Blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity
We study the blow-up phenomena arising in a p-laplacian equation with reaction and absorption terms. We show that there exists a unique blowing-up approximate self-similar solution which describe the asymptotic singular behaviour of a wide class of solutions. As a consequence, we conclude that in th...
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Інститут прикладної математики і механіки НАН України
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irk-123456789-1246332017-10-01T03:02:58Z Blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity Chaves, M. We study the blow-up phenomena arising in a p-laplacian equation with reaction and absorption terms. We show that there exists a unique blowing-up approximate self-similar solution which describe the asymptotic singular behaviour of a wide class of solutions. As a consequence, we conclude that in this class, the absorption became negligible in finite time in the competition between the reaction and the absorption terms. 2004 Article Blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity / M. Chaves // Український математичний вісник. — 2004. — Т. 1, № 4. — С. 583-597. — Бібліогр.: 22 назв. — англ. 1810-3200 1991 MSC. 35K55, 35K65 http://dspace.nbuv.gov.ua/handle/123456789/124633 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We study the blow-up phenomena arising in a p-laplacian equation with reaction and absorption terms. We show that there exists a unique blowing-up approximate self-similar solution which describe the asymptotic singular behaviour of a wide class of solutions. As a consequence, we conclude that in this class, the absorption became negligible in finite time in the competition between the reaction and the absorption terms. |
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Chaves, M. |
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Chaves, M. Blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity Український математичний вісник |
| author_facet |
Chaves, M. |
| author_sort |
Chaves, M. |
| title |
Blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity |
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Blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity |
| title_full |
Blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity |
| title_fullStr |
Blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity |
| title_full_unstemmed |
Blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity |
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blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity |
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Інститут прикладної математики і механіки НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/124633 |
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Blow-up phenomena arising in a reaction-absorption-diffusion equation with gradient diffusivity / M. Chaves // Український математичний вісник. — 2004. — Т. 1, № 4. — С. 583-597. — Бібліогр.: 22 назв. — англ. |
| series |
Український математичний вісник |
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AT chavesm blowupphenomenaarisinginareactionabsorptiondiffusionequationwithgradientdiffusivity |
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2025-07-09T01:45:39Z |
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2025-07-09T01:45:39Z |
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1837131933857873920 |
| fulltext |
Український математичний вiсник
Том 1 (2004), N 4, 583 – 597
Blow-up Phenomena Arising in a
Reaction-absorption-diffusion Equation with
Gradient Diffusivity
Manuela Chaves
(Presented by A. E. Shishkov)
Abstract. We study the blow-up phenomena arising in a p-laplacian
equation with reaction and absorption terms. We show that there exists
a unique blowing-up approximate self-similar solution which describe the
asymptotic singular behaviour of a wide class of solutions. As a conse-
quence, we conclude that in this class, the absorption became negligible
in finite time in the competition between the reaction and the absorption
terms.
1991 MSC. 35K55, 35K65.
Key words and phrases. Nonlinear heat propagation, asymptotic
behaviour, self-similarity, blow-up.
1. Introduction
In this paper we are concerned with the blowing-up behaviour of
solutions of the reaction-diffusion-absorption equation with gradient dif-
fusivity
ut = (|ux|m−1ux)x + up − uq, m > 1, p > 1, q > 1, (1.1)
in the range of parameters 1 < q < p < m. We consider initial data
u(x, 0) = u0(x) ∈ C0(R) such that blow-up in a finite time T occurs, in
the sense that the solution u(x, t) exists and is bounded for every t < T
and satisfies that
sup ‖u(x, t)‖∞ → ∞, as t→ T.
In the study of the blow-up asymptotic properties, similarity solu-
tions of the nonlinear partial differential equations (PDEs) are known
Received 15.12.2003
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
584 Blow-up Phenomena...
to play a fundamental role. In particular, this holds for typical models
from nonlinear diffusion and combustion/absorption theory, which was
the origin of a general classification of types of self-similarities (of the
first and second types) formulated in 1950s and various applications of
group-theoretical techniques and the renormalization group analysis. We
refer to G. I. Barenblatt’s book [1] where a list of other references is
available.
