We overcome them by de- veloping an appropriate decomposition of the two-parametric Mittag-Leffler function to obtain Mikhlin-type estimates and obtain our existence theorem

Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 250, pp. 1–20.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

SELF-SIMILAR SOLUTIONS FOR A SUPERDIFFUSIVE HEAT EQUATION WITH GRADIENT NONLINEARITY

MARCELO FERNANDES DE ALMEIDA, ARL ´UCIO VIANA

Abstract. This article studies the existence, stability, self-similarity and sym- metries of solutions for a superdiffusive heat equation with superlinear and gradient nonlinear terms with initial data in new homogeneous Besov-Morrey type spaces. Unlike in previous works on such time-fractional partial differ- ential equations of orderα ∈ (1,2), we take non null initial velocities into consideration, where new difficulties arise from. We overcome them by de- veloping an appropriate decomposition of the two-parametric Mittag-Leffler function to obtain Mikhlin-type estimates and obtain our existence theorem.

1. Introduction Let ∆x = PN

i=1

∂²

∂x²_i be the Laplace operator, u: R^1+N → R, and ∂_t^α be the Caputo’s fractional derivative of order 1 < α < 2 (see subsection 2.2). In this article, we study the equation

∂_t^αu= ∆xu+κ1|∇xu|^q+κ2|u|^ρ−1u, κ16= 0, κ2∈R, (1.1) subject to the initial data

u(0, x) =ϕ(x), ∂_tu(0, x) =ψ(x), (1.2) whereq >1 andρ >1. Note that the rescaled functionuγ(t, x) :=γ^ρ−12 u(γ^α2t, γx) solves (1.1) with initial data

ϕγ(x) =γ^ρ−12 ϕ(γx), ψγ(x) =γ^{ρ−12 +α2}ψ(γx), (1.3) provided that q = _ρ+1^2ρ and u(t, x) solves (1.1)-(1.2). Hence, we obtain ascaling map of solutions,

u(t, x)7→u_γ(t, x), for allγ >0, (1.4) and solutions invariant by (1.4) will be calledself-similar solutions, that is,

u(t, x) =uγ(t, x). (1.5)

In the study of self-similar solutions, the natural candidates to be initial data are the homogeneous functions,

ϕ(γx) =γ^−ρ−12 ϕ(x), ψ(γx) =γ^{−ρ−12 −2α}ψ(x).

2010Mathematics Subject Classification. 35A01, 35R11, 35R09, 35B06, 35C06, 35K05, 35L05.

Key words and phrases. Fractional partial differential equations; self-similarity;

radial symmetry; Sobolev-Morrey spaces.

c

2016 Texas State University.

Submitted July 8, 2016. Published Septembere 19, 2016.

1

In this work we are interested in existence of self-similar solutions to (1.1)-(1.2).

For this purpose, we study (1.1)-(1.2) through its integral formulation

u(t, x) =Gα,1(t)ϕ(x) +Gα,2(t)ψ(x) +Nα(u)(t, x), (1.6) where

G\α,j(t)f(ξ) =t^j−1Eα,j(−4π²t^α|ξ|²)fb(ξ), j= 1,2, f∈ S⁰(R^N), (1.7) Nα(u) =

Z t

0

Gα,1(t−s) Z s

0

rα(s−τ) κ2|u|^ρ−1u+κ1|∇xu|^q

dτ ds, (1.8)

r_α(t) =t^α−2/Γ(α−1). (1.9)

Hereafter a solution uwill be understood as a distribution u(t,·) satisfying (1.6), for eacht >0.

