Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 72, pp. 1–21.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2023.72

QUALITATIVE PROPERTIES OF SOLUTIONS TO A

REACTION-DIFFUSION EQUATION WITH

WEIGHTED STRONG REACTION

RAZVAN GABRIEL IAGAR, ANA I. MUÑOZ, ARIEL SÁNCHEZ

Abstract. We study the existence and qualitative properties of solutions to

the Cauchy problem associated to the quasilinear reaction-diffusion equation

∂tu = ∆um + (1 + |x|)σup,
posed for (x, t) ∈ RN × (0,∞), where m > 1, p ∈ (0, 1) and σ > 0. Initial data

are taken to be bounded, non-negative and compactly supported. In the range
when m + p ≥ 2, we prove existence of local solutions with a finite speed of

propagation of their supports for compactly supported initial conditions. We
also show in this case that, for a given compactly supported initial condition,

there exist infinitely many solutions to the Cauchy problem, by prescribing the

evolution of their interface. In the complementary range m+ p < 2, we obtain
new Aronson-Bénilan estimates satisfied by solutions to the Cauchy problem,

which are of independent interest as a priori bounds for the solutions. We

apply these estimates to establish infinite speed of propagation of the supports
of solutions if m + p < 2, that is, u(x, t) > 0 for any x ∈ RN , t > 0, even in

the case when the initial condition u0 is compactly supported.

1. Introduction

This article concerns the qualitative theory of the weak solutions to the Cauchy
problem for the reaction-diffusion equation

∂tu = ∆um + (1 + |x|)σup, (x, t) ∈ RN × (0,∞), N ≥ 1, (1.1)

supplemented with the initial condition

u(x, 0) = u0(x), x ∈ RN . (1.2)

The exponents in (1.1) are considered throughout the paper to belong to the range

m > 1, 0 < p < 1, 0 < σ <∞, (1.3)

although we also give alternative proofs or even slight improvements of known
results with σ = 0. We consider bounded, compactly supported, non-negative and
non-trivial initial conditions, more precisely

u0 ∈ L∞(RN ), suppu0 ⊆ B(0, R), u0(x) ≥ 0, ∀x ∈ RN , u0 6≡ 0, (1.4)

2020 Mathematics Subject Classification. 35B44, 35B45, 35K57, 35K59.
Key words and phrases. Reaction-diffusion equations; weighted reaction; strong reaction;

Aronson-Bénilan estimates.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted June 13, 2023. Published October 23, 2023.

1



2 R. G. IAGAR, A. I. MUÑOZ, A. SÁNCHEZ EJDE-2023/72

and further regularity assumptions (such as continuity) will be specified at the
points where they are needed. In nonlinear diffusion problems it is standard to
consider data belonging to the space L1

loc(RN ) and we are sure that some of our
results can be extended to this weaker space than L∞(RN ). However, for simplicity
and also as in the range of exponents (1.3) finite time blow-up of bounded solutions
is expected (as established, for example, for self-similar solutions in recent works
such as [18, 19, 16]), we decided to avoid possible pointwise singularities at finite
points and thus require (1.4).

A deep study of (1.1) in the range p > 1 has been performed by Andreucci and
DiBenedetto in [1] for weights with any σ ∈ R, that is, both positive and negative.
Properties such as existence of local weak solutions under optimal growth conditions
on the initial data, estimates, regularity of them and Harnack-type inequalities are
obtained. The authors also specify in [1] that some of their results can be extended
to the range 0 < p < 1 but only when σ < 0 and leave open the case σ > 0 with
0 < p < 1. We find this latter case very interesting to study due to the merging
of the non-Lipschitz reaction term (which does not produce finite time blow-up by
itself) with the unbounded weight.

The non-weighted equation

ut = ∆um + up, (1.5)

corresponding to σ = 0 in (1.1) is nowadays quite well understood (at least in
dimension N = 1) after a series of works by de Pablo and Vázquez [26, 27, 28, 25]
and the outcome of them is very interesting, despite the fact that we are dealing with
an ill-posed problem. It is shown that solutions exist always and they are global
in time if u0 satisfies a growth condition similar to the one required for existence
in the porous medium equation, namely u0(x) = o(|x|2/(m−1)) [27] and that there
is always a minimal and a maximal solution, both constructed via approximations.
The most interesting and surprising features of this problem with σ = 0 are related
with the uniqueness. This property depends strongly on two aspects: the sign of
the critical exponent m+ p− 2 and the positivity or not of the initial condition u0.
More precisely we have the following:
• If m + p − 2 ≥ 0, uniqueness of solutions to the Cauchy problem (1.5)-(1.2)

holds if and only if suppu0(x) = RN [26]. If u0 is compactly supported, then its
support has the property of finite propagation and interfaces appear. Moreover,
the extent of the non-uniqueness property in this case is addressed in [28] where it
is shown that giving a rather general compactly supported initial condition u0 and
a function of time ξ(t) advancing faster than the interface of the (unique) minimal
solution constructed in [27], there is always a solution to the Cauchy problem with
data u0 and interface at time t > 0 given by ξ(t). This is a very strong and sharp
non-uniqueness property, giving rise in fact to an infinity of solutions.
• If m + p − 2 < 0, things change radically because of the infinite speed of

propagation: even if u0 is compactly supported it is shown in [26] that u(x, t) > 0
for any x ∈ RN and t > 0, thus we have a property known as quasi-uniqueness:
there exists a unique solution to the Cauchy problem for any initial condition u0

except for the trivial one u0 ≡ 0, where two different solutions are constructed.
Considering weighted reaction terms came as a natural extension of the already

well developed knowledge on the “classical” reaction-diffusion equations with reac-
tion of the form up or more general functions resembling it (see [32, 34] as important
monographs on this subject). Many results were achieved for the semilinear case



EJDE-2023/72 REACTION-DIFFUSION WITH WEIGHTED STRONG REACTION 3

m = 1 and unbounded weights of the form |x|σup (and sometimes even more general
weights V (x) instead of pure powers), always with p > 1. Fujita-type exponents
and conditions on the data for finite time blow-up to occur were studied in cel-
ebrated papers by Baras and Kersner, Bandle and Levine, Pinsky et al., see for
example [6, 5, 29, 30]. More recently, still with m = 1, an interesting question was
addressed: considering the equation

ut = ∆u+ |x|σup, (1.6)

it is natural to ask ourselves whether x = 0 (or more generally, any zero of a power-
like weight V (x)) can be a blow-up point. Examples of both possible situations
(when x = 0 is a blow-up point and when it is not) were constructed (mostly for
Cauchy-Dirichlet problems posed in bounded domains) in the series of recent papers
by Guo and collaborators [11, 12, 13, 14]. Finer analysis on how blow-up occurs,
with rates and local asymptotic behavior in self-similar forms near the blow-up time
and points were performed by Filippas and Tertikas [10] and in the very recent work
by Mukai and Seki [24] for different ranges of the exponent p > 1.

Because of their further complexity and the fact that even for the non-weighted
case σ = 0 there are some difficult open problems (see for example [34, Chapter
4]), equations such as (1.1) or its close relative

ut = ∆um + |x|σup, (1.7)

with m > 1 have been less considered in literature. Apart from the quoted paper
[1], the Fujita-type exponent and rather sharp conditions on the initial data for the
finite time blow-up to hold have been obtained by Qi [31] and then Suzuki [35] for
the case p > m > 1 (including also a part of the fast diffusion range m < 1 in [31]).
We recommend Suzuki’s paper to the reader as a well-written basic work on the
qualitative theory for these equations, while blow-up rates as t→ T also for p > m
are proved in [2].