Such self-similar solutions are of special importance in the asymptotic
analysis of blow-up singularity formation for quasilinear heat equations,
i. e., those singularities which occur in finite time, see [22] and reference
there in. However, in many models from the combustion/absorption the-
ory, self-similar solutions of the related equation with the singular be-
haviour under study do not exist. In the analysis of such kind of sin-
gularities, it is shown that the behaviour of the solutions under study
is somehow hidden and it becomes necessary, in order to understand the
problem, to determine which terms of the equation become dominant and
which ones negligible in the analyzed singularity formation. Therefore,
a new simplified equation appears, which will play the key role in the
asymptotic analysis. Then, the equation under study is considered as a
perturbation of the main one, and an stability analysis in the context of
perturbed dynamical systems becomes necessary. The study of blow-up
problems has attracted a considerable attention during the last years.
Concerning the asymptotic blow-up analysis, we refer the works [5], [13],
[16], [19] and [15] where the asymptotic blow-up analysis has been done
for different semilinear heat equations and [7], [8], [9], [3] and [4] for
quasilinear equations. For an extensive list of other relevant results in
the theory of blow-up, see [22] and [10].
1.1. Blow-up phenomena for the reaction-diffusion equation
Before dealing with solutions of the reaction-absorption-diffusion
equation (1.1), we briefly comment the main properties of the solutions
of the reaction-diffusion equation with gradient diffusivity
ut = (|ux|m−1ux)x + up, m > 1, p > 1, (1.2)
in the range of parameters 1 < p < m. It is known that solutions of (1.2)
blow-up in a finite time T , and moreover, in the range of parameters
under study, the blow-up is global and it is proved that the solution
become unbounded for every x ∈ R.
The equation is invariant under a group of scaling transformations
M. Chaves 585
and admits self-similar solutions of the form
u∗(x, t) = (T − t)−αf(ξ), ξ = x/(T − t)β , α = 1/(p− 1), (1.3)
where β = (p−m)/(p− 1)(m+ 1) and f satisfies the ODE
A(f) ≡ (|f ′|m−1f ′)′ − βξf ′ − αf + fp = 0, (1.4)
with f positive and f(ξ) → 0 as ξ → ∞.
The problem of existence of such non-trivial similarity solutions is un-
derstood in greater detail. In fact, the existence result can be obtained
without relevant modifications, following the ideas in [22], Chapter 4,
where the analogous existence problem for the Porous medium equation
is treated. The uniqueness proof can be obtained by using an ODE ap-
proach, [2] or by means of PDE techniques via the Sturm Theorem on
zero sets for uniformly parabolic equations, see [9] for results concerning
the PME equation and [3] for a general approach. Comparison intersec-
tion theory, which strongly relies on the Sturm Theorems for PDEs, cf.
[21], has been widely used in the study of the classical properties of the
solutions of parabolic PDEs, including regularity and asymptotic anal-
ysis. For the application of Sturmian Intersection Theory in the study
of nonlinear parabolic equations with singularities, see [6] and the list of
references there in.
Concerning the stability of solutions, it usually holds that the non-
trivial profile f(ξ) ≥ 0 (or the profile of the simplest geometric shape in
several profiles are available) play a key role in the asymptotic analysis
of general solutions. Such similarity solutions are known to be asymptot-
ically stable as t→ T− in the corresponding rescaled variables,
θ∗(ξ, τ) = (T − t)−αu(ξ(T − t)β , t), τ = − ln(T − t) → ∞, (1.5)
where the rescaled solution θ∗ satisfies the rescaled parabolic equation
θ∗τ = A(θ∗) in R × R+. (1.6)
In [3] the authors stated a general stability result, which includes the
analysis of the blow-up behaviour of solutions of (1.2) as a particular
application. Next we deal with the analysis of the analogous stability
problem for solutions of equation (1.1).