The presence of the gradient requires suitable estimates in certain Sobolev- Morrey spaces M¹_r,µ and this motivated us to study the problem in the space Xβ of all bounded continuous functionsu: (0,∞)→ Mr,µendowed with the norm

kukXβ = sup

t>0

t^α2+βku(t)k_M¹_r,µ+ sup

t>0

t^βku(t)kMr,µ, (1.10) whereβ, randµwill be chosen later (see (3.1)). Assumingψ6≡0 brings new difficul- ties because we need to obtain suitable estimates for two-parametric Mittag-Leffler function Eα,2(4π²t^α|ξ|²). More precisely, we develop an appropriate decomposi- tion for Mittag-Leffler function to obtain a suitable estimate (see (4.10) and (4.14)) which enables us to introduce the space

I ={(ϕ, ψ)∈ S⁰× S⁰; (ϕ, ψ)∈D(α, β)×D(α, β)},˜ (1.11) where

D(α, β) :={ϕ∈ S⁰ :Gα,1(t)ϕ∈Xβ}, D(α, β) :=e {ψ∈ S :Gα,2(t)ψ∈Xβ}, for all t > 0. Hence, applying Lemma 5.1 we obtain (see Remark 3.2-(B)) that Mp,µ× M^−2/αp,µ ⊆D(α, β)×D(α, β). It is remarkable that the investigation of self-e similarity and symmetries for (1.1)-(1.2) allows us to deal with following prototype functions

ϕ(x) =1|x|^−ρ−12 , ψ(x) =2|x|^{−ρ−12 −α2},

which belong toD(α, β)×D(α, β) but can be arbitrarily large in the spacee L²(R^N)×

H˙^α2(R^N). See Remark 3.4-(A).

Our symmetry result, roughly speaking, says that if the initial dataϕandψare invariant on the orthogonal group acting on R^N so the solution is. In particular, we show the existence of radial self-similar solutions (see Remark 3.6-(A)).

We point out that our results hold for α= 1 and ψ= 0 and, in this case, the upper bound (γ₂−γ₁) +^N_p^−µ

1 −^N_p^−µ

2 in Lemma 5.1 can be removed. On the other hand, for 1 < α < 2 the Mikhlin theorem yields more restrictive constraints to Lemma 5.1 than the usual estimates for the heat semigroup in such a way that Theorem 3.1 cannot come near toα= 2.

Now, let us to review some works. Fujita [5] remarked that the linear counterpart of equation (1.1) has similarities with wave and heat equations and presents certain qualitative properties which qualifies it as a reasonable interpolation between these equations. When κ2 = 0, α = 1 and ψ = 0, (1.1)-(1.2) turns into the viscous Hamilton-Jacobi equation. Using scaling technique, Ben-Artzi et al [3] found the number r_c = ^N_2−q^(q−1) and showed that it is a critical exponent for existence of

solutions inL^r. In particular, the problem is well-posed whenr≥rcand 1< q <2.

In Remark 3.2-(C) we provide an existence result for this problem in Morrey spaces M_{p, N−}N

rcp which are strictly larger than the Lebesgue spaces, namely,

L^r(R^N)(Mp,µ(R^N)(D(1, β), (1.12) provided that ^N_r = ^N−µ_p andp < r. Our existence result is then compatible with [3, Theorem 2.1], in view of 1< p≤rc ≤r. In particular, the initial data taken in Theorem 3.1 is larger than those considered in [3].

Recently several authors have addressed the study of global existence, self- similarity, asymptotic self-similarity and radial symmetry of solutions for the semi- linear heat equation with gradient nonlinear terms. See e.g. [4, 7, 17, 18, 19]. In [17] it is assumed thatϕbelongs to homogeneous Besov space ˙B_r^−β_1,∞¹ and

kϕk_B˙^−β1 r1,∞ = sup

t>0

t^β1/2ke^t∆ϕkL^r1(R^N)≤, β₁= 2 ρ−1 −N

r1

.

By employing the Gagliardo-Nirenberg inequality, the authors studied the existence and asymptotic behavior of global mild solutions. On the other hand, our functional approach enables us to control the gradient estimates without making use of the Gagliardo-Nirenberg inequality and allows us to deal with a larger class of functions space for initial data.

Let us now review some works concerning to (1.1)-(1.2) withψ= 0 andκ₁= 0.