In recent years, the authors of the present work started a larger project of un-
derstanding the patterns (in self-similar form) that solutions may take either in the
case of global solutions, or close to the blow-up time if this occurs, for (1.7). This
is an important part of the study, since it is well-known that such patterns are a
prototype of the general behavior of the equation, in the form of asymptotic profiles
as t → ∞ or t → T and also bring a deeper understanding on the blow-up sets
and rates. A series of papers [17, 20, 22, 15] addressed the question of the blow-up
profiles to (1.7) for m > 1 and 1 ≤ p ≤ m, where interesting and rather unexpected
behaviors were established. In all these cases, solutions blow up in finite time when
σ > 0, but their specific blow-up behavior it is shown to depend strongly on the
magnitude of σ. Going back to the case of interest for us, p ∈ (0, 1), we have proved
that blow-up profiles exist if the following condition is fulfilled:

L := σ(m− 1) + 2(p− 1) > 0. (1.8)

We classified such blow-up profiles in [16] in general dimensions N ≥ 2, following
previous results restricted to dimension N = 1 in [19, 21], obtaining again that the
sign of m+ p− 2 is fundamental for their existence and behavior.
• When m + p − 2 > 0, all the blow-up self-similar profiles present an interface

(that is, they are compactly supported) and there are two different types of possible
interface behaviors. This is both a manifestation of non-uniqueness (expected for



4 R. G. IAGAR, A. I. MUÑOZ, A. SÁNCHEZ EJDE-2023/72

compactly supported data) and a suggestion of possible non-existence of solutions
when u0(x) > 0, as there is no pattern they can approach when t→ T .
•Whenm+p = 2, self-similar blow-up patterns present a rather similar panorama,

but with a single type of interface behavior.
•When m+p < 2, a rather striking non-existence of any kind of blow-up profiles

(either with interfaces or not) occurs [19]. Such an outcome gives the intuition of an
infinite speed of propagation and a complete non-existence of non-trivial solutions.

In this paper we thus begin the qualitative study of the Cauchy problem for
(1.1) posed in RN . We thus address the quite interesting (in view of precedents
such as [26, 28]) question of speed of propagation and non-uniqueness for compactly
supported data, deriving in the process some new Aronson-Bénilan estimates for
the solutions to (1.1) when m + p < 2. We stress here that uniform lower bounds
on the weight are essential in the forthcoming proofs, thus technical complications
introduced by the weight in a neighborhood of x = 0 (where it is no longer uniformly
positive) appear when trying to adapt the same proofs to the close relative (1.7).
But let us present below in more detail our main results.

Main results. To state our results concerning the qualitative theory of solutions
to (1.1), we first have to introduce the notion of weak solution that will be used
throughout the paper. Let us denote by u(t) the mapping x 7→ u(x, t) for t > 0 fixed.
We will slightly modify the functional framework from Andreucci and DiBenedetto
[1] by passing in our weak formulation the full Laplacian to the test function.

Definition 1.1. A non-negative function u : RN × (0, T ) 7→ [0,∞) is said to be
a weak solution to the Cauchy problem (1.1)-(1.2) with initial condition u0 as in
(1.4) if the following assumptions are satisfied by u:

(a) Regularity assumption: we have

u ∈ C(0, T ;L1
loc(RN )) ∩ L∞loc(RN × (0, T )). (1.9)

(b) Weak formulation of (1.1): For any test function η ∈ C∞0 (RN × (0, T )) and
for any t ∈ (0, T ) we have∫

RN
u(x, t)η(x, t) dx+

∫ t

0

∫
RN

(−u(x, τ)ηt(x, τ)− um(x, τ)∆η(x, τ)) dx dτ

=

∫ t

0

∫
RN

(1 + |x|)σup(x, τ)η(x, τ)dx dτ.

(1.10)

(c) Taking the initial condition: this is done in L1 sense, more precisely

lim
t→0

u(t) = u0, with convergence in L1
loc(RN ) (1.11)

This change relaxes the functional assumptions of regularity for a solution, mak-
ing it easier to obtain weak solutions by limiting processes. We will also need
throughout the paper the notions of (weak) sub- and supersolution to (1.1). We
say that u is a weak subsolution (respectively weak supersolution) to (1.1) if condi-
tion (a) is fulfilled and condition (b) is modified in the sense that the equal sign is
replaced by ≤ (respectively ≥) for any test functions η ∈ C∞0 (RN×(0, T )) such that
η ≥ 0 and for any t ∈ (0, T ). Moreover, the notions of weak solution, subsolution,
supersolution to (1.1) (without the initial condition) can be defined in an obvious
way on time intervals [t1, t2] ⊂ (0, T ) instead of (0, T ) by just removing assumption
(c).



EJDE-2023/72 REACTION-DIFFUSION WITH WEIGHTED STRONG REACTION 5

Once defined the notion of weak solution, we are in a position to give below the
main theorems of this work. As explained above, it is expected from the results in
papers such as [26, 27, 28] for equations with non-weighted reaction that, in our
range of exponents, ill-posedness of the Cauchy problem will still hold and both
existence and uniqueness of solutions are an issue. As we shall see below, if we
consider initial conditions u0 as in (1.4), these basic properties of existence and
uniqueness of weak solutions are strongly related to the finite or infinite speed of
propagation of their edge of the support. Throughout the paper, we will denote by
C0(RN ) the space of continuous, compactly supported functions on RN .

Range m + p ≥ 2. Let us first focus on the range of exponents for which m+p ≥ 2
and σ > 0. In this case, we shall prove that, at least for some interval of time
t ∈ (0, T ), there exist infinitely many weak solutions to (1.1). We begin with the
existence of at least one local (in time) weak solution, as stated in the following
result.

Theorem 1.2 (Existence of compactly supported solutions). In our framework
and notation, assume that m+ p ≥ 2 and let u0 be an initial condition as in (1.4)
satisfying moreover that u0 ∈ C0(RN ). Then there exists T > 0 and there exists
at least a weak solution u to the Cauchy problem (1.1)-(1.2) for t ∈ (0, T ) which
remains continuous and compactly supported: u(t) ∈ C0(RN ) for any t ∈ (0, T ).

The proof relies on the construction of so-called minimal solutions to the Cauchy
problem (1.1)-(1.2), which are obtained through a limit process from a family of
approximating (regularized) Cauchy problems. It will be shown that such a mini-
mal solution exists for any compactly supported condition u0 and stays compactly
supported for t ∈ (0, T ), a property known as finite speed of propagation. The
(local in time) finite speed of propagation will be proved with the aid of compari-
son with solutions and supersolutions in self-similar form introduced in the recent
works [19, 21, 16].

The statement of Theorem 1.2 and precedents in the non-weighted case [28] give
the idea that uniqueness does not hold. In fact, we infer from Theorem 1.2 that
at intuitive level the weight V (x) = (1 + |x|)σ is equivalent to a constant for times
t ∈ (0, T ) (that is, while the support of u(t) remains finite), thus a similar property
to the non-weighted case concerning non-uniqueness of solutions is expected. The
next result characterizes the extent of this non-uniqueness, by showing that we
can prescribe in infinitely many ways the evolution of the interface of a solution
stemming from the same initial condition.

Theorem 1.3 (Non-uniqueness of compactly supported solutions). In our frame-
work and notation, assume that m+ p ≥ 2 and let u0 be an initial condition as in
(1.4) satisfying moreover that u0 ∈ C0(R). Then there exists T > 0 and infinitely
many weak solutions to the Cauchy problem (1.1)-(1.2) for t ∈ (0, T ). The same
existence of infinitely many weak solutions holds true for radially symmetric initial
conditions u0 ∈ C0(RN ) satisfying (1.4).