2. Preliminaries and main results
2.1. Global existence and Blow-up phenomena for (1.1).
We begin by showing that due to the presence of absorption and
reaction terms in (1.1), there exist global solutions for small enough initial
586 Blow-up Phenomena...
data, and also blowing-up solutions of the Cauchy problem related to
(1.1). In fact, a straightforward calculation shows that the function
Wk1,k2(x, t) = (k1t+ k2)
−1/(q−1), t > 0,
is a supersolution of (1.1) if k1 is small and k2 is large enough. Therefore,
it follows from the Maximum Principle that any solution with initial data
satisfying u0(x) ≤ Wk1,k2(x, 0) is bounded for every value of time. On
the other hand, one can construct blowing-up subsolutions of (1.1) in the
following way. For a fixed 0 < ε < 1, consider fε(ξ) the profile satisfying
equation
A(fε) = (1 − ε)fpε ,
with A defined in (1.4). Define
Wε(x, t) = (h(t))−αfε(x(h(t))
−β),
where h(t) is a positive function satisfying,
h′(t) = −1 +
1
α
(h(t))(p−q)α, h(0) < 1 − ε.
It is not difficult to check that so defined Wε(x, t) is a subsolution of
(1.1) which blows up at finite time and therefore, the same holds for any
solution u(x, t) with initial data verifying u0(x) ≥Wε(x, 0).
In the next sections, we show by developing the method described in
[4], that the absorption term in (1.1) becomes negligible, in the blow-
up analysis, in finite time. Therefore the blowing up solutions are also
described by means of equation (1.2) and in the appropriate variables,
the rescaled solution converges to the self-similarity profile f∗ in (1.3).
In order to prove the result we proceed as follows. We first establish
the result for symmetric and decreasing (for x > 0) initial data. This
allow us to construct a family of solutions of (1.1) with some appropriate
geometric features such that the stability theorem in [4] applies and yields
to the stability result for a general initial data in the class under study.
2.2. Rescaled variables and a priori estimates
Following the self-similar structure described above, we introduce the
rescaled variables,
θ(ξ, τ) = (T − t)−αu(ξ(T − t)β , t), ξ = x(T − t)−β , τ = − log(T − t).
By substituting in (1.2), we arrive at the rescaled perturbed equation
θτ = A(θ) − e−(p−q)ατθq, τ > 0 (2.1)
M. Chaves 587
to be compared with the corresponding autonomous one (1.6). The sta-
bility theorem is now stated in terms of the rescaled variables in the
following way.
Theorem 2.1. Let θ(ξ, τ) be the rescaled solution of (2.1) with initial
data θ0(ξ) ∈ C0(R), corresponding to the blowing-up solution of (1.2)
u(x, t), with blow-up time T . Then, there holds
θ(τ) → f∗, as τ → ∞,
uniformly in R.
We begin with some a priori bounds for the solutions to be used later
on.
Lemma 2.1. Let u(x, t) a solution of (1.2) with initial datum u0 =
θ0 ∈ C0(R) and blowing up at time T . Then, the corresponding rescaled
solution θ(ξ, τ) satisfies:
‖θ(τ)‖ > c∗ for any τ ≥ 0.
Proof. It is strongly based on the existence of the family of blowing-up
homogeneous solutions of (1.2) of the type,
HT (t) = c∗(T − t)−α, c∗ = αα.
Assume that the result false and that ‖θ(τ0)‖ ≤ c∗ for some τ0 ≥ 0.
Then in the original variables we obtain that
u(x, t0) ≤ HT (t0) ≡ c∗(T − t0)
−α, with t0 = 1 − eβτ0 .
Hence, taking into account thatHT (t) is a supersolution of equation (1.1)
for every T ∈ R and by the strong Maximum Principle, we obtain that
for a fixed positive δ ≪ 1 and arbitrarily small ε > 0,
u(x, t0 + δ) ≤ HT+ε(t0 + δ).
By comparison, the same inequality holds for t ≥ t0 +δ and hence u(x, t)
does not blow-up at time T contradicting the assumption.
Lemma 2.2. Under the assumptions above, there exists a constant M>0
such that
θ(ξ, τ) ≤M, ∀ τ > 0.