In [8] the authors established L^p−L^q estimates for {G_α,1(t)}_t≥0 and showed a blowup alternative and local well-posedness in L^q(R^N)-framework for any ϕ ∈ L^q(R^N), whereq≥^{N α(ρ+1)}₂ . Using Mikhlin-Hormander’s type theorem on Morrey spaces, de Almeida and Ferreira [1] studied self-similarity, symmetry, antisymmetry and positivity of global solutions with small data ϕ ∈ Mp,µ, µ = N − _ρ−1^2p . In [2], the authors established existence, self-similarity, symmetries and asymptotic behavior of solutions in Besov-Morrey spacesN_p,µ,∞^σ and provided a maximal class of existence in the sense that there is no known results inX )N_p,µ,∞^σ . Indded, we notice that

Mp,λ(N_p,µ,∞^σ and B˙^k_r,∞⊂ N_p,µ,∞^σ , (1.13) where ^N−λ_p =−σ+^N−µ_p =−k+^N_r,σ=^N−µ_p −_ρ−1² ,k= ^N_r −_ρ−1² and 1≤p < r.

All spaces in (1.12) and (1.13) are invariant by scaling.

We still observe that problem (1.1)–(1.2) can be studied with a Fourier multiplier σ(D) in place of ∆x, where |σ(ξ)| ≤ C|ξ|^k due to estimates (4.10) and (4.14) into Propositions 4.2 and 4.3. Example of such an operator is the Riesz potential (−∆x)^k/2f =F⁻¹|ξ|^kFf, whereF denote the Fourier transform inS⁰.

This manuscript is organized as follows. Some basic properties of the Sobolev- Morrey spaces and Mittag-Leffler functions are reviewed in Section 2. We state and make some remarks on our results in Section 3 and their proofs are performed in Section 6. Sections 4 and 5 are reserved to a careful study of the several estimates which are crucial to yield our results.

2. Preliminaries

In this section we review some well-known properties of the Morrey spaces and Sobolev-Morrey spaces, more details can be found in [9, 10, 12, 13, 15]. Also, we

obtain an integral equation which is formally equivalent to (1.1)-(1.2) in the lines of [11].

2.1. Sobolev-Morrey spaces. Let Q_r(x₀) be the open ball in R^N centered at x₀ and with radius r > 0. Given two parameters 1 ≤ p < ∞ and 0 ≤ µ < N, the Morrey space M_p,µ =M_p,µ(R^N) is defined to be the set of all functions f ∈ L^p(Q_r(x₀)) such that

kfkMp,µ := sup

x₀∈Rⁿ, r>0

r^−µpkfkL^p(Q_r(x₀))<∞,

which is a Banach space endowed with this norm. For s ∈ R and 1 ≤ p < ∞, the homogeneous Sobolev-Morrey space M^s_p,µ = (−∆x)^−s/2Mp,µ is the Banach space of all tempered distributions f ∈ S⁰(R^N)/P modulo polynomials P withN variables. Ifs <^N−µ_p andp >1, from [9, Theorem 1.1] or [13], it holds that

kfkM_p,µ ∼

X

ν∈Z

|F⁻¹ψν(ξ)Ff|^21/2 _M

p,µ,

where∼denotes norm equivalence and{ψν}_ν∈Zis a homogeneous Littlewood-Paley resolution of unity, that is,

ψν(ξ) =φν(ξ)−φ_ν−1(ξ), φν(ξ) =φ0(ξ/2^ν),

for φ0 ∈ C₀^∞(R^N) such thatφ0 = 1 on the ballQ1(0) and suppφ0 ⊂ Q2(0). In particular, using (2.1) and that|ξ|^s∼2^sν on the suppψν(ξ)⊂ {ξ∈R^N : 2^ν−1<

|ξ|<2^ν+1}, we obtain

X

ν∈Z

|2^sνF⁻¹ψν(ξ)Ff|^21/2 _M

p,µ

∼

X

ν∈Z

|F⁻¹ψ_ν(ξ)|ξ|^sFf|²1/2

Mp,µ

=

X

ν∈Z

|2^νNψ(2ˇ ^ν·)∗(|ξ|^sfb)^∨|^21/2 _M

p,µ

∼ k(| · |^sfb)^∨kM_p,µ.

