Nonlinear PDEs mathematical models in biology chemistry and population genetics

Partial differential equations and, in particular, linear elliptic equations were ated and introduced in science in the first decades of the nineteenth century in or-der to study gravita

For further volumes:

http://www.springer.com/series/3733

Marius Ghergu

University College Dublin

School of Mathematical Sciences

PO Box 1-764BucharestRomaniavicentiu.radulescu@imar.ro

ISSN 1439-7382

ISBN 978-3-642-22663-2 e-ISBN 978-3-642-22664-9

DOI 10.1007/978-3-642-22664-9

Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2011939998

Mathematics Subject Classification (2010): 35-02; 49-02; 92-02; 58-02; 37-02; 35Qxx

c

 Springer-Verlag Berlin Heidelberg 2012

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or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,

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family who have always been there in hard times.

Vicent¸iu R˘adulescu dedicates this book to the memory of his beloved Mother,

Ana R˘adulescu (1923–2011)

Partial differential equations and, in particular, linear elliptic equations were ated and introduced in science in the first decades of the nineteenth century in or-der to study gravitational and electric fields and to model diffusion processes inPhysics The heat equation, the Navier–Stokes system, the wave equation and theSchr¨odinger equations introduced later on to describe the dynamic of heat con-duction, Newtonian fluid flows and, respectively, quantum mechanics are the basicequations of mathematical physics which are, in spite of their complexity, centeredaround the notion of Laplacian or, in other words, linear diffusion However, theseequations, which were primarily created to model physical processes, played an im-portant role in almost all branches of mathematics and, as a matter of fact, can beviewed as a chapter of applied mathematics as well as of so-called pure mathemat-ics In fact, the linear elliptic operators and, in particular, the Laplacian representwithout any doubt a bridge that connects a large number of mathematical fields andconcepts and provides the mathematical framework for physical theories as well asfor the theory of stochastic processes and some new mathematical technologies forimage restoring and processing The well posedness of the basic boundary valueproblems associated with the Laplace operator is a fundamental topic of the the-ory of partial differential equations It is instructive to recall that the well posed-ness of the Dirichlet and Neumann problem remained open and unsolved for morethan half a century until the turn of the nineteenth century, when Ivar Fredholmsolved it by a new and influential idea which is at the origin of a several branches

cre-of mathematics which will change the analysis cre-of the twentieth century; primarily,

I have in mind here functional analysis and operator theory A related problem, the

vii

Dirichlet variational principle, had a similar history, being rigorously proved only

in the fourth decade of the last century after the creation of Sobolev spaces Thisprinciple is at the origin of variational theory of elliptic problems and of the con-cept of weak or distributional solution, which fundamentally changed the basicsideas and techniques of PDEs in the second part of the last century The mathemati-cians of the nineteenth century failed to prove this principle because it is not wellposed in spaces of differentiable functions, but in functional spaces with energeticnorms that is in Sobolev spaces which were discovered later on Nonlinear ellipticboundary value problems arise naturally in the description of physical phenomenaand, in particular, of reaction-diffusion processes, governed by nonlinear diffusionlaws, or in geometry (the minimal surface equation or uniformization theorem inRiemannian geometry) The well posedness of most of these nonlinear problemswas treated by the new functional methods introduced in the last century such asthe Banach principle, Schauder fixed point theorem and Schauder–Leray degreetheorem and, in the 1960s, by the Minty–Browder theory of nonlinear maximalmonotone operators in Banach spaces It should be said that most of these func-tional approaches to nonlinear elliptic problems lead to existence results in spaceswith energetic norms (Sobolev spaces) and so quite often these are inefficient or toorough to put in evidence sharp qualitative properties of solutions such as asymptoticbehavior, monotonicity or comparison results Some classical methods such as themaximum principles, integral representation of solutions or complex analysis tech-niques are very efficient to obtain sharp results for new classes of elliptic problems

of special nature These techniques, which perhaps have their origins in the classicalwork of Peano on existence and construction of solutions to the Dirichlet problem