588 Blow-up Phenomena...
Proof. Let ±a be the interfaces of the symmetric equilibrium f∗(ξ). We
first prove that there exists τ0 ≥ 0 such that
θ(ξ, τ) ≤ f∗(0) + 1, for all τ ≥ τ0 and every |ξ| ≥ a. (2.2)
Assume that (2.2) is false for certain sequences {τj} → ∞ and {ξj} with
ξj ≥ a. Then, by the decreasing and symmetric hypotheses, the same
holds for every ξ ∈ [−ξ0, ξ0], with ξ0 = lim infj→∞ ξj . We construct a
subsolution in a similar way as in Subsection 1.1. For a fixed ε ∼ 1− we
define
Wε(ξ, τ) = (h(τ))−αfε(ξ(h(τ))
−β),
where h(τ) is a positive function satisfying,
h′(τ) = −1 + (h(τ))(p−q)α, h(0) = m0 ∼ 1−.
It is not difficult to check that so defined Wε(ξ, τ) is a subsolution of (2.1)
for every τ ≥ τj0 ≫ 1 which blows up at finite time. On the other hand,
by the contradiction assumption, we have for ε and m0 closed enough to
1 that θ(ξ, τj0) ≥Wε(ξ, 0). Then, by using invariance translation in time
and the Maximum Principle, one can prove that the θ(ξ, τ) also blows-up
in finite time and a contradiction follows.
Finally we deal with the upper bound in [−a, a]. On the one hand, one
has that for every M ≫ f∗(0), the symmetric and decreasing stationary
profile fM satisfying fM (0) = M is a supersolution of (2.1) in [−a, a]×R+
and fM → ∞ in [−a, a] as M → ∞. Then the result follows by applying
the Maximum Principle.
Lemma 2.3. Let θ(ξ, τ) be the solution of (2.1) with initial datum θ0(ξ)
and interfaces ±a(τ). There exists a constant c > 0 such that
a(τ) ≤ c for every τ ≥ 0.
Proof. We first note that the a priori bound in (2.2) can be improved
and give that
θ(ξ, τ) ≤ f∗(0) for all τ ≥ 0 and every |ξ| ≥ a. (2.3)
In fact, by repeating the arguments in Lemma 2.2 with no significant
modifications it follows that for every τ ≥ 0 there exists ξ0(τ) ≤ a such
that θ(ξ0(τ), τ) ≤ f∗(ξ0(τ)) whence (2.3) follows.
Concerning the interfaces, consider the function f∗(ξ−A) with shifted
argument, where A > 0 is large enough and such that
θ(A, τ) < f∗(0) for every τ ≥ 0, and θ(ξ, 0) ≤ f∗(ξ −A) for ξ ≥ A.
Since f∗(ξ − A) is a supersolution of (2.1) in [A,∞) × R+, the result
follows by comparison.
M. Chaves 589
3. The stability S-Theorem
In order to apply the main Stability Theorem in [4], we first remain
the main notions, properties and hypotheses and setting the appropriate
frame for the application above.
Let X be a complete metric space with the distance function d(·, ·).
In the application to parabolic blow-up problems, where the Sturm The-
orem on zero sets plays a key role, the space X = C1(I), where I ⊂ R
is a bounded closed interval, is a natural metric for using Sturmian in-
tersection properties. However, due to a special geometric structure of
solutions in this application and the standard regularity theory, we set
X = C(I).
We deal with a bounded class L of solutions θ ∈ C([0,∞) : X) of
(2.1) defined for every τ > 0 with values in X and by U we denote
the corresponding bounded subset of admissible initial data. Actually,
the analysis is based on metric-topology arguments applied to families of
curves {θ(τ)} and {θ∗(τ)}, which are formally treated as solutions of the
abstract equations (2.1) and (1.6) respectively.
Denote by ω(u0) the ω-limit set of an orbit {θ(τ), τ > 0} ⊂ L of
equation (2.1) with initial data θ0 ∈ U
ω(θ0) = {f ∈ X : there exists a sequence {τj}→∞ such that θ(τj)→f},
which is assumed to be compact subset of X.
By {ϕ∗
τ} we denote a continuous semigroup induced by the autono-
mous equation (1.6), globally defined on a bounded subset Ū∗ of admis-
sible initial data. The corresponding bounded class of solutions θ∗(τ) =
ϕ∗
τ (θ
∗
0) ∈ C([0,∞) : X) with θ∗0 ∈ Ū∗ is denoted by L̄∗. However, we
only deal with a “restricted” class L∗ ⊆ L̄∗ and its corresponding subset
U∗ of initial data. Both subsets are characterized later on. By ω∗(θ∗0)
with θ∗0 ∈ U∗, we denote the corresponding ω-limit set. Let f∗ be an
equilibrium
ϕ∗
τ (f
∗) ≡ f∗.
