VỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾN

VỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾNVỀ SỰ KHÔNG TỒN TẠI NGHIỆM CỦA MỘT SỐ PHƯƠNG TRÌNH ĐẠO HÀM RIÊNG PHI TUYẾN

MINISTRY OF EDUCATION AND TRAINING

HANOI NATIONAL UNIVERSITY OF EDUCATION

DAO MANH THANG

ON THE NONEXISTENCE OF SOLUTIONS OF SOMENONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

DISSERTATION OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

Ha Noi, 2024

MINISTRY OF EDUCATION AND TRAINING

HANOI NATIONAL UNIVERSITY OF EDUCATION

DAO MANH THANG

ON THE NONEXISTENCE OF SOLUTIONS OF SOMENONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

DISSERTATION OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

Speciality:Differential and Integral EquationsCode:9 46 01 03

Assoc.Prof Duong Anh TuanAssoc.Prof Dao Trong Quyet

Ha Noi, 2024

Committal in the dissertation

I assure that my scientific results are completed under the guidance of Assoc.Prof Duong Anh Tuan and Assoc Prof Dao Trong Quyet The results statedin the dissertation are completely honest and they have never been publishedin any scientific documents before I published All publications that work withother authors have been approved by them to include in the dissertation I takefull responsibility for my research results in the dissertation.

Full name: Dao Manh ThangSigned:

Date:

This dissertation has completed at Hanoi National University of Educationunder supervision of Assoc.Prof Duong Anh Tuan and Assoc.Prof Dao TrongQuyet I wish to acknowledge my supervisor’s instruction with greatest appre-ciation and thanks I would like to thank all Professors who have taught meat Hanoi National University of Education and my friends for their help I alsothank all the lecturers and PhD students at the seminar of Division of Mathe-matical Analysis for their encouragement and valuable comments I especiallyexpress my gratitude to my parents, my wife, my brothers, and my beloved sonsfor their love and support Finally my thanks go to Viet Tri education depart-ment support during my period of PhD study.

Hanoi, March, 2024

Dao Manh Thang

1.2 The Grushin operator 16

1.3 The fractional Laplacian 21

Chapter 2.On the nonexistence result for porous medium systems withsources 23

2.1 Problem setting and main results 23

2.1.1 Problem setting 23

2.1.2 Nonexistence results for porous medium equation/system 24

2.2 Proof of nonexistence results 26

2.2.1 Nonexistence result for the porous medium equation 26

2.2.2 Nonexistence result for the porous medium system 29

Chapter 3.On stable solutions of a weighted degenerate elliptic equationwith advection term 36

3.1 Problem setting and main result 36

3.1.1 Problem setting 36

3.1.2 Nonexistence of stable solutions 38

3.2 Proof of nonexistence of stable solutions 39

Chapter 4.On the nonexistence result for the nonlinear fractional Choquardequation .45

4.1 Problem setting and main results 45

4.1.1 Problem setting 45

4.1.2 Nonexistence results for fractional Choquard equation 47

4.2 Proof of nonexistence results 48

4.2.1 Nonexistence of positive solutions 48

4.2.2 Nonexistence of positive stable solutions 49

Conclusion 55

Future work 56

List of publications 57

References 58

List of symbols and acronyms

" R the set of real numbers

RNthe N -dimensional Euclidean space

Ck(Ω)the space of continuous differentiable functions of order k in

Ω ⊂ RN

C∞(Ω) the space of infinitely differentiable functions inΩ

Cc∞(Ω) the space of infinitely differentiable functions with compactsupport inΩ

Cck(Ω)the space of continuous differentiable functions of order k

with compact support inΩ

Cα(RN) the space of Hölder continuous function of order α, 0 < α <

1, on RN

Lp(RN) the space of integrable function of order p on RN

Ll ocp (RN) the space of locally integrable function of order p on RN

Hs(RN) the fractional Sobolev space {u ∈ L2(RN); ∥u∥2˙

Hs(RN)< +∞}Gα the Grushin operator

∆ the Laplace operator

(−∆)s the fractional Laplace operator∇ the gradient vector

∇α the gradient vector associated to the Grushin operatordiv ≡ ▽ the divergence operator

divG the divergence associated to the Grushin operator

utpartial derivative of u in variable t"

0.1 Literature review

The partial differential equation was first studied in the mid-18th century inthe works of such mathematicians as D’Alembert, Euler, Lagrange and Laplacewhich is an important tool for describing models of Physics and Mechanics.