by method of sub and supersolutions, are still largely used in the modern theory ofnonlinear elliptic equations This book is a very nice illustration of these techniques

in the treatment of the existence of positive solutions, which are unbounded to tier or for singular solutions to logistic elliptic equations as well as for the minimal-ity principle for semilinear elliptic equations Most of the elliptic equations studied

fron-in this book are of sfron-ingular nature or develop some “pathological” behavior whichrequires sharp and specific investigation tools different to the standard functional

or energetic methods mentioned above In the same category are the correspondingvariational problems which, in the absence of convexity, need some sophisticatedinstruments such as the Mountains Pass theorem, the Ekeland variational principle

or the Brezis–Lieb lemma A fact indeed remarkable in this book is the variety ofproblems studied and of methods and arguments The authors avoid formulations,tedious arguments and maximum generality, which is a general temptation of math-ematicians in favor of simplicity; they confine to specific but important problemsmost of them famous in literature, and try to extract from their treatment the essen-tial ideas and features of the approach The examples from chemistry and biologychosen to illustrate the theory are carefully selected and significant (the Brusselator,reaction-diffusion systems, pattern formation).

Marius Ghergu and Vicent¸iu D R˘adulescu, who are well-known specialists inthe field, have coauthored in this work a remarkable monograph on recent results

on nonlinear techniques in the theory of elliptic equations, largely based on theirresearch works The book is of a high scientific standard, but readable and accessible

to a large category of people interested in the modern theory of partial differentialequations

Among all mathematical disciplines the theory of differential equations is the most important It furnishes the explanation of all those elementary manifestations of nature which involve time.

Sophus Lie (1842–1899)

Much of the modern science is based on the application of mathematics It is tral to modern society, underpins scientific and industrial research, and is key to oureconomy Mathematics is the engine of science and engineering It also has an ele-gance and beauty that fascinates and inspires those who understand it

cen-Mathematics provides the theoretical framework for biosciences, for statisticsand data analysis, as well as for computer science New discoveries within math-ematics affect not only science, but also our general understanding of the world

we live in Problems in biological sciences, in physics, chemistry, engineering, and

in computational science are using increasingly sophisticated mathematical niques For this strong reason, the bridge between the mathematical sciences andother disciplines is heavily traveled

tech-Biosciences are some of the most fascinating of all scientific disciplines and is anarea of applied sciences we use to explore and try to explain the uncertain world inwhich we live It is no surprise, then, that at the heart of a professional in this field

is a fascination with, and a desire to understand, the ”how and why” of the materialworld around us

The purpose of this volume is to meet the current and future needs of the tion between mathematics and various biosciences This is first done by encouraging

interac-xi

the ways that mathematics may be applied in traditional areas such as biology, istry, or genetics, as well as pointing towards new and innovative areas of applica-tions Next, we intend to encourage other scientific disciplines (mainly oriented tonatural sciences) to engage in a dialog with mathematicians, outlining their prob-lems to both access new methods and suggest innovative developments within math-ematics itself

chem-The first chapter presents the main mathematical methods used in the book Suchtools include iterative methods and maximum principle, variational methods andcritical point theory as well as topological methods and degree theory

The second chapter deals with Liouville type results for elliptic operators in vergence form Since its appearance in the nineteenth century, many results in thetheory of Partial Differential Equations have been devoted to characterize all data

di-functions f such that the standard elliptic inequality L u ≥ f (x,u) admits only the

trivial solution We discuss such type of problems for elliptic operators of the form

L u = −div[A(|∇u |)∇u]

Chapter 3 is concerned with the study of solutions to the equationΔu=ρ(x) f (u)

in a smooth domain that blow-up at the boundary in the sense that limx →x0u (x) =

+∞, for all x0 ∈∂Ω; in caseΩ = RN, this condition can be simply formulated aslim|x|→∞u (x) = +∞ Here we emphasize the role played by the Keller–Ossermanintegral condition