Let us present the main hypotheses.
(H1) Compactness of the orbits of (2.1). We assume that, for any
data θ0 ∈ U , orbit {θ(τ), τ > 0} is relatively compact in X, and if
θs(τ) ≡ θ(τ + s), τ, s > 0,
then the sets {θs} are relatively compact in L∞
loc([0,∞) : X).
(H2) Convergence of equations. This means that B(·, τ) is a small
perturbation of A(·) in the sense that given a solution θ(τ) ∈ L of
590 Blow-up Phenomena...
(2.1), if for a sequence {τj} → ∞ the sequence {θ(τj + τ)} converges
in L∞
loc([0,∞) : X) as j → ∞ to a function θ∗(τ), then θ∗(τ) ∈ L∗ is a
solution of (1.6).
Next, we introduce key hypotheses including a topological (oriented
intersection, in applications) S-relation of partial ordering induced by the
non-perturbed evolution driven by equation (1.6).
We present first the hypotheses related to the autonomous equation.
(H3) Ordered invariant one-parametric family from domain of
stability. Let W s(f∗) be the domain of attraction (asymptotic stability)
of the equilibrium f∗
W s(f∗) = {θ∗0 ∈ U∗ : ϕ∗
τ (θ0) → f∗ as τ → ∞}.
We assume that there exists a one-parametric continuous set F ∗ = {fµ, µ
∈ (µ1µ2)} ⊂ W s(f∗) such that fµ∗ = f∗ for some µ∗ ∈ (µ1, µ2). Each
closed subinterval {fµ, µ1 < a ≤ µ ≤ b < µ2} is relatively compact in X.
The family F ∗ is one-parametric and we assume that it admits a total
ordering denoted by � in the sense that fµ � fν (or fν � fµ) for all µ ≤ ν.
Moreover, fµ ≺ fν for all µ < ν, i. e., fµ 6= fν .
For any µ ∈ (µ1, µ2), denote
Fµ(τ) ≡ ϕ∗
τ (fµ) → f∗ as τ → ∞. (3.1)
The invariance of the family means that for every µ ∈ (µ1, µ2) and τ > 0,
Fµ(τ) ∈ F ∗. i. e., Fµ(τ) ≡ fρ(τ) for some continuous function ρ(τ) with
ρ(0) = µ. This implies that for the autonomous equation (1.6) we need
to specify two orbits {F±(τ), τ ∈ R}, F±(τ) → f∗ as τ → ∞ satisfying
F−(τ1) ≺ F−(τ2) and F+(τ1) ≻ F+(τ2) for any τ1 < τ2 and F−(τ) ≺ f∗ ≺
F+(τ) for any τ ∈ R.
(H4) Asymptotic structural properties and intersection S-rela-
tion for (1.6). (i) Asymptotic transversality of F ∗: there exist µ1 < ν <
µ < µ2 such that for any θ∗0 ∈ U∗,
fν ≺ θ∗0 ≺ fµ. (3.2)
(ii) S-relation and S-semigroup. We assume that the total ordering in
F ∗ can be extended as a binary relation for solutions in L∗. As we have
mentioned, in the applications this S-relation is induced by the Sturmian
intersection property. It can be classified as a “restricted partial ordering"
of solutions θ∗(τ) ∈ L∗ with data from U∗ and elements of F ∗ = {fµ}.
The S-relation � satisfies two properties of partial ordering for any
v, v1, v2 ∈ U∗:
M. Chaves 591
(i) (reflexivity) v � v, and
(ii) (antisymmetry) v1 � v2 and v2 � v1 imply v1 = v2.
The constraint of S-relation induced by the subset F ∗ of particular
elements does not satisfy the transitivity property, i. e., v1 � fµ � v2
does not imply that v1 � v2 for any v1, v2 ∈ U∗. Actually, such relation
is defined relative to the elements of F ∗ only, and we do not define any
partial ordering in U∗ or L∗.