Today, after more than two centuries of development, partial differentialequations are used in many fields to model many problems in Physics, Mechan-ics, Chemistry, Biology, Economics Due to the complexity of the real problems,the established models are usually nonlinear partial differential equations Thestudy of the existence and qualitative properties of solutions of these classes ofequations is one of the main topics of applied mathematical analysis.

One research direction that has attracted the attention of many cians in recent years is to consider conditions for the existence or non-existenceof solutions through Liouville-type theorems They play a fundamental role andare considered to be the foundation for accessing insights into the structure ofthe solution set of boundary value problems Liouville-type theorems lead tomany particularly important consequences and applications, such as: a prioriestimation of the Dirichlet problem, singularity and decay estimates, Liouville-type theorem over half space, estimating universal quantifiers, Harnack-typeinequalities, initial burst rates and time decay rates of parabolic problems .

mathemati-The first topic in this thesis is the study of the porous medium tem with sources

equation/sys-ut− ∆um= up

ut− ∆um= vp

where p, q> m > 1.

In recent years, the Liouville property has emerged as one of the most erful tools in the study of qualitative properties for nonlinear equations Asconsequences of Liouville type theorems, one can establish a variety of results,for instance, universal, pointwise, a priori estimate; universal and singularityestimates; decay estimates, etc, see[54] and references therein In addition, in[6], the authors have obtained the existence of solutions of semilinear bound-ary value problems in bounded domains by exploiting Liouville type resultscombined with degree type arguments.

pow-Let us next review some related results in the literature The classificationof positive supersolutions of elliptic courterpart of (0.1) and (0.2) has beencompletely proved, see e.g [3, 17] More precisely, the elliptic counterpart of(0.2) is the Lane-Emden system

2p/m + 1pq/m2− 1,

2q/m + 1pq/m2− 1

In particular, when p= q, the Lane-Emden system−∆um= vp

in RN

has no positive supersolution if and only if N−2 ≤ p/m−12 On the other hand, theexistence and nonexistence of positive solutions to the Lane Emden equation(0.1) has been already established in[34] where the critical exponent is given

by pc(N) =N+2

N−2 However, the same problem for the Lane-Emden system (0.3)has not been completely solved It is known as the Lane-Emden conjecturesaying that the system (0.3) has no positive solution if and only if

We refer the readers to the proof of this conjecture in low dimensions N ≤ 3 in

[50, 57, 58] and in [60] for N = 4 The conjecture in the case N ≥ 5 has not

been confirmed.

For the parabolic model (0.1) in the semilinear case m= 1, the well-knownFujita result ensures the nonexistence of nontrivial nonnegative solutions inRN× (0, ∞) in the subcritical case 1 < p ≤NN+2, see [28], [51, Sec 26] In

the supercritical case p>N+2

N , problem (0.1) admits, see [37, Example 1], anonnegative supersolution in RN × R of the form

u(x, t) =

kt−p1−1e−γ1+|x|2tif t> 0, x ∈ RN

where k,γ are suitably chosen.

For the system (0.2) with m= 1, Duong and Phan [23] have recently lished optimal Liouville type theorems for nonnegative or positive supersolu-tions Among other things, it is shown in[23] that the system (0.2) with m = 1

estab-has no nonnegative supersolution if and only if

≥ N

We next consider the problems (0.1) and (0.2) in the quasilinear case m> 1.

For solutions in RN×(0, T ) of the equations of type (0.1), some local solvability

and general regularity results have been studied in [1, 29, 30, 59, 62] It wasshown in[30, 56] that when p ≤ m+2

N, the solution u of (0.1) in RN×(0, +∞)

with bounded, continuous initial data u0 ̸≡ 0 does not exist globally and blow

up in a finite time, i.e there is T> 0 such that

This type of estimate has been then proved in [1] for a large range of p, i.e.

p< p0(m, N) where p0(m, N) is explicitly computed and satisfies

m+ 2

N< mp0(m, N) < mpSfor N ≥ 2.

Here pS denotes the Sobolev exponent Remark, in addition, that the proofof this result is based on the Liouville type theorem established in the samepaper[1] In fact, the author showed that the equation (0.1) has no nontrivialnonnegative weak solution in the whole space RN× R when m < p < p0(m, N).This nonexistence result is conjectured to be true in the range m< p < mpS in[1] However, this has been not confirmed yet.

Motivated by the papers [1, 23, 30] and recent progress on the study ofporous medium equation[62], we propose to study the existence and nonexis-tence of nonnegative weak supersolutions on the whole space of problems (0.1)and (0.2).