This result is rather similar to the non-weighted case σ = 0, although the proof
will be technically more involved: it is not clear whether there exists when σ > 0
a maximal solution, in order to get almost for free a different, second solution (as
it holds true for σ = 0, see [26, 27]), thus we have to use a different approach. The
statement of Theorem 1.3 will be thus enforced and made more precise in Section
3, where we show that the existence of infinitely many solutions is linked to a



6 R. G. IAGAR, A. I. MUÑOZ, A. SÁNCHEZ EJDE-2023/72

prescribed evolution of their interface in time, adapting but also slightly improving
techniques from [28].

Range m + p < 2. In this complementary range, our main goal is to prove that
solutions to the Cauchy problem for (1.1) with initial conditions as in (1.4) have
infinite speed of propagation, that is, they become immediately positive at any
point x ∈ RN . This fact extends to the non-homogeneous range σ > 0 a similar
result holding true for σ = 0, established in [26], but we use a different approach
which gives, along the way, a result of independent interest in the study of the
homogeneous equation (1.5). We begin by establishing the following Aronson-
Bénilan estimates (whose name stems from the celebrated short note [3] on solutions
to the porous medium equation) for solutions to the homogeneous case σ = 0. We
thus introduce the pressure function

v =
m

m− 1
um−1.

Theorem 1.4 (Aronson-Bénilan estimates when m+ p < 2 and σ = 0). Assume
that m+ p < 2. Let u be a weak solution to (1.5) in RN × (0, T ) with a continuous
initial condition u0(x) = u(x, 0) satisfying (1.4) and let v be the pressure variable
introduced above. Then the inequality

∆v ≥ −K
t
, K =

N

N(m− 1) + 2
, (1.12)

holds in the sense of distributions in RN , that means,∫ T

0

∫
RN

(
v(x, t)∆ϕ(x, t) +

K

t
ϕ(x, t)

)
dx dt ≥ 0, (1.13)

for any test function ϕ ∈ C∞0 (RN × (0, T )) such that ϕ ≥ 0 in RN × (0, T ).

Remark. Theorem 1.4 is expected to hold also for any σ > 0, provided N ≥ 2.
We give a formal proof of this fact at the end of Section 4. However, transforming
it into a rigorous proof is impossible by now for σ > 0, as we are lacking a well-
posedness result for the Cauchy problem (1.1)-(1.2) in the range m + p < 2. This
is why, we introduce this formal proof as a remark.

We then employ the Aronson-Bénilan estimates established in Theorem 1.4 and
some consequences of them in order to prove the infinite speed of propagation of
the supports of solutions to (1.1) when m+p−2 < 0, which is strongly contrasting
to the results established in the range m + p ≥ 2. We include the range σ = 0 in
the statement, as our approach gives an alternative proof to the one in [26].

Theorem 1.5 (Infinite speed of propagation when m + p < 2). If the exponents
m and p satisfy m + p < 2, Equation (1.1) has the property of infinite speed of
propagation for any σ ≥ 0, that is, every weak solution (if it exists) to the Cauchy
problem associated to (1.1) with continuous initial condition u0 as in (1.4) satisfies
u(x, t) > 0 for any x ∈ RN and t > 0.

The rest of this article is devoted to the proofs of the main theorems, following
the outlines explained in the comments near their statements. We then give at the
end of the paper a list of open problems and possible extensions of our study that
we believe to be interesting for future developments.



EJDE-2023/72 REACTION-DIFFUSION WITH WEIGHTED STRONG REACTION 7

2. Existence and finite speed of propagation when m+ p ≥ 2

The goal of this section is to prove Theorem 1.2 in the range of parameters
m + p ≥ 2. Let u0 be a continuous initial condition as in (1.4). The scheme of
the proof is based on the construction of a minimal solution via an approximation
process and showing that this minimal solution is a weak solution to the Cauchy
problem (1.1)-(1.2) with compact support at any time t ∈ (0, T ) for some T > 0.

Proposition 2.1. There exists some T > 0 and a continuous weak solution u
defined and compactly supported for t ∈ (0, T ) to the Cauchy problem (1.1)-(1.2)
with u0 as above such that for any other weak solution u to the Cauchy problem
(1.1)-(1.2) (if it exists), we have

u(x, t) ≤ u(x, t), for all (x, t) ∈ RN × (0, T ).

Proof. We divide the proof into three steps for the reader’s convenience. Let us
stress at this point that, while Step 1 of the proof is an adaptation of an analogous
construction in [27], the idea in Steps 2 and 3 strongly departs from the one used
in the previously mentioned work and employs results on self-similar solutions to
(1.7) published recently by the authors.

Step 1. Construction of the minimal solution. We approximate the Cauchy prob-
lem (1.1)-(1.2) by the following sequence of Cauchy problems for any positive integer
k ≥ 1

wt = ∆wm + min{(1 + |x|)σ, k}fk(w), (x, t) ∈ RN × (0,∞),

w(x, 0) = u0(x), x ∈ RN ,
(2.1)

where

fk(w) =

{
( 1
k )p−1w, if 0 ≤ w ≤ 1

k ,

wp, if w ≥ 1
k .

Since the nonlinearity in (2.1) is of the form g(x)h(w) with g ∈ L∞(RN ), g(x) ≥ 1
for any x ∈ RN and h is a Lipschitz function, we infer by standard results for quasi-
linear parabolic equations (see for example [9, 33]) that the Cauchy problem (2.1)
admits a unique solution wk defined for (x, t) ∈ RN × (0,∞), which is compactly
supported and continuous. The comparison principle is in force for the Cauchy
problem (2.1) and wk+1 is a supersolution to the problem (2.1), thus wk+1 ≥ wk
for any k ≥ 1. This allows us to introduce the pointwise (and monotone increasing)
limit (which might become infinite starting from some finite time)

u(x, t) = lim
k→∞

wk(x, t) <∞, (x, t) ∈ RN × (0, T∞),

which is well defined provided that T∞ > 0. This fact will follow from the construc-
tion of a “universal” family of supersolutions in self-similar form which is postponed
to Step 2 (in dimension N = 1) and Step 3 (in dimension N ≥ 2) below. Moreover,
the solutions wk are thus uniformly bounded on RN×(0, T ) for some T = T (u0) > 0
depending on u0, and this uniform boundedness, together with classical results in
[9, 33], imply that the family (wk)k≥1 is uniformly equicontinuous in RN × [0, T ],
hence there exists a subsequence (relabeled also wk for simplicity) which converges
locally uniformly to the same function u(x, t).

Then the fact that u is a continuous weak solution to (1.1)-(1.2) for (x, t) ∈
RN×(0, T∞) follows now readily from the previous convergences (assuming for now
the outcome of Steps 2 and 3 below): indeed, Lebesgue’s monotone convergence



8 R. G. IAGAR, A. I. MUÑOZ, A. SÁNCHEZ EJDE-2023/72

theorem ensures the assumptions (b) and (c) in Definition 1.1, while the uniform
bound by the supersolutions constructed in Steps 2 and 3 below, together with the
equicontinuity, give that u satisfies the regularity assumption (a) in Definition 1.1.
Moreover, if u is another weak solution to the Cauchy problem (1.1)-(1.2), then it
is a supersolution to the problem (2.1) for any k ≥ 1, whence u(x, t) ≥ wk(x, t)
for any k ≥ 1 and (x, t) ∈ RN × (0,∞) and by passing to the limit u ≥ u in
RN × (0, T∞), proving the minimality of u.

