature; Ф is an increasing function: R + ↦ R +, and usually f = 0. The diffusion coefficient is now D(u) = Ф′(u), and the condition Ф′(u) ≥ 0 is needed to make the equation formally parabolic. Whenever Ф′(u) = 0 for some u ∈ R, we say that the equation (p. 8 ) degenerates at that u-level, since it ceases to be strictly parabolic. This is the cause for more or less serious departures from the standard quasilinear theory, as we have already explained in the PME case.

An important role in the development of the topic of the filtration equation has been played by the Stefan problem, a simple but powerful model of phase transition, developed in the study of the evolution of a medium composed of water and ice. It can written as a filtration equation with (StE) $Display mathematics$ More generally, we can put Ф(u) = c 1(uL)+ for u ≥ 0, and Ф(u) = c 2 u for u < 0, where c 1, c 2 and L are positive constants. The Stefan problem and the PME have had a somewhat parallel history.

Note Due to the interest of other GPME models, we will develop a large part of the basic existence and uniqueness theory of this book for the GPME, and we will then specialize to the PME in the detailed analysis of the last part of the book.

(iii) p-Laplacian evolutions. There is another popular nonlinear degenerate parabolic equation: (PLE) $Display mathematics$ called the p-Laplacian evolution equation, PLE, which has also attracted much attention from researchers. It is part of a general theory of diffusion with diffusivity depending on the gradient of the main unknown. It has a parallel, sometimes divergent, sometimes convergent theory. We can combine PME and PLE to get the so-called doubly nonlinear diffusion equation (DNDE) $Display mathematics$ Though these equations have many similarities with the PME, we will not deal with them in this book.

(iv) PME with lower order terms. These are equations of the form (1.13) $Display mathematics$ We have written the general filtration diffusion, but Ф(s) = |s| m−1 s gives the PME. The lower order term takes several forms in the applications. The best known are:

1. (1) the form B = f(u) is a homogeneous reaction term, and the full equation is then a PME-based reaction–diffusion model; when f ≤ 0 we have the nonlinear diffusion-absorption model that has been studied extensively;

2. (2) when B = a. ∇u q we have a convection term; a famous example is the Burgers equation u t + uu x = μ u xx ;

3. (3) when B = |∇u|2 we have a diffusive Hamilton–Jacobi equation.

(p. 9 ) We can see these latter equations as particular cases of the complete PME, but this could be misleading: their theory is quite rich. Of particular interest are the equations of the form (1.14) $Display mathematics$ called Fokker-Planck equations. The extra term stands for a confining effect due to a potential V. In the case V(x) = c|x|2 these equations are closely connected to the study of the asymptotic behaviour of the plain PME/HE/FDE after a convenient rescaling (see details in Chapter 18).

# 1.4 Contents

In a classical mathematical style, the foundation of the book is the study of existence, uniqueness, stability and practical construction of suitably defined solutions of the equation plus appropriate initial and boundary data. This theory uses the machinery of nonlinear functional analysis, as developed extensively in the last century. In the spirit of this theory, classical concepts of solution do not suffice, which leads to the introduction of suitable concepts of generalized solution, in the concrete form of weak, limit, strong and mild solution, among others.

## 1.4.1 The main problems and the classes of solutions

There are three main problems that are posed in parabolic theories:

• Problem A is the initial value problem in the whole space, x ∈ R d , d ≥ 1, for a time 0 < t < T with T finite or infinite. It is usually called the Cauchy problem, CP, and is considered the reference problem in the literature about the PME. It is usually posed for non-negative solutions without a forcing term (u ≥ 0 and f = 0), but we will also study it for signed solutions, and with a forcing term.

• Problem B is posed in a subdomain Ω or R d , and the additional data include initial conditions and boundary conditions of Dirichlet type, u(x, t) = g(x, t) for x ∈ ∂Ω and 0 < t < T. The same observations on the sign of u and on f apply. By default Ω is bounded, u ≥ 0, f = 0, and g = 0.

• Problem C is similar to Problem B, but the data on the lateral boundary are Neumann data, ∂ n u m (x, t) = h(x, t). By default, Ω is bounded and f = 0, h = 0.

There is a number of other problems posed on spatial domains Ω with more general conditions of mixed or nonlinear type. In one space dimension a typical problem is posed in a semi-infinite domain Ω = (0, ∞). Typical data in that case are u(0, t) = C or (u m ) x (0, t) = 0.

Once the problems are shown to be well-posed in suitable functional settings, the next question is the study of the main qualitative properties. Prominent (p. 10 ) among them is the phenomenon of finite propagation and its consequences in the form of free boundaries. The emphasis shifts now into dynamical considerations and differential geometry.