The second topic is to study the nonexistence of stable solutions for ellipticequation

|x|2α∆y, α > 0, is the Grushin operator, the weight

function h(z) is continuous Here, c(z) is a smooth vector field satisfying

divGc= 0 and β := sup

|z|G|c (z)|

|∇α|z|G| < ∞, (0.5)with the Grushin norm

|z|G= |x|2(1+α)+ (1 + α)2

| y|2

and the Grushin gradient

∇α= (∇x,(1 + α)|x|α∇y).Ifα = 0, c = 0 and h = 1, the problem (0.8) reduces to

which is known as Gelfand equation This equation can be derived from thethermal self-ignition model in low dimension [32] On the other hand, (0.6)also describes the diffusion phenomena induced by nonlinear sources[36] Re-cently, the classification of solutions to the equation (0.6) has attracted much

attention of mathematicians Among other things, it was shown that the

prob-lem (0.6) has no stable solution if and only if N ≤ 9, see [26] Roughly ing, stable solutions are those that make the energy of the system attain a local

speak-minimum In other words, a solution u is stable if the second variation at u of

the energy functional is nonnegative.

Recently, the elliptic problems with advection terms have been much studied[10, 11, 38] and references given there In [10], the author obtained someclassification of stable positive solutions to the equation

−∆u + c · ∇u = up

in RN,

where c is a smooth, divergence free vector field satisfying|c(x)| ≤ 1+|x|ϵ , ϵ is

small enough The technique used in [10] is a combination of test functionmethod and the generalized Hardy inequality[9].

In the case of exponential nonlinearity, by exploiting the technique in [10],the authors in[38] established the nonexistence of stable solutions to

when 3≤ N ≤ 9 and c satisfies the same conditions as in [10].

With some relax on the smallness condition of the advection term, the thors in[19] proved that the equation (3.1.1) has also no stable solution when3 ≤ N ≤ 9 and c is a smooth, divergence free vector field satisfying |c(x)| ≤

1+|x| Here the constant 0< ϵ <p(N − 2)(10 − N).

We next consider the elliptic problems involving the Grushin operator Recall

that Gαis elliptic for x ̸= 0 and degenerates on the manifold {0} × RN2 Thisoperator was introduced in[35] (see also Baouendi [4]) and has attracted theattention of many mathematicians The classification of stable solutions to theelliptic equation involving the Grushin operator with power-type nonlinearitywas obtained in [18] For the Gelfand equation involving Grushin operator,some nonexistence results of stable solutions was proved in[2].

The third topic is devoted to the study of the fractional Choquard equation

on the whole space RN

up(y)|x − y|N−2sd y.

Let 0< s < 1, the fractional Laplacian (−∆)sis initially defined on the Schwartzspace of rapidly decaying functions by

u: RN → R;Z

(1 + |x|)N+2sd x< ∞

We refer the readers to the monograph [52] for elementary properties of thefractional Laplacian.

In the case s= 1, the problem (0.8) becomes

−∆u =

1

mathematical treatment to the Choquard equations is provided in the surveyarticle[53].

Concerning the classification of positive solutions in the local case s= 1, Lei[44] proved that (0.8) does not possess positive solutions under the condition1 ≤ p <NN+2−2 Recently, Le [42] has established a stronger result saying that

(0.8), with s= 1, has no positive solution when N ≤ 2 or N ≥ 3 and −∞ <

N−2 One of the contributions in [42] is that the nonexistence of positive

solutions is still true for p≤ 1 In the nonlocal case, i.e 0 < s < 1, the

nonexistence of positive solutions of (0.8) has been proved when 1≤ p <NN+2s−2s,see[12, 39, 49].

Besides many results were established for the Lane-Emden equation or Gelfandequation, the classification of stable solutions to the Choquard equation is less

understood There are only some partial results for (0.8) with s= 1, see e.g.[44, 66] for the classification of positive stable solutions and [43] for the classi-fication of sign-changing stable solutions The techinique used in these papers isa combination of the test function method and the energy estimates originatedfrom the idea of Farina[25] Among other things, the following classificationwas proved in[44].

Theorem A([44]) "Let s = 1 and N ≥ 3 Suppose that p > 1 and

N< 6 +4(1 + pp2− p)p− 1 .

Then, the problem(0.8) has no positive stable solution."