Step 2. Supersolutions in dimensions N = 1. We are left with the task of obtaining
a uniform bound from above for all solutions wk to the Cauchy problems (2.1), k ≥
1, at least up to some (short) finite time. This follows by comparison with suitable
supersolutions in self-similar form constructed in recent works by the authors such
as [16] for N ≥ 2 and m + p ≥ 2, respectively [19, 21] in dimension N = 1 and
either m+ p > 2 or m+ p = 2. If we restrict ourselves only to dimension N = 1 for
this step, it is shown in the previously quoted works that, in our range of exponents
together with the extra condition σ > 2(1−p)/(m−1), there are radially symmetric
blow-up self-similar supersolutions to (1.7) in the form

u(x, t) = (T − t)−αf(|x|(T − t)β), α =
σ + 2

L
, β =

m− p
L

, (2.2)

where L > 0 has been defined in (1.8), such that their self-similar profiles f solve
the differential equation

(fm)′′(ξ)− αf(ξ) + βξf ′(ξ) + ξσf(ξ)p = 0,

ξ = |x|σ(T − t)β .
(2.3)

Moreover, it is established in [19, Proposition 4.1] if m+ p > 2 and in [21, Propo-
sition 3.1] if m + p = 2 that the self-similar profiles f of the supersolutions in the
form (2.2) fulfill the following two additional properties:
• f is strictly decreasing until reaching the zero level: f(0) = A > 0, f ′(ξ) < 0

at points ξ ≥ 0 where f(ξ) > 0.
• f presents an interface at some finite point ξ0 ∈ (0,∞), that is, f(ξ0) = 0,

f(ξ) > 0 for any ξ ∈ (0, ξ0) and (fm)′(ξ0) = 0.
Here the blow-up time T > 0 is a free parameter and the functions defined in

(2.2) are actually weak solutions to (1.7) except at the point x = 0 where the
condition (fm)′(0) = 0 is not fulfilled in order to be a weak solution. We adapt
these supersolutions to our equation (1.1) by defining

z(x, t) = (T − t)−αf((1 + |x|)(T − t)β), (2.4)

with α, β and f as in (2.2). Since f is a supersolution to the differential equation
(2.3), it is straightforward to check that z is a supersolution to (1.1). The amplitude
s(t) of the support of z(t) at some t ∈ (0, T ) is given by ξ0 = (1 + s(t))(T − t)β or
equivalently

s(t) = (T − t)−βξ0 − 1→∞, as t→ T.

Moreover, the above supersolutions have been established in [19, 21] only under
the condition σ > 2(1 − p)/(m − 1). However, if 0 < σ ≤ 2(1 − p)/(m − 1), it
is obvious from the fact that 1 + |x| ≥ 1, that (1 + |x|)σ ≤ (1 + |x|)σ1 for any
σ1 > 2(1 − p)/(m − 1) and for any x ∈ R. It thus follows that the supersolutions
constructed for (1.1) with such an exponent σ1 > 2(1 − p)/(m − 1) as above, will
serve also as supersolutions to (1.1) with exponents σ smaller. Since T is a free



EJDE-2023/72 REACTION-DIFFUSION WITH WEIGHTED STRONG REACTION 9

parameter and u0 is bounded and compactly supported, we can choose some T0 > 0
sufficiently small such that

z(x, 0) = T−α0 f((1 + |x|)T β0 ) ≥ ‖u0‖∞ ≥ u0(x) (2.5)

for any x ∈ suppu0. This, together with the fact that any function z as in (2.4) is
a supersolution to (1.1), gives that z is also a supersolution to the Cauchy problem
(2.1) for any k ≥ 1 for which the comparison principle applies to give that

wk(x, t) ≤ z(x, t), for all (x, t) ∈ R× (0, T0).

In particular we infer on the one hand that T∞ ≥ T0 > 0 as claimed, and on the
other hand by passing to the limit as k →∞ we obtain

u(x, t) ≤ z(x, t), for all (x, t) ∈ R× (0, T0),

which proves the finite speed of propagation of the support of u at least for some
short interval of time.

Step 3. Supersolutions in dimension N ≥ 2. In this case, there are no longer
bounded and decreasing supersolutions in the self-similar form used in Step 2. We
thus construct suitable supersolutions to (1.7) by joining two different self-similar
profiles. We begin again by fixing σ > 2(1 − p)/(m − 1) as a first case. We look
for self-similar solutions to (1.7) in the same form (2.2) as above, with the same
exponents α and β, but whose profiles solve the differential equation

(fm)′′(ξ) +
N − 1

ξ
(fm)′(ξ)− αf(ξ) + βξf ′(ξ) + ξσf(ξ)p = 0,

ξ = |x|σ(T − t)β .
(2.6)

On the one hand, the analysis in [16, Proposition 4.1] if m + p > 2, respectively
[16, Proposition 4.2 and Lemma 4.3] if m + p = 2, ensure that, given ξ0 ∈ (0, ξ∗)
sufficiently small, there exists a decreasing self-similar profile f2(ξ) solution to the
differential equation (2.6) having an interface exactly at ξ = ξ0 and a vertical
asymptote as ξ → 0 with local behavior

f2(ξ) ∼

{
Cξ−(N−2)/m, if N ≥ 3,

C(− ln ξ)1/m, if N = 2,

as it follows from [16, Lemma 3.2 and Lemma 3.5], where C > 0 designs a positive
constant that might change from one case to another. On the other hand, the
analysis performed in [16, Lemma 3.1] implies that, for any A > 0, there exists a
profile f1(ξ;A) local solution to (2.6) such that

f1(0;A) = A, f1(ξ;A) ∼
[
Am−1 +

α(m− 1)

2mN
ξ2
]1/(m−1)

, as ξ → 0,

which is increasing in a right-neighborhood of the origin up to some maximum
point ξ1(A) > 0. Thus, given an initial condition u0 as in (1.4), one can choose for
example A = ‖u0‖∞ and fix some ξ0 ∈ (0, ξ1(A)) ∩ (0, ξ∗) such that there exists
a decreasing profile f2(ξ) as above with vertical asymptote as ξ → 0 and edge of
the support at ξ = ξ0. The profiles f1(·;A) and f2 have to cross at some point
ξ ∈ (0, ξ1(A)). We finally define a self-similar supersolution to (1.1) as follows:

z(x, t) = (T − t)−αf((1 + |x|)(T − t)β), f(ξ) =

{
f1(ξ;A), ξ ∈ [0, ξ],

f2(ξ), ξ ≥ ξ,
(2.7)



10 R. G. IAGAR, A. I. MUÑOZ, A. SÁNCHEZ EJDE-2023/72

and notice that this is indeed a supersolution for any T > 0, as it can be described
alternatively as

z(x, t) = min{z1(x, t), z2(x, t)}, zi(x, t) = (T − t)−αfi((1 + |x|)(T − t)β),

for i = 1, 2, and z1, z2 are in fact solutions. The same considerations about the
magnitude of σ as in the end of Step 2 ensure that the supersolution defined in (2.7)
works also for values of σ smaller than 2(1−p)/(m−1). We thus complete the proof
by choosing a sufficiently small T0 > 0 such that (2.5) holds true on the support
of u0, and notice that z is a supersolution to the approximating problems (2.1)
leading to the minimal solution. This again implies that T∞ ≥ T0 > 0, completing
the proof. �

The solution u constructed in the proof of Proposition 2.1 will be referred as the
minimal solution to the Cauchy problem (1.1)-(1.2) and denoted by M(u0) in the
sequel. Notice that the above proof does not imply that necessarily the minimal
solution blows up in finite time. In fact, it might blow up or not according to
whether σ is larger or smaller than 2(1− p)/(m− 1), but this is not easy to prove
once we miss a comparison principle, and we refer the reader to the section of open
problems at the end.