A third important subject related to both previous ones is optimal regularity. Let us illustrate it on the source-type solution. We have seen that it is continuous in its domain of definition Q = R d × R +. However, it is not smooth at the free boundary, again a consequence of the loss of the parabolic character of the equation when u vanishes. In fact, the function u m−1 is Lipschitz continuous in Q with jump discontinuities on Γ (i.e., there exists a regularity threshold). On the contrary, the solution is C -smooth in 𝓟 u . And we are interested in noting that though u is not smooth on Γ, nevertheless the free boundary is a C smooth surface given by the equation (1.10). However, not all free boundaries of solutions of the PME will be so smooth.

## 1.4.2 Chapter overview

The book is organized as follows. After this Introduction, we review the main applications in Chapter 2. This pays homage to the fundamental role played by these applications in motivating the mathematical research and supplying it with problems, intuitions, concepts and conjectures.

We continue with two preparatory chapters. In Chapter 3 we review the main facts and introduce the basic estimates we will need later in a classical framework. Chapter 4 examines the fundamental examples, and we use the opportunity to present in a simple and practical context some of the main topics of the theory, like the property of finite propagation, the appearance of free boundaries, the need for generalized solutions and the question of limited regularity. It even shows cases of blow-up and the evolution of signed solutions.

This gives way to the study of the classical problems of existence, uniqueness and regularity of a (generalized) solution for the tree main problems mentioned above. There have been two basic approaches to the existence theory for the PME in the literature: one of them is the so-called semigroup approach based on posing the problem in the abstract setting of ODEs in Banach spaces; the other one uses a priori estimates, approximation by related smooth problems (to which the estimates apply uniformly), and passage to the limit. Though both approaches have been fruitful, we have chosen to give priority to the latter, which uses as a cornerstone the preparatory work of Chapter 3. It is used in Chapters 5, 6 and 8 to study the Dirichlet boundary value problem, and in Chapter 9 to treat the Cauchy problem. An intermediate Chapter 7 establishes the continuity of the constructed solutions. Chapter 10 presents the semigroup approach which is very different in spirit and has had a fundamental importance in the historical development of the whole subject. The whole set of ideas is used Chapter 11 to treat the Neumann problem as well as the problems posed on Riemannian manifolds. This completes the first half of the book. Three remarks are in order: (p. 11 )

1. (i) At this general level, there is an interest in considering not only the PME but rather a wider class of equations to which most of the methods apply. This is why a large part of the material is derived for the class of complete generalized porous medium equations, $Display mathematics$

2. (ii) For reasons of simplicity at this stage, most of the treatment is restricted to integrable data, a sound assumption on physical grounds, though not necessary from the point of view of mathematical analysis, as the sequel will show.

3. (iii) A main point of the study is the introduction of the different types of generalized solution that appear in the literature and are natural to the problem, and the careful analysis of their scope and mutual relationships.

With this foundation, the second part of the book enters into more peculiar aspects of the theory of the PME; existence with optimal data, free boundaries, self-similar solutions, higher regularity, symmetrization and asymptotics; though relying on the previous foundation, the new material is not necessarily more difficult, and the aspects it covers can probably be more attractive for many active researchers, both for theoretical or practical purposes.

Let us examine the contents of the different chapters in this part. The existence and uniqueness theory is complemented with two beautiful chapters on solutions for general classes of data, i.e., data that are not assumed to be either integrable or bounded. Chapter 12 covers the theory of solutions with so-called growing data. Optimal growth conditions are found that allows for a theory of existence and uniqueness. Chapter 13 extends the analysis to solutions whose initial value (so-called trace) is a Radon measure.

We are now ready for the main topics of the qualitative theory, which are covered in the next block of four chapters. The propagation properties, another fundamental topic in the PME theory, are discussed in detail in Chapter 14, including all questions related to finite propagation, free boundaries and evolution of the support.

The PME theory in several space dimensions presented many difficulties and was developed at a slow pace. Much of the earlier progress focused on understanding the basic questions in a one-dimensional setting. Actually, we have a much more detailed knowledge in that case, and we devote Chapter 15 to present the main features, like the 1D free boundary.

Chapter 16 contains the full analysis of self-similarity, which plays a big role in the theory of the PME.

Chapter 17 deals with the principles of symmetrization and concentration and their applications.