Later on we will use � as a standard ordering relation, so that v ≺ fµ
means that v � fµ and v 6= fµ. The S-relation is assumed to be closed
meaning that for any convergent sequence {vn} ⊂ U∗ there holds
vn � fµ and vn → f̄ =⇒ f̄ � fµ. (3.3)
Let us present the main hypothesis on the autonomous evolution: the
semigroup ϕ∗
τ on U∗ induced by equation (1.6) preserves the S-relation
relative to the set F ∗ (and is called an S-semigroup) in the following
sense: given a µ > 0,
v0 ≺ fµ (v0 ≻ fµ) =⇒ v(τ) ≺ Fµ(τ) (v(τ) ≻ Fµ(τ)) for all τ > 0.
(3.4)
Moreover, although is not always necessary for the asymptotic analysis,
in main applications the semigroup ϕ∗
τ is strong S-semigroup, i. e., for
any given µ > 0, τ0 ≥ 0 and arbitrarily small τ > 0, there exists a δ > 0
such that
v(τ0) ≺ Fµ(τ0) (v(τ0) ≻ Fµ(τ0))
=⇒ v(τ0 + τ) ≺ Fµ−δ(τ0 + τ) (v(τ0 + τ) ≻ Fµ+δ(τ0 + τ)).
Dynamical systems generating order-preserving semigroups satisfy a
number of fundamental properties, and their asymptotic behaviour is well
understood, see books [14] ,[20] and the papers [17], [18] and [10]. For
the study of perturbed dynamical systems see the book [12] where a list
of relevant references in this subject is available.
Our applications to blow-up singularities in reaction-diffusion or reac-
tion-absorption equations deal with dynamical systems admitting no par-
tial ordering between solutions having the same blow-up time. One can
see that the S-relation of restricted partial ordering in (H4) which mimics
the Sturm Theorem on zero sets for linear parabolic equations, expresses
purely geometric intersection properties of curves and cannot be extended
to any partial ordering of solutions. Moreover, we note that even inter-
section comparison theory do not apply when comparing solutions of
different equations, difficulty solved with the approach in [4].
We now state the main stability theorem (the S-Theorem, for short).
592 Blow-up Phenomena...
Theorem 3.1. Under hypotheses (H1)–(H4), for any θ0 ∈ U there hold
ω(θ0) = f∗.
3.1. Application of the S-Theorem. Blowing-up behaviour for
equation (1.2).
As we mention above, we consider X = C(I) be the space of bounded
and continuous compactly supported functions defined on sufficiently
large closed symmetric interval I = [−k, k], k ≫ 1 defined taking into
account the analysis of the interfaces of the solutions in Lemma 2.3. We
consider the distance function given by the L∞-norm ‖·‖. The convenient
solution class L is given by
L = {θ(τ) ∈ X satisfying (2.1) and c1 ≤ ‖θ(ξ, τ)‖ ≤ c2 ∀ τ ≥ τ0},
(3.5)
where c1 = c1(k) > 0 is a sufficiently small constant and the constant
c2 = c2(k) > c1 is large enough, both defined taking into account the a
priori bounds of the solution stated in Lemmas 2.1 and 2.2. Then, we
denote by U the class of such smooth compactly supported initial data
θ0 such that the corresponding solution θ ∈ L. Finally, the class L∗ of
solutions θ∗(τ) of the unperturbed equation (1.6) is defined in the same
way, and by U∗ we denote the bounded set of the corresponding initial
data θ∗0.
We next prove via the S-Theorem that L ⊂ W s(f∗) and hence, the
result stated in Theorem 2.1 follows. We need to check hypotheses (H1)–
(H4). The proof is similar to that in [4] for the Porous medium equa-
tion. We include it for the convenience of the reader. To begin with,
we note that interior regularity results for parabolic equations guarantee
the compactness and convergence hypotheses (H1) and (H2). To see that
ω(θ0) ∈ U∗ for every θ0 ∈ U we argue as follows. Assume for contradic-
tion that there exists f = limj→∞ θ(τj) ∈ ω(θ0) which does not belong
to U∗. This means that one of the bounds in the definition of L∗ is not
true for the corresponding solution θ∗(ξ, τ) and some τ1 > τ0. By the
convergence θ(ξ, τ + τj) → θ∗(ξ, τ) uniformly on [τ0, τ1] × R we deduce
that the assumption holds for θ(ξ, τ0 + τj) for any τj ≫ 1 contradicting
the definition of L.