3. Non-uniqueness for m+ p ≥ 2

A natural question raised by the previous section is whether in the case m+ p ≥
2 and for compactly supported and continuous data u0 the minimal solution u
constructed in Proposition 2.1 is the only solution to the Cauchy problem (1.1)-
(1.2). For σ = 0 non-uniqueness follows easily from the construction of a different
solution called maximal, which is shown to be strictly positive for any t > 0 and
thus different from u, see [27]. We cannot construct such a maximal solution to
(1.1), since strictly positive solutions might not even exist at all in some ranges of
σ (see a comment with a formal intuition for such non-existence in the final Section
of this work). In order thus to prove the existence of multiple solutions (and in
fact an infinite number of them) we adapt to our case the deeper results in [28],
where infinitely many solutions to the Cauchy problem (1.5)-(1.2) are constructed,
based on a prescribed evolution of the interface of them. As a preliminary fact,
let us recall that (1.5) admits in dimension N = 1 an absolute minimal solution in
self-similar form

E(x, t) = t1/(1−p)ϕ(|x|t−γ), γ =
m− p

2(1− p)
(3.1)

with zero initial condition E(x, 0) = 0 for any x ∈ R, according to [26]. It is also
shown in [26] that such solution lies below any solution (and supersolution) to (1.5)
and consequently also below any solution to (1.1) (which is a strict supersolution
to (1.5)). Moreover, the profile ϕ of E is non-increasing and compactly supported,
thus suppE ⊆ [−%0t

γ , %0t
γ ] where %0 > 0 is the right-interface point of the profile

ϕ.
With these elements and notation in mind, we can prove Theorem 1.3 as an

immediate consequence of a stronger result adapting a construction from [28]. More
precisely, by prescribing the behavior of the interface (under some limitations) we
can obtain a solution to the Cauchy problem (1.1)-(1.2) with exactly that given
interface at every (small) time t ∈ (0, T ). We formalize this below in dimension



EJDE-2023/72 REACTION-DIFFUSION WITH WEIGHTED STRONG REACTION 11

N = 1, but the result for radially symmetric solutions in dimension N ≥ 2 will be
then completely analogous.

Proposition 3.1. Let u0 ∈ C0(R) be a compactly supported initial condition such
that suppu0 = [r0, R0] for some r0, R0 ∈ R. Let M(u0) be the minimal solution
to the Cauchy problem (1.1)-(1.2) with initial condition u0 defined for t ∈ (0, T0)
and denote by sl(t), sr(t) the left and right interfaces of M(u0)(t) for t ∈ (0, T0),
that is, suppM(u0)(t) = [sl(t), sr(t)]. Let ξl(t), ξr(t) be two continuous functions
of time such that ξl(0) ≤ r0, ξr(0) ≥ R0 and

ξl(t1)− ξl(t2) ≥ sl(t1)− sl(t2), ξr(t2)− ξr(t1) ≥ sr(t2)− sr(t1),

for any t1, t2 ∈ (0, T0) such that t1 < t2. Then there exists at least a shorter time
interval (0, T ) ⊂ (0, T0) and a solution u to the Cauchy problem (1.1)-(1.2) with
initial condition u0 defined for t ∈ (0, T ) such that its left and right interfaces are
given exactly by ξl(t) and ξr(t) for any t ∈ (0, T ).

Proof. We divide the proof into two steps.

Step 1. Fundamental extension. This step adapts to our problem Step 1 in [28,
Proof of Corollary 1.2] and at its end in fact we already have the proof of Theorem
1.3. The goal here is to construct a solution to our Cauchy problem whose support
has an instantaneous jump to the right from R0 = sr(0) to R0 + r at time t = 0 for
a given (fixed) r > 0 (and of course, a perfectly similar construction can be done
for the left interface). Let then r > 0 be given. For any ε > 0 sufficiently small (it
is enough to start for example from ε < ‖u0‖∞/2) there exists a last, closest point
(that we denote by Rε) to the right interface R0 of u0 such that u0(Rε) = ε. Let
us introduce the following compactly supported and continuous initial condition

uε,0(x) =


u0(x), for x ∈ (−∞, Rε),
ε(R0+r−x)
R0+r−Rε , for x ∈ [Rε, R0 + r],

0, for x ∈ (R0 + r,∞),

(3.2)

that is, adding a linear extension to u0 from Rε to the new edge of the support
R0 + r. Let uε = M(uε,0) be the minimal solution associated to this new condition.
By comparison between minimal solutions (which holds since it is obvious that
any solution to (1.1) with a larger initial condition than the given u0 becomes
a supersolution to any of the problems (2.1) approximating the minimal solution
M(u0)) it readily follows that uε2(x, t) ≥ uε1(x, t) for any ε2 > ε1 > 0 and at any
(x, t) ∈ R × (0, T ) where T > 0 is for example the lifetime of the solution with
the biggest ε chosen. Recalling the construction of the absolute minimal solution
E(x, t) to (1.5) (see the details in [26, Theorem 4]) and the fact that any non-trivial
solution to our Eq. (1.1) is a strict supersolution to (1.5) we easily conclude that
for any (x0, t0) ∈ R× (0, T ) with x0 ∈ [Rε, R0 + r] we have

uε(x, t) ≥ E(x− x0, t− t0), t > t0. (3.3)

We can then define

u(x, t) = lim
ε→0

uε(x, t), (x, t) ∈ R× (0, T ),

which is a monotone limit and it is easy to see that u is a weak solution to (1.1) using
the Monotone Convergence Theorem (in fact we even have uniform convergence by
Dini’s Theorem since the limit is continuous). Let us stress here that we do not



12 R. G. IAGAR, A. I. MUÑOZ, A. SÁNCHEZ EJDE-2023/72

need in this construction a bound from above to guarantee that the limit is finite,
since we are dealing with a decreasing limit. It is then obvious that

u(x, 0) = lim
ε→0

uε,0(x) = u0(x)

and the comparison from above with E in (3.3) transfers to the limit u, proving
that for any t > 0 we have u(x, t) > 0 for x ∈ (R0, R0 + r).

Step 2. Iterative construction. We perform now an iterative construction based
on a discretization in time and the application of Step 1 in each interval of this
discretization to perform an instantaneous jump in the supports. Let then ξl(t)
and ξr(t) be two continuous, increasing functions of time as in the statement of
Proposition 3.1. Fix a time t ∈ (0, T ) for a T > 0 sufficiently small (that will be
chosen later). For any positive integer n we construct an approximate solution un
as in [28, Proof of Corollary 1.2]. We briefly and sketchy describe the construction
here for the sake of completeness. Consider a partition

0 = t0 < t1 < t2 < · · · < tn−1 < tn = t, tj =
jt

n
and construct the function un by induction. More precisely, assume first that un
is already constructed for t ∈ [0, tj ] and we want to pass to tj+1. To this end,
we begin from the edges of the support of un(tj) and we perform a jump of them
by applying Step 1 both to the right (with r = ξr(tj+1) − ξr(tj)) and to the left
(with r = ξl(tj) − ξl(tj+1)), using here the fact that the speed of advance of the
prescribed interfaces to the left and to the right is higher than the ones of the
minimal solution with initial condition u0. The precise details are easy and given
in the above mentioned proof of [28, Corollary 1.2]. To pass to the limit as n→∞ in
the iteration we need an uniform bound from above to show that the limit solution
does not escape to infinity. We cannot use a translation of a minimal solution or a
construction based on it (as it was done in [28, Lemma 2.4]) since our equation is
not invariant to translations, but instead we can bound uniformly from above the
iterated solutions by a sufficiently big non-increasing supersolution in self-similar
form