Furthermore, the sharpness of Theorem A was also shown in [44] In ticular, the Joseph-Lundgren exponent was explicitly computed as

par-pjl(N) =

N−4−2pN−1 if N ≥ 11which allows us to transform the condition

N< 6 +4(1 + pp2− p)p− 1

|x|2α∆y,α > 0, is the Grushin operator.

• The fractional Choquard equation on the whole space RN

|x|N−2s∗ up

up−1,where 0< s < 1 and p ∈ R.

0.3 Scope of research

• Establish some sharp Liouville type theorems for nonnegative weak persolutions in the whole space of porous medium equation with sources andporous medium systems with sources.

su-• Prove the nonexistence of stable solutions of the elliptic equation involvingthe Grushin operator and advection term under some conditions of dimension.• Show the nonexistence results of positive solutions (stable or not) of thefractional Choquard equation.

0.4 Methodology

• In order to prove the nonexistence of nontrivial nonnegative weak lutions for the porous medium equations/systems with sources in the subcriticalcase, we use the test function method and nonlinear integral estimates.

superso-The construction of nontrivial nonnegative weak solutions to the tem is based on the heat kernel of the porous medium equation.

equation/sys-• Applying the test function method combined with stability inequality andnonlinear integral estimates, we establish the nonexistence of stable solutionsof the elliptic equation involving the Grushin operator, advection term and ex-ponential nonlinearity.

• Taking into account a comparision principle and a result in [20, Theorem1.1], we prove the nonexistence of positive solutions (without stability) of thefractional Choquard equation in the sublinear case.

In the superlinear case, by exploiting again the comparision principle gether with the nonlinear integral estimates in[21], we were able to establishthe nonexistence of positive stable solutions of the fractional Choquard equa-tion.

nonnegative supersolutions of porous medium equation/system with sources.• Chapter 3 is devoted to study of the nonexistence of stable solutions ofdegenerate elliptic equation with advection term.

• Chapter 4 provides results on the nonexistence of positive solutions of thenonlinear fractional Choquard equation.

Chapter 2, 3 and 4 are based on the papers[P1], [P2] and [P3] in the Listof Publications.

These results have been presented at:

• Seminar of Division of Mathematical Analysis at Hanoi National Universityof Education.

• Seminar at Vietnam Institute for Advanced Studies in Mathematics.

Chapter 1Auxiliary results

In this chapter, we present some auxiliary results on the inequalities, theGrushin operator and the fractional Laplacian based on[13, 24, 31, 52].

| f (x)g(x)| d x ≤

1.2 The Grushin operator

We begin this section by recalling some notations Denote by z= (x, y) a

generic point of RN= RN1× RN2 The usual gradient and the Grushin gradientare respectively denoted by∇ = (∇x,∇y) and ∇G = (∇x,|x|α∇y) The norm

of z associated to the Grushin distance is given by|z|G= |x|2(1+α)+ (1 + α)2

| y|2

12(1+α).

With this norm, we define the open ball and the sphere of radius R centered at

B(z0, R) = {z ∈ RN;|z − z0|G< R}

∂ B(z0, R) = {z ∈ RN;|z − z0|G= R}.

We writeBR and∂ BRinstead ofB(0, R) and ∂ BR(0, R).

In what follows, we use the weight function

w(z) :=|x|2α|z|2αG

where c(Nα,α) > 0 depends on Nα andα.

In addition, the co-area formula (see e.g.,[31, Formula 2.4]) implies that|BR| =

∂ BR

|∇(|z|G)|d HN−1= c(Nα,α)NαRNα−1.

These preparations combined with the idea of Garofalo and Lanconelli [31]

lead to the definition of the spherical average of a function V∈ C(R) as follows

V(R) = 1|∂ BR|

∂ BR

|∇(|z|G)|d HN−1, for R> 0. (1.3)Note that whenα = 0, this formula becomes the usual spherical average on the

Euclidean ball.

The following proposition was known, see[31] However, we give here theproof for the completeness.

Proposition 1.1 Let V∈ C2(RN) Then for every R > 0 and z0∈ RNwe have

GαV(z) |z|2−Nα

G− R2−Nα
dz, (1.5)

where c(Nα,α) is given in (1.2).Proof. From now on, we denote by

Γ (z) = |z|2−Nα

which is the fundamental solution of Gα, see[31].

It is enough to prove (1.5), (1.4) is then deduced from (1.5) by a simpletranslation Letϵ > 0 be small enough Using integration by parts, we arrive at

∂ (BR\Bϵ)

(∇x,|x|2α∇y)V Γ (z).νdHN−1

∂ BR

(∇x,|x|2α∇y)V Γ (z).νdHN−1

+Z

|∂ BR|Z

The following proposition shows the relation between the derivative of Vand GαV.