(2.1)

Givenf ∈ M^s_p,µ, the quantity (2.1) define two equivalent norms on Sobolev-Morrey space, namely,

kfk_M^s_p,µ =k(| · |^sfb)^∨k_Mp,µ, kfk_M^s_p,µ =

X

ν∈Z

|2^sνF⁻¹ψν(ξ)Ff|²1/2

Mp,µ

. (2.2)

It follows from Littlewood-Paley decomposition of the Lebesgue spaceL^p(R^N) and homogeneous Sobolev space H_p^s(R^N) thatMp,0 =L^p(R^N) and M^s_p,0 = ˙H_p^s(R^N), respectively. Also, Morrey and Sobolev-Morrey spaces present the following scaling

kf(γ·)kM_p,µ =γ^−N−µp kfkM_p,µ and kf(γ·)kM^s_p,µ =γ^s−N−µp kfkM^s_p,µ, where the exponentss ands−^N−µ_p are calledscaling index and regularity index, respectively.

Lemma 2.1. Suppose that s∈R,1≤p₁, p₂, p₃<∞and0≤µ_i< N,i= 1,2,3.

(i) (Inclusion) If ^N−µ_p ¹

1 =^N−µ_p ²

2 andp1≤p2,

M_p2,µ2 ⊂ M_p1,µ1. (2.3)

(ii) (Sobolev-type embedding) Let p1≤p2, M^s_p1,µ⊂ M^s−(

N−µ p1 −^N−µ_p

2 )

p₂,µ . (2.4)

(iii) (H¨oder inequality) Let _p¹

3 = _p¹

2+_p¹

1 and ^µ_p³

3 =^µ_p²

2+^µ_p¹

1. Iffj∈ Mp_j,µ_j with j= 1,2, thenf1f2∈ Mp₃,µ₃ and

kf1f2kp₃,µ₃ ≤ kf1kp₁,µ₁kf2kp₂,µ₂. (2.5) Finally, notice that the following homogeneous functions, of degree−dands−d, belong to Morrey and Sobolev-Morrey spaces, respectively:

ρ0(x) =Yk(x)|x|^−d−k∈ Mp,µ and ρs(x) =Yk(x)|x|^s−d−k∈ M^s_p,µ, (2.6) where Y_k(x)∈L^p(S^N−1) is a harmonic homogeneous polynomial of degreek, µ= N −dp, 0 < d−s < N and 1< p < N/d. Indeed, using [20, Theorem 4.1] we obtainρb_s(ξ) =γ_k,sY_k(ξ)|ξ|^{d−s−k−N} provided 0< d−s < N, whereγ_k,sis a positive constant. It follows from (2.2) that

kρsk_M^s_p,µ =

^+∞X

ν=−∞

|2^sνF⁻¹ψν(ξ)γk,sYk(ξ)|ξ|^{d−s−k−N}|^21/2 _M

p,µ

∼

X^+∞

ν=−∞

|F⁻¹ψ_ν(ξ)|ξ|^sγ_k,sY_k(ξ)|ξ|^{d−s−k−N}|^21/2 Mp,µ

=kρ0kMp,µ,

which is finite. In fact, polar coordinates in R^N and homogeneity of Y_k(x) ∈ L^p(S^N−1) yield

kρ₀k^p_Lp(Q

r)= Z

S^N−1

|Y_k(x⁰)|^p Z r

0

t^N−dp−1dt dσ(x⁰) =kY_kk^p_Lp(

S^N−1)r^µ, whereµ=N−dp, 1< p < N/d.