Chapters 4 and 5 deal with some related singular elliptic problems This time,the solution is bounded but the nonlinearity appearing in the problem is unboundedaround the boundary of the domain Particular attention is paid to the Lane–Emdenequation and the associated system in this singular framework Chapter 4 is devoted

to the model equation−Δu = au + u −α, 0<α< 1 and the associated system In

Chap 5 we study singular elliptic problems in exterior domains Here we point out

the role played by the geometry of the domain in the existence of a C2solution Inparticular we completely describe the solution set of the equation−Δu = |x|αu −p

by showing that all the solutions are radially symmetric and characterized by twoparameters

Chapter 6 presents two classes of quasilinear elliptic equations The approach

in this chapter is variational and combines some tools in this field such as land’s variational principle and mountain pass theorem The lack of compactness of

Eke-Sobolev embeddings or the presence of p-Laplace operator are the main features of

the chapter

In Chap 7 we are concerned with three classes of higher order elliptic problemsinvolving the polyharmonic operator By adopting three different approaches we un-derline the complex structure of such problems in which the higher order differentialoperator and the type of conditions imposed on the boundary play an important role

in the qualitative study of solutions

The last two chapters are devoted to reaction diffusion systems In their mostgeneral form, the models we intend to study can be stated as



u t =d uΔu + f (u,v) (x ∈Ω, t > 0),

v t =d vΔu + g(u,v) (x ∈Ω, t > 0). (0.1)

These equations describe the evolution of the concentrations, u = u(x,t), v = v(x,t)

at spatial position x and time t, of two chemicals due to diffusion, with different constant diffusion coefficients d u , d v , respectively, and reaction, modeled by the typically nonlinear functions f and g that can be derived from chemical reaction

formulas by using the law of mass action and other physical conditions

In Chap 9 several reaction-diffusion models are studied Oscillating chemical actions have been a rich source of varied spatial-temporal patterns since the discov-ery of the oscillating wave in the Belousov–Zhabotinsky reaction in 1950s Thesephenomena and observations have been transferred to challenging mathematicalproblems through various models, especially reaction-diffusion equations Amongthese mathematical models, we present:

re-• The Brusselator model introduced by Prigogine and Lefever in 1968 as a model

for an autocatalytic oscillating chemical reaction This corresponds to

f (u,v) = a − (b + 1)u + u2v , g(u,v) = bu − u2v

• The Schnackenberg model for chemical reactions with limit cycle behavior

f (u,v) = a − u + u2v , g(u,v) = b − u2v

• The Lengyel–Epstein model for the chlorite–iodide–malonic acid (CIMA)

reac-tion This corresponds to (0.1) with the nonlinearities f and g given by

The few examples we have provided illustrate the great alliance between matics and biosciences This is recognized universally and both disciplines thrived

mathe-by supporting each other The prerequisite for this book includes a good uate course in functional analysis and Partial Differential Equations This book isintended for advanced graduate students and researchers in both pure and appliedmathematics

undergrad-Our vision throughout this volume is closely inspired by the following words ofV.I Arnold (1983, see [8, p 87]) on the role of mathematics in the understanding of

real processes: In every mathematical investigation the question will arise whether

we can apply our results to the real world Consequently, the question arises of choosing those properties which are not very sensitive to small changes in the model and thus may be viewed as properties of the real process.