Proposition 1.2 Let V∈ C2(RN) Then for every R > 0 we have

d HN−1.This combined with (1.5) imply that

c(Nα,α)Nα(Nα− 2)Z

1−Nαd HN−1,where in the last equality, we have usedΓ (z) = |z|2−Nα

G= R2−Nα on ∂ BR Thisequality and the co-area formula imply (1.12).

The following result is the Hardy type inequality involving the Grushin erator, see[41].

op-Lemma 1.1 " Let r, s∈ R be such that Nα+ 2 > r − s and N1> 2α − s Then foreveryϕ ∈ C1

|z|2(α+1)−rG|x|s−2α|∇Gϕ|2dz.”

1.3 The fractional Laplacian

Assuming 0 < s < 1, the fractional Laplace operator (−∆)s is defined on aSchwartz space of rapidly decreasing functions, as follows:

1− cos(x1)

|x|N+2sd x

and B(x, ϵ) is a sphere with center x ∈ RN and radius ϵ This operator is

extended, in a distribution sense, to a wider spaceLs(RN) =

u: RN → R;Z

(1 + |x|)N+2sd x< ∞

According to the above definition, we have the following properties.lim

u(x + ξ) + u(x − ξ) − 2u(x)

It is known that the two definitions above are equivalent.

We define a fractional Sobolev space ˙Hs(RN) through the following normal:

semi-∥u∥2H˙s(RN):=Z

Lemma 1.2 "Let u∈ C2σ(RN)∩Ls(RN), σ > s Then, there exists U ∈ C2(RN+1

−div(t1−2s∇U) = 0in RN++1U= uon∂ RN+1

− lim

t→0t1−2s∂tU= κs(−∆)suon∂ RN+1+

and p(N, s) is the normalization constant "

This chapter is written based on the paper[P1] in the list of publications.

2.1 Problem setting and main results

In spite of the simplicity of the equation and of its applications, and due

perhaps to its nonlinear and degenerate character, a mathematical theory forthe PME has been developed only very recently.

For solutions in RN×(0, T ) of the equations of type (2.1), some local

solvabil-ity and general regularsolvabil-ity results have been studied in[1, 29, 30, 59, 62] It wasshown in[30, 56] that when p ≤ m+2

N, the solution u of (2.1) in RN×(0, +∞)

with bounded, continuous initial data u0 ̸≡ 0 does not exist globally and blowup in a finite time Motivated by the papers[1, 23, 30] and recent progress onthe study of porous medium equation[62], we propose to study the existenceand nonexistence of nonnegative weak supersolutions on the whole space ofproblems (2.1) and (2.2).

Definition 2.1 We say that u∈ Llocp (RN

× R; [0, ∞)) is a nonnegative weaksupersolution of (2.1) if for anyφ ∈ C2,1

c (RN× R; [0, +∞)), there holds−

2.1.2 Nonexistence results for porous medium equation/system

Our first result is the following nonexistence result for (2.1).

Theorem 2.1 Assume that p> m > 1 The problem (2.1) has no nontrivialnonnegative weak supersolution in RN× R if and only if p ≤mNN+2.

These exponents were found in[23] In this case, our results coincide with thatin[23].

Remark 2.2 From our result, we want to address a question on the

nonex-istence of nontrivial nonnegative solutions of (2.1) and (2.2) when p< m orq< m Inspired by the result in [23], we conjecture that there has no nontrivial

nonnegative solution in this case Nevertheless, this question is still open.

2.2 Proof of nonexistence results

Let us now state the idea of the proof of our main results The nonexistenceresult in Theorem 2.1 is more or less known However, we cannot find anysatisfactory reference Then, we give a complete proof of this result by usingthe test-function method It is worth to notice that Theorem 2.2, to the bestof our knowledge, is the first result classifying nonnegative supersolutions ofthe porous medium system (2.2) The nonexistence part in Theorem 2.2 isproved by exploiting again the test-function method Nevertheless, due to the

presence of m> 1, this method becomes more complicated and it also requires

a suitable choice of parameters in scaling argument Recall that our results aresharp thanks to the constructions of nontrivial nonnegative weak supersolutionsof (2.1) (resp (2.2)).