2.2. Duhamel formula. We consider the partial fractional differential equation

∂_t^αu(t, x) = ∆_xu(t, x)−f(t, x), x∈R^N, t >0, u(t, x)

_t=0=ϕ(x), ∂

∂tu(t, x)

_t=0=ψ(x),

(2.7) forα∈(1,2) and∂_t^αstands for partial fractional derivative given by

∂_t^αf(t, x) = 1 Γ(m−α)

Z t

0

∂^m_s f(s, x)

(t−s)^α+1−mds, m−1< α≤m, m∈N. Formally, applying the Fourier transform in (2.7), we obtain the fractional ordinary differential equation

∂_t^αu(t, ξ) + 4πb ²|ξ|²bu(t, ξ) =fb(t, ξ), u(t, ξ)|b _t=0=ϕ(ξ),b ∂_tbu(t, ξ)

_t=0=ψ(ξ)b

which, by [11, Example 4.10], is equivalent to

bu(t, ξ) =E_α,1(−4π²t^α|ξ|²)ϕ(ξ) +b tE_α,2(−4π²t^α|ξ|²)ψ(ξ)b +

Z t

0

Eα,1(−4π²(t−s)^α|ξ|²) Z s

0

rα(s−τ)fb(τ, ξ)dτ ds,

(2.8)

whereEα,β(z) denotes the two-parametric Mittag-Leffler function E_α,β(z) =

∞

X

k=0

z^k

Γ(αk+β) and E_α(z) :=E_α,1(z), for allα, β >0. (2.9) Hence, in original variables, we have

u(t, x) =G_α,1(t)ϕ(x) +G_α,2(t)ψ(x) +N_α(u)(t, x), whereG\α,j(t)f(ξ) is defined by (1.7) andNαis defined by (1.8).

Note thatG2,2(t) is the wave group ^sin(4π_4π2t|ξ|^2t|ξ|)

^∨

,G2,1(t) = cos(4π²t|ξ|)^∨ and G1,1(t) = (e^{−4π2t|ξ|2})^∨ is the heat semigroup.

3. Functional setting and theorems

Before starting our theorems, letβ >0 and 0≤µ < N be such that β =α

2

N−µ

p −N−µ r

and µ=N− 2p

ρ−1, (3.1)

which makek · kX_β invariant by scaling map (1.4).

3.1. Existence of solutions. Given a Banach space Y, we will denoteBY(ε) a closed ball of radius ε centered at the origin of the space Y. Our existence and stability result is stated as follows.

Theorem 3.1. Let N ≥2,1< α <2,q=_ρ+1^2ρ , and 0≤µ=N−_ρ−1^2p , forp >1.

Suppose that ^N−µ_p −^N−µ_r <2,r > ρ >1 +α, p

r < 1 α−1

2, α

2−α< q < 2

α, 1−p r

< ρ−1 α

1 q−α

2

. (3.2)

(i) (Global solution) There exist ε >0 such that if kϕk_D(α,β)+kψk

D(α,β)e ≤ε, then problem (1.1)-(1.2) has a unique global-in-time mild solution u ∈ B_Xβ(2ε) satisfying

ku(t,·)kMr,µ≤Ct^−β and k∇xu(t,·)kMr,µ≤Ct^−β−α/2. (3.3) (ii) (Stability in X_β) The solution u in Theorem 3.1(i) is stable with respect to the initial data ϕ and ψ, that is, the data-map solution (ϕ, ψ) 7→ u is locally Lipschitz continuous from D(α, β)×D(α, β)e intoXβ:

ku−uk˜ X_β ≤C kϕ−ϕk˜ D(α,β)+kψ−ψk˜

D(α,β)e

, (3.4)

where uand u˜ are solutions of (1.1) with initial values(ϕ, ψ) and( ˜ϕ,ψ), respec-˜ tively.

Remark 3.2. Let us compare our theorem with some previous results.

(A) If ψ = 0, we may take ϕ ∈ Mp,µ in Theorem 3.1-(i) with smallness on kϕkMp,µ.

(B) Theorem 3.1-(i) holds forα= 1 andψ= 0. Hence, the spaceD(1, β) strictly includes the space N(ϕ) taken in [17]. Indeed, letr < r₁ andµ₂ = 0 in Lemma 2.1-(i) to get

kϕk_D(1,β)= sup

t>0

t^βke^t∆ϕk_Mr,µ+ sup

t>0

t^12+βke^t∆ϕk_M1 r,µ,

≤sup

t>0

t^βke^t∆ϕkL^r1+ sup

t>0

t^12+βke^t∆ϕkH˙_r¹

1

=kϕk_N(ϕ).