1 Overview of Mathematical Methods in Partial Differential

Equations 1

1.1 Comparison Principles 1

1.2 Radial Symmetry of Solutions to Semilinear Elliptic Equations 6

1.3 Variational Methods 9

1.3.1 Ekeland’s Variational Principle 9

1.3.2 Mountain Pass Theorem 11

1.3.3 Around the Palais–Smale Condition for Even Functionals 12

1.3.4 Bolle’s Variational Method for Broken Symmetries 14

1.4 Degree Theory 15

1.4.1 Brouwer Degree 15

1.4.2 Leray–Schauder Degree 16

1.4.3 Leray–Schauder Degree for Isolated Solutions 17

2 Liouville Type Theorems for Elliptic Operators in Divergence Form 19

2.1 Introduction 19

2.2 Some Related ODE Problems 21

2.3 Main Results 26

3 Blow-Up Boundary Solutions of the Logistic Equation 29

3.1 Singular Solutions of the Logistic Equation 30

3.1.1 A Karamata Regular Variation Theory Approach 43

xv

xvi Contents

3.2 Keller–Osserman Condition Revisited 60

3.2.1 Setting of the Problem 61

3.2.2 Minimality Principle 65

3.2.3 Existence of Solutions on Some Ball 70

3.2.4 Existence of Solutions on Small Balls 72

3.2.5 Existence of Solutions on Smooth Domains 73

3.2.6 Blow-Up Rate of Radially Symmetric Solutions 74

3.2.7 Blow-Up Rate of Solutions on Smooth Domains 75

3.2.8 A Uniqueness Result 77

3.2.9 Discrete Equations 79

3.2.10 Numerical Computations 86

3.3 Entire Large Solutions 91

3.3.1 A Useful Result: Bounded Entire Solutions 91

3.3.2 Existence of an Entire Large Solution 93

3.3.3 Uniqueness of Solution 98

3.4 Elliptic Equations with Absorption 100

3.5 Lack of the Keller–Osserman Condition 106

4 Singular Lane–Emden–Fowler Equations and Systems 117

4.1 Bifurcation Problems for Singular Elliptic Equations 117

4.2 Lane–Emden–Fowler Systems with Negative Exponents 130

4.2.1 Preliminary Results 132

4.2.2 Nonexistence of a Solution 139

4.2.3 Existence of a Solution 142

4.2.4 Regularity of Solution 149

4.2.5 Uniqueness 153

4.3 Sublinear Lane–Emden Systems with Singular Data 155

4.3.1 Case p > 0 and q > 0 155

4.3.2 Case p > 0 and q < 0 158

4.3.3 Case p < 0 and q < 0 162

4.3.4 Further Extensions: Superlinear Case 163

5 Singular Elliptic Inequalities in Exterior Domains 167

5.1 Introduction 167

5.2 Some Elliptic Problems in Bounded Domains 168

5.3 An Equivalent Integral Condition 174

5.4 The Nondegenerate Case 175

5.4.1 Nonexistence Results 175

5.4.2 Existence Results 180

5.5 The Degenerate Case 188

5.6 Application to Singular Elliptic Systems in Exterior Domains 203

6 Two Quasilinear Elliptic Problems 211

6.1 A Degenerate Elliptic Problem with Lack of Compactness 211

6.1.1 Introduction 211

6.1.2 Auxiliary Results 214

6.1.3 Proof of the Main Result 222

6.2 A Quasilinear Elliptic Problem for p-Laplace Operator 227

7 Some Classes of Polyharmonic Problems 245

7.1 An Eigenvalue Problem with Continuous Spectrum 245

7.2 Infinitely Many Solutions for Perturbed Nonlinearities 251

7.3 A Biharmonic Problem with Singular Nonlinearity 258

8 Large Time Behavior of Solutions for Degenerate Parabolic Equations 267

8.1 Introduction 267

8.2 Superlinear Case 268

8.3 Sublinear Case 275

8.4 Linear Case 283

9 Reaction-Diffusion Systems Arising in Chemistry 287

9.1 Introduction 287

9.2 Brusselator Model 288

9.2.1 Existence of Global Solutions 290

9.2.2 Stability of the Uniform Steady State 293

9.2.3 Diffusion-Driven Instability 295

9.2.4 A Priori Estimates 296