We devote the next three chapters to the questions of asymptotic behaviour as t goes to infinity and higher regularity. Chapter 18 does the asymptotics for the Cauchy problem, and Chapter 20 for the homogeneous Dirichlet problem. (p. 12 ) The former contains the famous result on stabilization of the integrable solutions of the PME towards the ZKB profile which is the analogue for m > 1 of the convergence towards the Gaussian profiles of the solutions of the heat equation. Since this convergence is a way of expressing the central limit theorem of probability theory, the convergence of the PME flow towards the ZKB is a nonlinear central limit theorem.

Chapter 19 examines the actual regularity of the solutions of the Cauchy problem; it concentrates on describing two of the main results for non-negative and compactly supported solutions: the Lipschitz continuity of the pressure and the free boundary for large times and the lesser regularity for small times of the so-called focusing solutions (or hole-filling solutions). Partial C regularity is also shown according to Koch, and the concavity properties according to Daskalopoulos and Hamilton and Lee and Vázquez.

The last two chapters gather complements on the previous material. We devote Chapter 21 to collect further applications to the physical sciences.

We will use notations that are rather standard in PDE texts, like Evans [229], Gilbarg-Trudinger [261] or equivalent, which we assume known to the reader. A detailed summary of the main basic concepts and notations of real and functional analysis is contained in the Appendix. This chapter also contains a number of technical appendices on material that is used in the book and was considered not to have a place in the main flow of the text. One of these results is the proof of the lack of contractivity of the PME flow in L p spaces with p large, which answers a question raised by some experts and posed some open problems.

## 1.4.3 What is not covered

This is a basic book on a very rich subject that keeps growing in many exciting directions. We list here some of the topics where much progress has been made and have been nevertheless left out of the presentation.

1. (1) The theory of the so-called limit cases of the PME. First, the limit m → 1, where we can get either the heat equation or the eikonal equation, u t = |∇u|2, depending on the scaling of the data [50, 375]. We also have the limit as m → ∞, leading to the famous Mesa problem [85, 141, 242, 463].

2. (2) The detailed treatment of the fast diffusion equation. The reader can find an expository account at a rather advanced level in the author's Lecture Notes [515]. A whole set of references is given.

3. (3) The more detailed study of the behaviour for large times, using recent work on gradient flows, optimal transportation and the entropy-entropy dissipation method [155, 413].

Also, the question of asymptotic geometry, in particular the question of asymptotic concavity, cf. [196, 197, 365].

4. (4) The theory of viscosity solutions for the PME developed by Caffarelli and Vázquez, [144], see also [125, 332].

5. (p. 13 )
6. (5) More general boundary value problems: the general Dirichlet problem, and then Neumann and mixed problems.

7. (6) The Lagrangian approach and particle trajectories, as developed in [279, 389]. See also [515].

8. (7) Numerical computation of PME flows, see [232].

9. (8) Stochastic versions of the porous medium equation, as in the work of da Prato et al. [192].

10. (9) The porous medium equation posed on a Riemannian manifold [121, 413]. See Section 11.5 below.

Of course, we have left out the developments for parallel equations and models, though their mathematical development has been closely connected to that of the PME, like

1. (i) The combination of nonlinear diffusion and reaction or absorption. This is a classical area where a wide literature exists.

2. (ii) The combined models involving nonlinear diffusion and convection, like u t = ΔФ(u) + ∇ · F(u). This has been a very active area of research in recent years.

3. (iii) Gradient flows and p-Laplacian equations, and their relation with the PME in 1D.

4. (iv) The detailed study of the so-called dual equation, v t = (Δu) m .

The whole book is aimed at providing a comprehensive coverage that hopes to be useful both to the beginning researcher as a text, and to the specialist as a reference. For that purpose, it is organized in blocks of different difficulty and scope.

While trying to present the most relevant basic results with whole proofs in each chapter, a parallel effort has been made to present an informative panorama of the relevant results known about the topics of the chapter. However, and especially in the second part of the book, many interesting results that can be easily traced and read in the sources were discussed more briefly by evident reasons of space. The more advanced sections have been marked with a star, *. On the other hand, we have included the proof of many new results that the author felt were needed to complete the presentation and were not reported in the literature. Chapters contain detailed introductions where the topics to be covered are announced and commented upon, and are supplied with a final section of Notes (comments, historical notes or recommended reading) and a list of problems. Problems contain many bits of proofs and some are used in later chapters. Solving them is recommended to the reader, since we believe that the best way of reading mathematics is active reading. We also include some advanced problems; they are marked with a star, *.

The first part of the book has been devised as an introductory course on nonlinear diffusion centred on the PME and the GPME. Selections of the text centred on the PME and versions of it have been taught as such to PhD students (p. 14 ) having previously followed courses in classical analysis, functional analysis and PDEs. Knowing some physics of continuous media or studying the subject in parallel is useful, but not required. Several selections are possible for one semester courses, the simplest one consisting of Chapters 211 plus 14, maybe jumping over most over 6 and 7. Relevant and elementary material is also contained in Chapters 15, 16, and 18. We will give extensive references when the material used is not standard.