Consider the crucial hypotheses (H3)–(H4) dealing with the class of
solutions L∗ for the unperturbed equation (1.6) (cf.[3]).
(H3) Let us introduce the family F ∗ ⊂ W s(f∗). We define functions fµ
for every µ ∈ R by translation
fµ(ξ) = f∗(ξ − µ) in R.
M. Chaves 593
The corresponding solution Fµ(τ) of the unperturbed parabolic equation
(1.6) is given explicitly
Fµ(τ) = ϕ∗
t (fµ) ≡ f∗(ξ − µeβτ ) in R × R+.
Since β < 0, we have that for any µ ∈ R, Fµ(τ) → f∗ as τ → ∞
uniformly on compact subsets in R, so that F ∗ ⊂W s(f∗). Thus,
F ∗ = {fµ, µ ∈ R}
is a continuous one-parametric family of functions satisfying fµ∗ ≡ f∗ for
µ∗ = 0 and
fµ(·) → 0 = f∞ as µ→ ±∞. (3.6)
The total ordering � in F ∗ is straightforward. It characterizes the
number and the character of intersection of profiles from F ∗. We say
that fλ ≺ fν if these profiles intersect each other exactly once and the
difference fν(ξ)− fλ(ξ) has a change of sign from − to + at the intersec-
tion. Then fλ � fν means that either fλ ≺ fν or fλ = fν .
(H4) (i) Asymptotic transversality of F ∗. The property (3.2) fol-
lows from obvious geometric structure of the family F ∗. In particular, it
suffices to observe that the interfaces aµ < bµ of the function fµ(ξ) satisfy
aµ → ±∞, bµ → ±∞ as µ→ ±∞.
This implies that the asymptotic transversality assumption holds for any
initial data with compact support.
(ii) S-relation and S-semigroup. We keep the same definition of ≺ and
� as in (H3) for functions θ∗0 ∈ U∗. Obviously, both the reflexivity and
antisymmetry properties of the S-relation hold. In order to show that
the S-relation is closed in the sense of (3.3) and that the S-semigroup
property (3.4) holds, we need to prove special intersection properties
characterizing solutions in L∗. The proof relies on the existence of a
two-parametric family of functions G∗ = {fλµ} from the local domain of
unstability W u(f∗) of the equilibrium f∗. These functions are defined as
follows:
fλµ (ξ) = λ−αf∗((ξ − µ)λ−β), µ ∈ R, λ ∈ R.
The corresponding solutions of the PDE (1.6) are given explicitly
F λµ (ξ, τ) = (1 − (1 − λ)eτ )−αf∗((ξ − µeβτ )(1 − (1 − λ)eτ )−β). (3.7)
These rescaled solutions are obtained from the self-similar ones for
the original PDE by translations in both independent variables x and t.
We have that for any fixed µ, solutions (3.7) stabilize as τ → ∞ to the
594 Blow-up Phenomena...
trivial one f∞ ≡ 0 if λ > 1 and blow-up in finite time if λ < 1. For λ = 1
we are given the family F ∗ from the domain of stability of f∗.
Next we introduce the main intersection property of the evolution in
L∗. Given a solution θ ∈ L∗, we denote by Iµ(τ) ≡ Int(θ(τ), fµ) the
number of intersections of θ(ξ, τ) and the function fµ(ξ).
Lemma 3.1. Let θ(τ) ∈ L∗ and θ0 6∈ F ∗. Then
(i) for any µ ∈ R, Iµ(τ) > 0 for τ ≥ 0, and
(ii) Int(g, f∗) > 0 for any g ∈ ω(θ0).
Proof. The proof of (i) is quite similar to the stated for Lemma 2.2 by
using special subsolutions of the type (3.7) and we omit the details. For
(ii), given a sequence {τk} → ∞, we pass to the limit τk+τ → ∞ so that
θ(τk + τ) → θ̃(τ) and repeat the same argument applied to the solution
θ̃ ∈ L∗ of the equation (1.6) with initial data g ∈ ω(θ0).