9.2.5 Nonexistence Results 299

9.2.6 Existence Results 302

xviii Contents

9.3 Schnackenberg Model 306

9.3.1 The Evolution System and Global Solutions 307

9.3.2 A Priori Estimates 310

9.3.3 Nonexistence of Nonconstant Steady States 314

9.3.4 Existence Results 317

9.4 Lengyel–Epstein Model 322

9.4.1 Global Solutions in Time 323

9.4.2 Turing Instabilities 326

9.4.3 A Priori Estimates for Stationary Solutions 328

9.4.4 Nonexistence Results 330

9.4.5 Existence 332

10 Pattern Formation and the Gierer–Meinhardt Model in Molecular Biology 337

10.1 Introduction 337

10.2 Some Preliminaries 340

10.3 Case 0≤ p < 1 347

10.3.1 Existence 347

10.3.2 Further Results on Regularity 354

10.3.3 Uniqueness of a Solution 356

10.4 Case p < 0 362

10.4.1 A Nonexistence Result 362

10.4.2 Existence 364

A Caffarelli–Kohn–Nirenberg Inequality 369

B Estimates for the Green Function Associated to the Biharmonic Operator 373

References 377

Index 387

Overview of Mathematical Methods in Partial Differential Equations

Mathematics may be defined as the subject in which we never know what

we are talking about, nor whether what

we are saying is true.

Bertrand Russell (1872–1970)

In this chapter we collect some results in Nonlinear Analysis that will be quently used in the book The first part of this chapter deals with comparison prin-ciples for second order differential operators and enables us to obtain an orderedstructure of the solution set and, in most of the cases, the uniqueness of the solution

fre-In the second part of this chapter we review the celebrated method of moving planesthat allows us to deduce the radial symmetry of the solution The third part of thischapter is concerned with variational methods The final section contains some re-sults in degree theory that will be mostly used to derive existence and nonexistence

of a stationary solution to some reaction-diffusion systems

M Ghergu and V Rˇadulescu, Nonlinear PDEs, Springer Monographs in Mathematics,

DOI 10.1007/978-3-642-22664-9 1, c Springer-Verlag Berlin Heidelberg 2012 1

2 1 Overview of Mathematical Methods in Partial Differential Equations

and w (x0) = maxΩw, then g (x0,w(x0)) ≥ 0.

(ii) If w ∈ C2(Ω) ∩C1(Ω) satisfies

Δw + g(x,w) ≤ 0 in Ω, ∂w

∂n ≥ 0 on ∂Ω,

and w (x0) = minΩw, then g (x0,w(x0)) ≤ 0.

Proof We shall prove only part (i) as (ii) can be established in a similar way There

are two possibilities for our consideration

Case 1: x0∈Ω Since w (x0) = maxΩw we haveΔw (x0) ≤ 0 and now from the first

inequality in (1.1) we obtain g (x0,w(x0)) ≤ 0.

Case 2: x0∈∂Ω Assume by contradiction that g (x0,w(x0)) < 0 By the continuity

of g and w, there exists a ball B ⊂Ω with∂B ∩∂Ω= {x0} such that

g (x,w(x)) < 0 for all x ∈ B.

Thus, from (1.1) we findΔw > 0 in B Since w(x0) = maxB w, it follows from the

Hopf boundary lemma that∂w /∂n (x0) > 0 which contradicts the boundary

condi-tion in (1.1) This completes the proof of Theorem1.1 

Basic to our purposes in this book we state and prove the following result which

is suitable for singular nonlinearities

Theorem 1.2 LetΨ:Ω×(0,∞) → R be a H¨older continuous function such that the

mapping (0,∞)  t −→Ψ(x,t)/t is decreasing for each x ∈Ω Assume that there

exist v1, v2∈ C2(Ω) ∩C(Ω) such that

(a) Δv1+Ψ(x,v1) ≤ 0 ≤Δv2+Ψ(x,v2) inΩ;

(b) v1,v2> 0 inΩ and v1≥ v2on∂Ω;

(c) Δv1∈ L1(Ω) or Δv2∈ L1(Ω).

Then v1≥ v2inΩ.