This is a book in PDEs and analysis at a theoretical level but covering the interests of what is usually called applied analysis. We will pay a serious attention to some, say, classical applications, but the reader need not be an expert in any physical or natural science or engineering, since all relevant concepts will be clearly defined.

The reader will notice that the subject is rich in methods and results, but also in concepts and denominations, many taken from different branches of the applied sciences, others from different areas of mathematics. We will underline all new concepts by writing them in italics the first time they are precisely defined and referencing the relevant ones in the index.

We hope that the material will make it easier for the interested reader to delve into deeper or more specific literature. We have already mentioned that, although we concentrate most of our effort in examining the non-negative solutions of the PME, the natural functional framework leads the mathematician to work with the signed PME. A number of important issues are still open for signed solutions.

# Notes

## Some historical notes

We have seen the important contribution of Zel'dovich and Kompaneets [532], 1950, who found the source solutions in a particular case, and Barenblatt [60], who performed a complete study of these solutions in 1952. After the work in the decade by Barenblatt et al. on self-similar solutions and finite propagation, cf. [71] and the book [63], the systematic theory of the PME can be said to have begun with the fundamental work of Oleĭnik and her collaborators Kalashnikov and Czhou around 1958 [408], who introduced a suitable concept of generalized solution and analysed both the Cauchy and the standard boundary value problems in one space dimension. The work was continued by Sabinina, [457], who extended the results to several space dimensions. The qualitative analysis was advanced by Kalashnikov and many authors followed. The survey of the last author contains a very complete reference list on the literature concerning different aspects of the PME and related equations at the time. For earlier history see the Notes of the next chapter.

Since the 1970s, the interest in the equation has touched many other scholars from different countries. Here are some important landmarks. Bénilan [79] and Crandall et al. [178, 180] constructed mild solutions, Brezis developed the theory of maximal monotone operators [128], Aronson studied the properties of the free (p. 15 ) boundary [35, 36, 37], Kamin began the analysis of the asymptotic behaviour [319, 320], and Peletier et al. studied self-similarity [54]. In the 1980s well-posedness in classes of general data was established in Aronson-Caffarelli [42] and Bénilan-Crandall-Pierre [91], and the study of solutions with measures as data was initiated in Brezis-Friedman [131] and advanced by Pierre [434] and Dahlberg-Kenig [187]. Basic continuity of solutions and free boundaries was proved by Caffarelli and Friedman [138, 139, 140] and refined by DiBenedetto [206, 207], Sacks [461] and a number of authors.

There exists today a relatively complete theory covering the subjects of existence and uniqueness of suitably defined generalized solutions, regularity, properties of the free boundary and asymptotic behaviour, for different initial and boundary-value problems. Their names will appear in the development.

## Previous reports on the PME and related equations

The text has as a precedent the notes prepared on the basis of the course taught at the Université de Montréal in June-July of 1990, aimed at introducing the subject and its techniques to young researchers [508]. The material has been also used for graduate courses at the Universidad Autónoma de Madrid. It has several earlier precedents. A short survey was published by Peletier [425] in 1981 and has been much used. A much longer survey paper is due to Aronson [38], written in 1986. Another often cited contribution, more in the form of a summary but including a discussion of related nonlinear parabolic equations and a very extensive reference list is due to Kalashnikov [317] in 1987. These have been main references during these years. In his book on Variational Principles and Free-Boundary Problems [240], 1982, Friedman devotes a chapter to the PME because of its strong connection with free boundary problems. Recently, the book by four Chinese authors Wu, Yin, Li and Zhao [527], 2001, about nonlinear diffusion equations is worth mentioning.

Both PME and p-Laplacian equations are tied together as degenerate diffusions in DiBenedetto's book [209]. The book [469] by Samarski et al. is mainly devoted to reaction diffusion leading to blow-up but has wide information about PME, specially related to self-similarity. A similar observation applies to [255] by Galaktionov and the author which concentrates on asymptotic methods based on self-similarity and dynamical systems ideas. This book contains a chapter with the main facts about the PME that appear in the asymptotic studies.

A reference to the mathematics of diffusion is Crank [182] which contains a bulk of basic information on the classical applied topics and results. Conduction of heat in solids is treated by Carslaw and Jaeger [159]. A general text on reaction-diffusion equations is Smoller's [482]. The Stefan problem is covered in the already mentioned books by Rubinstein [454] and Meirmanov [388]. (p. 16 )