Let us show that the S-relation is closed in the sense of (3.3). In-
deed, if f̄ � fµ does not hold, then, by construction, the only possibility
is that f̄ 6≡ fµ does not intersect fµ. Since f̄ ∈ U∗, this contradicts
Lemma 3.1, (i).
Finally, we prove that {ϕ∗
t } is an S-preserving semigroup preser-
ving the S-relation. This is a consequence of the Sturm Theorem and
Lemma 3.1. Assume that θ∗0 � Fµ(0). This means that Iµ(0) = +1,
i. e., there exists a unique intersection of the profiles and the difference
Fµ(ξ, 0) − θ∗0(ξ) changes sign from − to + at the intersection. For dege-
nerate equations admitting weak solutions, intersections can be a point
or an interval. By the Sturm Theorem and Lemma 3.1, it follows that
Iµ(τ) ≡ +1 for every τ > 0 and hence, the same local character of the
intersection is preserved in time. Therefore θ∗(τ) ≺ Fµ(τ) for any τ > 0.
By repeating the arguments relative to the profile fν and the solution
Fν(τ), we obtain the opposite estimate θ∗(τ) ≻ Fν(τ) for all τ > 0.
Hence, (H1)–(H4) hold and this provide us with the following conclu-
sion.
Theorem 3.2. Any bounded and compactly supported rescaled solution
of the parabolic equation (2.1), θ(ξ, τ) ∈ L , satisfies
θ(τ) → f∗ as τ → ∞
uniformly in R.
As a straightforward consequence of the analysis, we can also con-
clude that any bounded and compactly supported solution θ(τ) of (2.1)
stabilize either to the trivial solution or to the profile f∗. The analysis
M. Chaves 595
performed shows how the invariance properties of the semigroup ϕt can
be used in order to get estimates for a “good" set of initial data, namely
symmetric and decreasing initial data. This set provides us with a fam-
ily F̂ ∗ satisfying (H3) and (H4) with respect to the perturbed equation
(2.1). Hence, although this equation is not autonomous, the classical in-
tersection comparison arguments apply to any solution of (2.1) and the
solutions with initial data in the family F̂ ∗. We note that now we are
comparing solutions of the same equation (2.1). Then, the stability the-
ory in [3] applies without changes, and the convergence result follows for
a wide class of initial data. Such an approach allows to avoid rather
delicate calculations and estimates for general orbits.
End of the Proof of Theorem 2.1 Theorem 3.2 guarantees that any
solution θ ∈ L of the perturbed equation (2.1) has the asymptotic self-
similar behaviour with the unique rescaled similarity profile f∗. It is
clear from the a priori bounds stated in Lemmas 2.1, 2.2 and 2.3, that
any solution with symmetric and decreasing initial data belongs to L and
therefore the result in Theorem 2.1 follows.
On the other hand, consider an initial datum θ0 satisfying the previous
hypotheses. Define a one-parametric family F̂ ∗ = {f̂λ(ξ) = θ0(ξ+λ), λ ∈
R} belonging to the domain of stability of f∗. Then, following the main
lines of the stability analysis of unperturbed equations in [2], Section 4,
we see that this family F̂ ∗ satisfies the crucial hypotheses (H3)–(H4)
relative to the perturbed equation (2.1) and the sets L and U . Therefore,
the stability result [3] applies and proves the convergence of any rescaled
solution from L (with blow-up time T ) to the unique profile f∗. The
proof is completed.
We note that the same stability analysis applies to more general per-
turbations under the assumptions that the associated semigroup admits
translations in both x and t, as well as spatial symmetry and monotonic-
ity properties. It could be also an interesting problem to analyze the
behaviour of the global bounded solutions. Our conjecture is that also
in this analysis asymptotic simplification occurs and the reaction term
becomes negligible in the asymptotic behaviour.
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M. Chaves 597
Contact information
M. Chaves Departamento de Matemáticas
Universidad Autónoma de Madrid
28049 Madrid,
Spain
E-Mail: manuela.chaves@uam.es
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