On the one hand (see [14, (2.56)]), homogeneous Besov-Morrey spaces can be de- fined by

N_r,µ,∞^−2s =

f ∈ S⁰ :kfk_N−2s

r,µ,∞= sup

t>0

t^−ske^t∆fk_Mr,µ <∞ , s >0.

Hence, the space D(1, β) is a kind of Besov-Morrey spaces. On the other hand, whenα6= 1 the norms kϕkD(α,β)=kGα,1(t)ϕkX_β andkψk

D(α,β)e =kGα,2(t)ψkX_β

satisfy

kϕk_D(α,β)≤Ckϕk_Mp,µ and kψk

D(α,β)e ≤Ckψk

M^−2/αp,µ

in view of Lemma 5.1. SoMp,µ⊂D(α, β) andM^−2/αp,µ ⊂D(α, β).e

(C) (Viscous Hamilton-Jacobi) Letκ2= 0, ψ= 0 in (1.1)-(1.2),µ=N −^q−1_2−qp and kϕkD(α,β) small enough. Using the proof of Theorem 3.1, the problem (1.1)- (1.2) has a unique solutionu∈C((0,∞);Mr,µ)∩C((0,∞);M¹_r,µ) such that

sup

t>0

t^{(N−µ)2 α(1p−1r)}ku(t)kM_r,µ≤C, sup

t>0

t^α2+

(N−µ) 2 α(¹_p−¹_r)

k∇u(t)kM_r,µ≤C,

under the assumptions in Theorem 3.1 with the changeρ= _2−q² . In other words, we obtain a version of Theorem 2.1 and Proposition 2.3 of [3] when 1< α <2. If α= 1, the assumption ^N−µ_p −^N−µ_r <2 is not necessary because of the smoothing effect of the heat semigroup inMp,µ (see e.g. [10]).

3.2. Self-similar solutions. As we commented before, a necessary condition for initial data to produce self-similar solutions is homogeneity and simplest candidates are the radial functions

ϕ(x) =ε1|x|^−ρ−12 and ψ(x) =ε2|x|^{−ρ−12 −α2}. (3.5) Hence, we needD(α, β) andD(α, β) to satisfye

kψγk

D(α,β)e =kψk

D(α,β)e and kϕγk_D(α,β)=kϕk_D(α,β), (3.6) and it comes from the scaling invariance ofXβ. Moreover, (2.6) and Remark 3.2- (B) permit us to take thesingular functions(3.5) as initial data, sinceϕ∈ Mp,µ⊂ D(α, β) andψ ∈ M^−2/αp,µ ⊂D(α, β) providede µ=N−_ρ−1^2p , ρ >max{1 +_N²,1 +

2α

αN−2} and 1< p < r.

Theorem 3.3(Self-similarity). Under the assumptions of Theorem 3.1, letϕand ψbe homogeneous functions of degree −_ρ−1² and−_ρ−1² −_α², respectively. Then the solution uof Theorem 3.1-(i) is self-similar.

Remark 3.4. Let us remark some consequences of this theorem.

(A) (Infinity energy data) In Theorem 3.3 we can build singular initial data (ψ, ϕ) which can be arbitrarily large inL²(R^N)×H˙^2/α(R^N), provided that _α²+_ρ−1² < ^N₂ and 1< p < ^N(ρ−1)₂ . Indeed, letϕ∈ S⁰(R^N) andψ∈ S⁰(R^N)/P be given by (3.5).