U(x, t) = (T − t)−αf((1 + |x|)(T − t)β), α =
σ + 2

L
, β =

m− p
L

similar to the ones introduced in (2.2) with a profile f(ξ) solving (2.3) and having
a right interface at ξ0 ∈ (0,∞). Indeed, comparison from above with such a super-
solution U can be performed as the iterative construction of un is based on adding
up at each iteration step only minimal solutions, provided that at our fixed time
t > 0 for which we have built un we have ordered supports between un and U , that
is

ξr(t) < ξ0(T − t)−β , ξl(t) > −ξ0(T − t)−β , (3.4)

which also gives a limitation for the lifetime T > 0 depending on the two prescribed
functions ξr, ξl. We notice that for “faster” advancing interface functions ξl(t),
ξr(t), smaller lifetime T is expected. Once satisfied this condition, we can pass to
the limit in the discretization as n→∞ and obtain the desired weak solution

v(x, t) = lim
n→∞

un(x, t), t ∈ (0, T ),

with T > 0 chosen sufficiently small according to (3.4). The bound from below by
E at every point of positivity (as done in Step 1) together with the construction
and the continuity of the prescribed functions ξl, ξr ensure that the left and right



EJDE-2023/72 REACTION-DIFFUSION WITH WEIGHTED STRONG REACTION 13

interfaces of v at every time t ∈ (0, T ) are given exactly by the continuous functions
ξl(t) and ξr(t). In the meantime, the universal bound from above by the supersolu-
tion U in self-similar form ensures that at least in the time interval (0, T ) we have
v(x, t) <∞ for any x ∈ R. �

Remarks. (1) Step 1 above already completes the proof of Theorem 1.3 in di-
mension N = 1. Indeed, for every r > 0 we can construct a different solution to
the same Cauchy problem (1.1)-(1.2). A totally analogous extension can be con-
structed for radially symmetric initial conditions u0 ∈ C0(RN ) by replacing x by
r = |x|, which allows us to work with the radial variable exactly as in dimension
N = 1 but using for comparison from above the supersolutions introduced in Step
3 of the proof of Proposition 2.1, thus completing the proof of Theorem 1.3 also in
dimension N ≥ 2.

(2) Our Proposition 3.1 also holds for σ = 0 and improves slightly the result
for (1.5) in [28, Theorem 1.1 and Theorem 5.1] since we do not need to consider
the non-increasing majorant ũ0 of the initial condition as considered in the above
mentioned work.

4. Aronson-Bénilan estimates when m+ p < 2

This section is devoted to the deduction of the Aronson-Bénilan estimates (1.12)
in the homogeneous case σ = 0. Let us recall here the pressure function

v =
m

m− 1
um−1,

since most of the forthcoming work will be performed on this function.

Proof of Theorem 1.4. The proof is inspired from the one for [36, Proposition 9.4]
but technically more involved. We first derive after rather straightforward calcula-
tions the pressure equation which will be used further in the present work, that is,
the parabolic PDE satisfied by the function v introduced above:

vt = (m− 1)v∆v + |∇v|2 +K(m, p)v(m+p−2)/(m−1),

K(m, p) = m
(m− 1

m

)(m+p−2)/(m−1)

.
(4.1)

Let us notice here the strong influence of the sign of m + p − 2 on this equation,
because of its last term. To go further, we set w = ∆v and we next derive the partial
differential equation solved by w. To this end, we calculate the terms separately.
On the one hand, for the reaction term we obtain

∂

∂xi
K(m, p)v(m+p−2)/(m−1) = K(m, p)

[m+ p− 2

m− 1
v(p−1)/(m−1) ∂v

∂xi

]
hence

∂2

∂x2
i

K(m, p)v
m+p−2
m−1

= K(m, p)
[m+ p− 2

m− 1
v
p−1
m−1

∂2v

∂x2
i

+
(2−m− p)(1− p)

(m− 1)2
v
p−m
m−1

( ∂v
∂xi

)2]
.

The contribution of the reaction term thus gives

N∑
i=1

∂2

∂x2
i

K(m, p)v
m+p−2
m−1



14 R. G. IAGAR, A. I. MUÑOZ, A. SÁNCHEZ EJDE-2023/72

= K(m, p)
m+ p− 2

m− 1
v
p−1
m−1w +K(m, p)

(2−m− p)(1− p)
(m− 1)2

v
p−m
m−1 |∇v|2

= R1(x, v)w +R2(x, v),

where

R1(x, v) = K(m, p)
m+ p− 2

m− 1
v
p−1
m−1 < 0,

R2(x, v) = K(m, p)
(2−m− p)(1− p)

(m− 1)2
v
p−m
m−1 |∇v|2 ≥ 0.

On the other hand, the diffusion term can be worked out as in [36, Proposition 9.4]
to obtain

wt = (m− 1)v∆w + 2m∇v · ∇w + (m− 1)w2 + 2

N∑
i,j=1

( ∂2v

∂xi∂xj

)2

+R1(x, v)w +R2(x, v)

≥ (m− 1)v∆w + 2m∇v · ∇w +
(
m− 1 +

2

N

)
w2 +R1(x, v)w +R2(x, v),

after a straightforward application of the Cauchy-Schwartz inequality. This can be
written equivalently as Lw ≥ R2(x, v), where

Lw := wt − (m− 1)v∆w − 2m∇v · ∇w −
(
m− 1 +

2

N

)
w2 −R1(x, v)w

is a uniformly parabolic operator. Since R2(x, v) ≥ 0, we deduce that Lw ≥ 0 and
we aim to find a subsolution for L depending only on time. We thus take for t > 0

W (x, t) = −C
t
, C =

N

N(m− 1) + 2

and calculate

LW =
C

t2
− C

t2
+
CR1(x, v)

t
< 0,

since m+ p− 2 < 0 in our range of exponents. Applying the comparison principle
to the parabolic operator L we infer that

∆v(x, t) = w(x, t) ≥W (t) = − N

(N(m− 1) + 2)t
,

for any (x, t) ∈ RN × (0,∞), as stated.
All the previous calculations and the application of the maximum principle to

the operator L are fully justified for solutions u such that (in the pressure variable)
L is uniformly parabolic, that is, when v, ∇v are bounded and v > 0 uniformly.
In order to extend the Aronson-Bénilan estimates to general weak solutions, we
proceed by approximation. Let us first consider u0 to be an initial condition as in
(1.4) and continuous. Then, there exists a unique solution u to the Cauchy problem
(1.5)-(1.2) in a time interval (0, T ), according to [27]. Let uk be the solution to the
Cauchy problem with initial condition

u0,k(x) = u0(x) +
1

k
,

for each positive integer k. We infer from [27, Theorem 2.1] and its proof that
there exists a unique solution uk to (1.5) with initial condition u0,k, and it satisfies
uk(x, t) ≥ 1/k for all x ∈ RN , t > 0. We further find from [23, Theorem 8.1,



EJDE-2023/72 REACTION-DIFFUSION WITH WEIGHTED STRONG REACTION 15

Chapter V] (which applies for our approximating solutions uk since they are now
bounded from below by a positive constant) that uk has the regularity required for
(1.5) to hold in a classical sense and all the space derivatives of uk up to the second
order and time derivatives of uk up to order one are uniformly bounded. In this
case, the previous calculation applies rigorously for uk and we obtain that

∆vk(x, t) ≥ −K
t
, vk =

m

m− 1
um−1
k . (4.2)