Proof Suppose by contradiction that v ≤ w is not true in Ω Then, we can find

ε0,δ0> 0 and a ball B ⊂⊂Ω such that



B vw

Considerθ ∈ C1(R) a nondecreasing function such that 0 ≤θ≤ 1, θ(t) = 0, if

t ≤ 1/2 andθ(t) = 1 for all t ≥ 1 Define

4 1 Overview of Mathematical Methods in Partial Differential Equations

A direct consequence of Theorem1.2is the result below

Corollary 1.3 Let k ∈ C(0,∞) be a positive decreasing function and a1,a2∈ C(Ω)

with 0 < a2≤ a1inΩ Assume that there existβ ≥ 0, v1,v2∈ C2(Ω) ∩C(Ω) such

where A ∈ C(0,∞) is positive such that the mapping t → tA(t) is increasing.

and f ∈ C(R) Assume that u,v ∈ C2(Ω) ∩C(Ω) satisfy

6 1 Overview of Mathematical Methods in Partial Differential Equations

with equality if and only if∇u=∇v Using this fact in (1.6) it follows that u ≤ v in

L u :=∂t u − a(x,t,u)Δu + f (x,t,u),

where a , f :Ω× [0,∞) × [0,∞) → R are continuous functions such that a ≥ 0 in

Ω× [0,∞) Assume that there exist u1,u2∈ C2,1(Ω× (0,T )) ∩C(Ω× [0,T ]) such

that:

(i) L u1≤ L u2inΩ× (0,T).

(ii) u1 ≤ u2onΣT := (∂Ω× (0,T)) ∪ (Ω× {0}).

(iii) at least for one i ∈ {1,2} we have |D2u i | ∈ L∞(Ω× [0,T]) and the functions

a and f are Lipschitz with respect to the u variable in the neighborhood of

K : = u i(Ω× [0,T]).

Then u1≤ u2inΩ× [0,T].

1.2 Radial Symmetry of Solutions to Semilinear Elliptic

Equations

An important tool in establishing the radial symmetry of a solution to elliptic PDEs

is the so-called moving plane method that goes back to A.D Alexandroff and J

Ser-rin It was then refined by Gidas, Ni and Nirenberg in the celebrated paper [97] Therequirements on the regularity of the domain were further simplified by Berestyckiand Nirenberg [16] We follow here the line in [16] and [25] to provide the readerwith a simple and instructive proof of the radial symmetry of solutions to semilinearelliptic PDEs in bounded and convex domainsΩ that vanish on∂Ω

Theorem 1.6 LetΩ ⊂ R N be a convex domain which is symmetric about the x1

axis Assume that f : R → R is a Lipschitz continuous function andρ:[0,∞) → R

is a decreasing function If u ∈ C2(Ω) ∩C(Ω) satisfies

−Δu=ρ(|x|) f (u),u > 0 inΩ,

then u is symmetric with respect to the x1axis.

Proof We first need a version of the maximum principle for small domains as stated

that is, ifωis small, then w ≥ 0 inω.

Proof We multiply the first inequality in (1.8) by w − = max{−w,0} Integrating

Using (1.9), the above inequality implies − L 2N /(N−2)( ω )= 0, so w ≥ 0 inω 

Let us now come back to the proof of Theorem1.6 For any x = (x1,x2, ,x N ) ∈ R N

we write x = (x1,x ), where x1∈ R and x ∈ R N −1 Let

8 1 Overview of Mathematical Methods in Partial Differential Equations

λ0= max{x1:(x1,x ) ∈Ω}.

We claim that

u (x1,x ) < u(y1,x ), (1.10)for all(x1,x ) ∈Ω with x1 > 0 and all y1∈ R with |y1| < x1.

Then (1.10) implies u (x1,x ) ≤ u(x1,−x ) and similarly u(x1,x ) ≥ u(x1,−x ... and a partial ordering to obtain a point where

a linear functional achieves its supremum on a closed bounded convex set In itsoriginal form, Ekeland’s variational principle can be stated... function defined in

RN , N ≥ 1, is constant The interested reader may find a valuable overview in the

linear elliptic inequalityΔu ≥ρ(|x|) f (u) in R N... Mountain Pass Theorem

The mountain pass theorem was established by Ambrosetti and Rabinowitz in [7]

It is a powerful tool for proving the existence of critical points