Usingϕ(ξ) =b γ_0,0ε₁|ξ|^{ρ−12 −N}, we see thatϕandψare arbitrarily large in ˙H^2/αand L² in view of

kψk²_L2(R^N)=ε²₂ Z

R^N

|x|^{−ρ−14 −α4}dx

=ε²₂ lim

ω₂→∞

Z ω2

0

Z

S^N−1

r^{−ρ−14 −α4}r^N−1dσdr

=C lim

ω₂→∞ω⁻

4 ρ−1−_α⁴+N

2 = +∞

and

kϕk²

H˙α²(R^N)= Z

R^N

|ξ|^4/α|ϕ(ξ)|b ²dξ=γ²_0,0ε²₁ Z

R^N

|ξ|^{4/α+ρ−14 −2N}dξ

=C lim

ω₁→0

Z ∞

ω1

Z

S^N−1

r^{4/α+ρ−14 −N−1}dσdr

=C lim

ω₁→0ω

4

α+_ρ−1⁴ −N

1 = +∞.

Then, even the initial data ϕand ψ are in the Morrey spaces Mp,µ and M^−2/αp,µ , respectively, they may be arbitrarily large in ˙H^2/α(R^N) andL²(R^N).

(B) Inspired by [16], we use a Littlewood-Paley decomposition of the Sobolev- Morrey spaces (see subsection 2.1) to build general singular functions for Theorem 3.3. In fact, let Yk₁(x), Yk₂(x) be homogeneous harmonic polynomials of degree k1 and k2, respectively. Consider S(ϕ, ψ) the set of functions (ϕ, ψ) ∈ S⁰(R^N)× S⁰(R^N)/P such that

ϕ(x) =1

Y_k1(x)

|x|^{ρ−12 +k1} and ψ(x) =2

Y_k2(x)

|x|^{ρ−12 +α2+k2}.

By (2.6), the setS(ϕ, ψ) gives us a class of data for existence of self-similar solutions for (1.1)-(1.2).

3.3. Symmetries. This subsection concerns with symmetries of solutions obtained in Theorems 3.1 and 3.3. It is straightforward to check that Eα,1(4π²t^α|ξ|²) and tEα,2(4π²t^α|ξ|²) are invariant by the setO(N) of all rotations inR^N. It follows that Gα,1(t) andGα,2(t) are O(N)− invariant. Hence, it is natural to ask whether or not the solutions of the above theorems present symmetry properties under certain qualitative conditions on the initial data.

Let A be a subset of O(N). A function h is said symmetric under action A when h(x) = h(T(x)) for all T ∈ A. If h(x) = −h(T(x)), the function h is said antisymmetric under the action ofA.

Theorem 3.5. Let the hypotheses of Theorem 3.1 be satisfied. The solutionu(·, t) is symmetric for allt >0, wheneverϕandψ are symmetric under actionA.

Remark 3.6. A radially symmetric solution is a self-similar solution, if the profile ω depends only on r = |x|, that is, there is a function U such that u(t, x) = t^−ρ−1α U(|x|/t^α2),t >0.

(A) LetA=O(N) in Theorem 3.5. Ifϕandψare radial and homogeneous func- tions of degree−_ρ−1² and−_ρ−1² −²_α, respectively (see Remark 3.4), then Theorems 3.1, 3.3 and 3.5 imply that (1.1)-(1.2) have a unique self-similar solution u∈ X_β which is radially symmetric inR^N.

(B) Unlike the caseκ1= 0, antisymmetry does not hold in general, forκ16= 0.

4. Technical estimates

In this section we prove some Mikhlin-type estimates for Mittag-Leffler functions.

In spite of the fact that these estimates are necessary in the proof of Lemma 5.1, they are of independent interest. We start the section with a suitable decomposition ofEα,β(z).

4.1. Decompositions ofE_α,β(z).

Proposition 4.1. Let z∈Cbe such thatRe(z)>0and define ω_α,β(z) = z^1−βα

α h

exp a_α(z) +1−β α πi

+ exp b_α(z)−1−β α πii

, lα,β(z) =

Z ∞

0

Hα,β(s)e^{−z1/αs1/α}z^α1(1−β)ds, where

Hα,β(s) = 1 απ

sin[(α−β)π]−ssin(βπ)

s²+ 2scos(απ) + 1 s^1−βα , (4.1) a_α(z) =z^1/αe^πiα, b_α(z) =z^1/αe^−πiα.