Moreover, the comparison principle for (1.5) (see [27, Theorem 2.1]) entails that
solutions uk for k ≥ 1 form a non-increasing sequence of functions, thus there exists
a limit

u(x, t) = lim
k→∞

uk(x, t), (x, t) ∈ RN × (0, T ),

and the uniform bound of uk and their derivatives up to second order together
with the Arzelá-Ascoli Theorem imply that uk → u locally uniformly and the same
holds true for their first order derivatives with respect to the space variables. Then,
the uniform boundedness of uk and ∂tuk gives the continuity with respect to the
time variable over (0, T ) of the limit function u, while the fact that u belongs
to L∞loc is obvious, as it is bounded from above by any uk. We thus fulfill the
regularity assumption (a) in Definition 1.1. The monotone convergence theorem
then easily gives that u satisfies assumptions (b) and (c) in Definition 1.1, hence,
since uk(x, 0) = u0,k(x)→ u0(x) as k →∞ uniformly on RN , we readily infer that
u is a weak solution to the Cauchy problem (1.5)-(1.2). Uniqueness of solutions to
the latter Cauchy problem, established in [26] for continuous and bounded initial
conditions, then proves that u = u. We then come back to (4.2), which, after
multiplication by a non-negative test function and integration by parts, reads∫ T

0

∫
RN

(
vk(x, t)∆ϕ(x, t) +

K

t
ϕ(x, t)

)
dx dt ≥ 0, (4.3)

for any ϕ ∈ C∞0 (RN × (0, T )), ϕ ≥ 0. We pass to the limit in (4.3) as k → ∞,
taking into account that vk → v locally uniformly as k → ∞, and obtain the
claimed distributional form (1.13). �

We end this section with a corollary which will be used in the sequel.

Corollary 4.1. In the same conditions as in Theorem 1.4, we have

ut ≥ −
Ku

t
, K =

N

N(m− 1) + 2
,

in the sense of distributions in RN .

Proof. At a formal level, we infer from (4.1) that vt ≥ (m− 1)v∆v, hence

(m− 1)
ut
u

=
vt
v
≥ (m− 1)∆v ≥ − N(m− 1)

(N(m− 1) + 2)t

and we reach the conclusion with the same constant K as in (1.12). For general
weak solutions the estimate is proved by using the same approximation as in the
proof of Theorem 1.4. �

Remark. Formal proof of Aronson-Bénilan estimates for σ > 0. At a
formal level, the Aronson-Bénilan estimates (1.12) or (1.13) hold also for σ > 0 and
N ≥ 2. Indeed, a slightly longer but straightforward calculation along the first few



16 R. G. IAGAR, A. I. MUÑOZ, A. SÁNCHEZ EJDE-2023/72

lines of the proof of Theorem 1.4 by computing the derivatives up to second order
of the reaction term, but applied to (1.1) with σ > 0, gives

N∑
i=1

∂2

∂x2
i

K(m, p)(1 + |x|)σv
m+p−2
m−1

= K(m, p)
m+ p− 2

m− 1
(1 + |x|)σv

p−1
m−1w

+K(m, p)(1 + |x|)σ−2v
p−m
m−1

[ (2−m− p)(1− p)
(m− 1)2

(1 + |x|)2|∇v|2

− 2σ
2−m− p
m− 1

(1 + |x|)v x
|x|
· ∇v + σ

(
σ − 1 + (N − 1)

1 + |x|
|x|

)
v2
]

= R1(x, v)w +R2(x, v),

where

R1(x, v) = K(m, p)
m+ p− 2

m− 1
(1 + |x|)σv

p−1
m−1 < 0

and R2(x, v) gathers the rest of the terms. In order to proceed with the comparison
principle as we did in the body of the proof of Theorem 1.4, we still need to have
R2(x, v) ≥ 0. To this end, we write R2 as a square and we examine the remainders.
More precisely, using once more a standard Cauchy-Schwarz inequality for the scalar
product ∇v · x/|x| we find that

R2(x, v)

K(m, p)(1 + |x|)σ−2v
p−m
m−1

≥
[2−m− p

m− 1
(1 + |x|)|∇v| − σv

]2
+

2−m− p
m− 1

(1 + |x|)2|∇v|2

+ σ
[
(N − 1)

1 + |x|
|x|

− 1
]
v2

and taking into account that we are in the range m + p − 2 < 0, it follows that
R2(x, v) ≥ 0 provided that

σ
[
(N − 1)

1 + |x|
|x|

− 1
]
≥ 0,

which holds for σ > 0 if N ≥ 2. We thus conclude, at a formal level, that (1.13)
should hold in this case. However, we left this part out of the statement of Theorem
1.4 since the final approximation argument leading to the rigorous proof for σ = 0
cannot be performed, as we are missing an existence and uniqueness result for
solutions to the Cauchy problem (1.1)-(1.2) when σ > 0.

5. Infinite speed of propagation when m+ p < 2

In this part we use the Aronson-Bénilan estimates in Theorem 1.4 to establish the
infinite speed of propagation of the supports of solutions to (1.1) when m+p−2 < 0
and thus complete the proof of Theorem 1.5. Let us stress again here that Theorem
1.5 has been proved in [26, Lemma 2.4] for σ = 0. We give here an independent
proof, based on a completely different argument, and extend it to exponents σ > 0.

Proof of Theorem 1.5. In a first step, let σ = 0 and assume for contradiction that
for some compactly supported initial condition u0 as in (1.4), u(t) remains com-
pactly supported for t ∈ (0, T0). Recalling the pressure equation (4.1), at a formal



EJDE-2023/72 REACTION-DIFFUSION WITH WEIGHTED STRONG REACTION 17

level, since m + p − 2 < 0, one reaches easily a contradiction. Indeed, picking
an arbitrary t ∈ (0, T0), since ∆v(x, t) ≥ −K/t and |∇v(x, t)|2 ≥ 0 at any point
x ∈ RN , it follows that at the interface point x = s(t) we obtain vt(s(t), t) = +∞ in
order to compensate the negative power (m+p− 2)/(m− 1) in the last term of the
right-hand side. This is obviously equivalent to the infinite speed of propagation
of the supports. More rigorously, since m + p < 2 we multiply by v(2−m−p)/(m−1)

in (4.1) and also by a test function ϕ ∈ C∞0 (RN ), ϕ ≥ 0, then we integrate on RN
and on any time interval (τ0, τ1) ⊂ (0, T0) and we drop the second term in the right
hand side (which is always positive) to obtain

m− 1

1− p

∫ τ1

τ0

∫
RN

(v(1−p)/(m−1))tϕdx dt

≥ (m− 1)

∫ τ1

τ0

∫
RN

v(1−p)/(m−1)∆vϕ dx dt+K(m, p)

∫ τ1

τ0

∫
RN

ϕdx dt

≥ − N(m− 1)

N(m− 1) + 2

∫ τ1

τ0

∫
RN

1

t
v(1−p)/(m−1)ϕdx dt

+K(m, p)

∫ τ1

τ0

∫
RN

ϕdx dt.

(5.1)

We now consider a sequence of test functions (ϕn)n≥1 defined as follows

ϕn(x) = 1, for |x| ≤ n, 0 ≤ ϕn(x) ≤ 1 ∀x ∈ RN , suppϕn ⊆ B(0, 2n),

where B(0, 2n) = {x ∈ RN : |x| ≤ 2n}, and let ϕ = ϕn in (5.1) for any positive
integer n ≥ 1. Since the support of v is uniformly localized for t ∈ [τ0, τ1] (as
τ1 < T0), it follows that the right-hand side of (5.1) tends to +∞ as n → ∞ due
to the last integral only of ϕn. We thus infer that

lim
n→∞

∫ τ1

τ0

∫
RN

(v(1−p)/(m−1))tϕn dx dt = +∞

or equivalently

lim
n→∞

∫
RN

[
v(1−p)/(m−1)(τ1)− v(1−p)/(m−1)(τ0)

]
ϕn dx = +∞,

which is a contradiction with the localization of the supports of v(t) for t ∈ [τ0, τ1].
Let us notice here that the chosen range of exponents p ∈ (0, 1) and m+p < 2 were
decisive, as after multiplication by a positive power v(2−m−p)/(m−1), we got in the
left-hand side also a positive power v(1−p)/(m−1) of v, thus we do not create new
singularities at the edges of the supports.