2.2.1 Nonexistence result for the porous medium equation

Let us begin this section by proving the nonexistence result In what follows,

we denote by C a generic constant which may change from line to another and

independent of solutions Set

β =N(m − 1) + 2pp− m + 1 .For r> 0, define

Br= {(x, t); |x| < r and |t| < rβ}.Letψ ∈ C∞

c (RN

×R; [0, 1]) be a test function satisfying ψ = 1 on B1andψ = 0

outside B2 For r> 0, put ψr= ψl(x

r,rtβ) where l is chosen later on.

Suppose that u is a nonnegative weak supersolution of (2.1) By the

defini-tion of supersoludefini-tions above withφ = ψr, we have−

u∂tψrd x d t−Z

um∆ψrd x d t ≥Z

upψrd x d t (2.4)Moreover, a straightforward computation gives

| − ∆ψr| ≤ C

r2ψl−2lr

|∂tψr| ≤ C

rβψl−1lr

This observation combined with (2.4) and the fact that 0≤ ψr ≤ 1 leads toZ

rd x d t ≤‚Z

upψ(l−1)plrd x d t

umψl−2lrd x d t

upψ(l−2)pmlrd x d t

Œmp ‚Z

d x d t

(2.7)

Let us choose l large enough such that(l−2)pml> 1 and(l−1)pl> 1 Thus,

substi-tuting (2.6) and (2.7) into (2.4), we obtainZ

upψrd x d t

(2.9)

We next consider two cases ofκ.

Case 1. κ < 0.

It is easy to see that this condition is equivalent to

p<mN+ 2N

By simplifying the inequality (2.9) and then letting r→ +∞, one hasZ

upd x d t is finite However, this consequently

yields the fact that the right hand side of (2.9) tends to 0 as r→ +∞ fore, we again obtain from (2.9) that

upd x d t = 0.

This is the case if only if u= 0.

The rest of the proof is devoted to the existence result For p>mN+2

N , weconstruct a nontrivial nonnegative weak supersolution of the form

u(x, t) =

hereϵ is a small positive constant, γ =p−m

2(p−1)and t+ = max(t, 0) Indeed, onthe range t> 0 and ϵ −γ(m−1)2m|x|2+1

t2γ> 0, we compute∂tu= − 1

= (γN − 1

p− 1)ϵ−mp−1−1up(x, t).Since p>mN+2

N , there holdsγN − 1

p−1 > 0 Let us choose ϵ small enough such

that(γN − 1

p−1)ϵ−mp−1−1 > 1 Then, we have shown that u(x, t) constructed above

is a nontrivial nonnegative weak supersolution of (2.1).

2.2.2 Nonexistence result for the porous medium system

In this section, we divide the proof of Theorem 2.2 into two parts: tence result and existence of solutions.

nonexis-Step 1 Nonexistence result

The nonexistence proof is also based on the rescaled test-function method.

Nevertheless, as mentioned above, for the quasilinear case m> 1, the

test-function method becomes more complicated and requires delicate tions Let (u, v) be a nonnegative weak supersolution of (2.2) and let ψ ∈

computa-Cc∞(R; [0, 1]) be a test function satisfying ψ = 1 on [−1, 1] and ψ = 0 outside[−2, 2] For r > 0, denote by ψr= ψl(x

rβ)ψl( t

rα) where l is chosen later on and

α, β > 0 In what follows, we use the notation

Br= {(x, t); |x| ≤ rβ and|t| ≤ rα}.

Suppose that (u, v) is a nonnegative weak supersolution of (2.2) Then we

u∂tψrd x d t−Z

um∆ψrd x d t ≥Z

vpψrd x d t (2.10)By using similar arguments as in (2.8), we also obtain from (2.10) that

(2.11)It results from the definition of supersolutions above with the test functionψk

k= (l−2)qml thatZ

rd x d t−Z

rd x d t ≥Z

rd x d t (2.12)Again, similar to (2.8), we also arrive at

vpψrd x d t.

For simplicity of notation, we put

Suppose that the four powers of r in the right hand side of (2.15) are

nonposi-tive, i.e.,κi≤ 0 for i = 1, 2, 3, 4 Then, arguing as in the proof of Theorem 2.1,

we obtain from (2.15) thatZ

vpd x d t = 0.

Consequently, v= 0 and this implies u = 0.

The rest of the proof of nonexistence result is devoted to showing that, withsuitable α, β > 0, the four powers of r in the right hand side of (2.15) are

nonpositive We first have some elementary properties of these powers.

Without loss of generality, we may assume that p≥ q Then,