We pass now to σ > 0. Assume that there is a weak solution u to the Cauchy
problem Eq. (1.1)-(1.2) defined for t ∈ (0, T ) with some T > 0. Since (1+ |x|)σ ≥ 1
for any x ∈ RN we deduce that u is a supersolution to the Cauchy problem (1.5)-
(1.2). We infer from the comparison principle (which holds true for (1.5) and
non-trivial initial data in the range m + p < 2, [27]) that u(x, t) > 0 for any
(x, t) ∈ RN × (0, T ). �

Extensions and open problems

We gather here some extensions related to the previous results, that we consider
interesting.



18 R. G. IAGAR, A. I. MUÑOZ, A. SÁNCHEZ EJDE-2023/72

1. Finite time blow-up. A natural question is whether any solution to (1.1) blows up
in finite time or there are some initial conditions u0 producing (minimal) solutions
that are global in time. Our conjecture is that, if L > 0, each non-trivial solution
is expected to blow up in finite time, while if L ≤ 0, there are initial conditions
producing solutions that are global in time (we recall that L is defined in (1.8)). A
formal argument about general finite time blow-up if L > 0 is based on comparison
with subsolutions in self-similar form obtained in our recent papers [19, 21, 16].
More precisely, it goes by contradiction as follows: assume that there exists u0 ∈
C0(RN ) as in (1.4) such that the minimal solution M(u0) is defined for t ∈ (0,∞).
We infer by comparison with the absolute minimal solution E defined in [26] and
(3.1) that for large t, solution M(u0)(t) is as large as we want both in amplitude and
support. We can thus find a blow-up self-similar solution to (1.7) as in [19, 21, 16]
(which is a subsolution to (1.1)) below it. If comparison would be allowed, then we
would get an easy contradiction proving that any minimal solution (and then any
other solution) blows up in finite time.

But as we see by considering the subsolution U with initial condition U(x, 0) = 0
for any x ∈ RN , which is defined explicitly by

U(x, t) =
( 1

1− p

)1/(p−1)

t1/(1−p)(1 + |x|)σ/(1−p), (5.2)

comparison does not hold in general, as otherwise any solution (even the ones with
compact support) could have been compared to U in order to force it to become
positive everywhere, contradicting the finite speed of propagation at least in the
range m+ p ≥ 2. This opens the question on whether the self-similar solutions to
(1.1) are minimal solutions in the sense of the construction performed in Proposition
2.1.

2. Non-existence of positive solutions if σ > 2(1− p)/(m− 1). Strongly connected
to the first comment, and expecting (at a formal level) that we might compare a
strictly positive solution to (1.1) with the subsolution U introduced in (5.2), we
conjecture that, if L > 0 (that is, σ > 2(1 − p)/(m − 1)) there are no solutions to
(1.1) such that u(x, t) > 0 for any x ∈ RN . Indeed, assuming that the comparison
can be performed rigorously if u(x, t) > 0 for any x ∈ RN , we would get a solution
with local behavior

u(x, t) ≥ C(1 + |x|)σ/(1−p), as |x| → ∞, for all t > 0.

But any solution to (1.1) is a supersolution to the standard porous medium equation
and classical results on the porous medium equation (see for example [4, 7, 8]) state
that there are no solutions (and it seems to us that the proofs can be extended to
supersolutions) to it increasing at infinity faster than |x|2/(m−1). Since σ/(1− p) >
2/(m − 1) if L > 0, we would be in this case. In particular, since infinite speed
of propagation is in force for m + p < 2 by Theorem 1.5, we expect complete
non-existence of non-trivial solutions if L > 0 and m+ p < 2.

3. Establishing which self-similar solutions are minimal. An immediate adaptation
of the result in [19] proves that (1.1) with m+ p > 2 presents two types of blow-up
self-similar solutions

U(x, t) = (T − t)−αf((1 + |x|)(T − t)β), α =
σ + 2

L
, β =

m− p
L

,



EJDE-2023/72 REACTION-DIFFUSION WITH WEIGHTED STRONG REACTION 19

which differ with respect to the local behavior of the profile f(ξ) near the interface
point ξ0 ∈ (0,∞), namely

f(ξ) ∼ (ξ0 − ξ)1/(m−1) (Type I) or f(ξ) ∼ (ξ0 − ξ)1/(1−p) (Type II),

both taken in the limit ξ → ξ0, ξ < ξ0. It is also shown at least at a formal level that
these solutions satisfy two different interface equations. Thus, a natural question
would be whether any of these self-similar solutions is minimal in the sense of the
construction in Proposition 2.1, which would allow us to compare and conclude on
the finite time blow-up. This is not an easy question and our intuition suggests
that minimality has to do with the interface equation: we might expect that the
solutions with interface of Type I are minimal, while the other ones are not. This
conjecture is supported by the analogy with the minimality of the traveling wave
solutions to (1.5), see for example [27, Theorem 4.1].

4. Connection between non-uniqueness and blow-up time. A much deeper open
question is related to whether, in the range σ > 2(1−p)/(m−1), prescribing a blow-
up time T and a function u0, there exists a unique solution to the Cauchy problem
(1.1)-(1.2) with condition u0 blowing up in finite time exactly at the given time T .
More precisely, taking u0 ∈ C0(R) satisfying (1.4), there exists a minimal solution
M(u0) which (assuming that point 1 in this enumeration of open problems holds
true, as we strongly expect) comes with a finite blow-up time T0 ∈ (0,∞). We have
also proved in Section 3 that the Cauchy problem (1.1)-(1.2) has an infinite number
of compactly supported solutions with interfaces advancing faster than the minimal
one, and estimate (3.4) shows that faster advancing speed of the interface implies
shorter lifetime before blow-up. Thus one can wonder naturally whether, given
T ∈ (0, T0), there exists one solution to the Cauchy problem (1.1)-(1.2) blowing up
exactly at this time T . We do not have by now a suggestion of how to approach
this problem, but it is in our opinion an interesting and deep open question.

Acknowledgments. R. G. I. and A. S. were partially supported by the Project
PID2020-115273GB-I00 and by the Grant RED2022-134301-T (Spain).

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EJDE-2023/72 REACTION-DIFFUSION WITH WEIGHTED STRONG REACTION 21

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Razvan Gabriel Iagar

Departamento de Matemática Aplicada, Ciencia e Ingenieria de Materiales y Tecnolo-
gia Electrónica, Universidad Rey Juan Carlos, Móstoles, 28933, Madrid, Spain

Email address: razvan.iagar@urjc.es

Ana I. Muñoz

Departamento de Matemática Aplicada, Ciencia e Ingenieria de Materiales y Tecnolo-

gia Electrónica, Universidad Rey Juan Carlos, Móstoles, 28933, Madrid, Spain
Email address: anaisabel.munoz@urjc.es

Ariel Sánchez
Departamento de Matemática Aplicada, Ciencia e Ingenieria de Materiales y Tecnolo-

gia Electrónica, Universidad Rey Juan Carlos, Móstoles, 28933, Madrid, Spain

Email address: ariel.sanchez@urjc.es


	1. Introduction
	2. Existence and finite speed of propagation when m+p2
	3. Non-uniqueness for m+p2
	4. Aronson-Bénilan estimates when m+p<2
	5. Infinite speed of propagation when m+p<2
	Extensions and open problems
	Acknowledgments

	References