Suppose that1< α <2 and1≤β≤2, then

E_α,β(−z) =ω_α,β(z) +l_α,β(z). (4.2) Proof. Recall that Mittag-Leffler function can be written as

E_α,β(−z) = 1 2πi

Z

Ha

t^α−βe^t

t^α+zdt, (4.3)

whereHais the Hankel path, i.e. a path starts and ends at−∞and encircles the disk |t| ≤ |z|^1/α positively. The integrand Φ(t) = ^tα−β_tα+z^et of (4.3) has two poles aα(z) andbα(z), because 1< α <2. Proceeding as in [5, Lemma 1.1], the residues theorem yields

2πiE_α,β(−z) = Z R

∞

Φ(te^−πi)d(te^−πi) + 2πi(Res(Φ, a_α(z)) + Res(Φ, b_α(z)))

− Z R

Φ(te^−πi)d(te^−πi)− Z

R

Φ(te^πi)d(te^πi)

− Z π

−π

Φ(e^θi)d(e^θi) + Z ∞

R

Φ(te^πi)d(te^πi)

=:I₁(R) + 2πi(Res(Φ, a_α(z)) + Res(Φ, b_α(z)))−I₂(, R)

−I3(R, )−I4() +I5(R).

We first get

lim

R→∞I₁(R) = lim

→0⁺I₄() = lim

R→∞I₅(R) = 0.

An easy computation yields lα,β(z) =− 1

2πi lim

R→∞,→0⁺I2(, R) +I3(R, ).

Indeed, 1

2πi lim

→0⁺,R→∞

I2(, R) +I3(R, )

= 1 2πi

Z ∞

0

e^−tt^α−βe^(α−β)πi t^αe^απi+z dt−

Z ∞

0

e^−tt^α−βe^{−(α−β)πi} t^αe^−απi+z dt

=− 1 2πi2i

Z ∞

0

e^−tt^α−βzsin[(α−β)π]−t^αsin(βπ)

t^2α+ 2t^αzcos(απ) +z² dt (4.4)

=− Z ∞

0

Hα,β(s) exp (−z^1/αs^1/α)z^α1(1−β)ds, (4.5)

=−lα,β(z),

where the changet7→z^1/αs^1/α was used from (4.4) to (4.5). Also, we obtain Res(Φ, a_α(z)) = z^1−βα

α exp(a_α(z) +πi(1−β)/α), Res(Φ, b_α(z)) = z^1−βα

α exp(b_α(z)−πi(1−β)/α).

These give us the desired decomposition.

In particular, forβ= 1 andβ = 2 in Proposition 4.1, we have the decompositions E_α,1(−z) =ω_α,1(z) +l_α,1(z) (4.6) in [5, Lemma 1.1], and

Eα,2(−z) =ωα,2(z) +lα,2(z), (4.7) in [6, Lemma 1.2-(IV)]. Notice thatωα,1(z) oscillates with frequency sin(π/α) and amplitude decaying exponentially with rate|cos(π/α)|, in view of

ωα,1(z) = 2

αexp(z^1/αcos(π/α)) cos(z^1/αsin(π/α)).

On the other hand, the function lα,1(z) exhibits the relaxation phenomena of Eα,1(−z), namely,

lα,1(z) = Z ∞

0

Hα,1(s) exp(−s^1/αz^1/α)ds= Z ∞

0

exp(−s^1/αz^1/α)dµα(s), where

Hα,1(s) = sin(απ) απ

1

s²+ 2scos(απ) + 1

and dµα(s) = Hα,1(s)ds is a finite measure in R+ such that µα(R+) = 2− _α². Furthermore, whenβ=α, the decomposition (4.2) is useful to show that the map G_α,β(·),β= 1,2, is differentiable for t >0. Indeed, see (5.4) and (5.6) below.

4.2. Mikhlin estimates for Eα,β(−σ(ξ)). We provide estimates for Eα,1(σ(ξ)), Eα,2(σ(ξ)) andEα,α(σ(ξ)), where σ∈C^∞(R^N\{0}; (−∞,0)) is the symbol of the Fourier multiplier

σ(D)f =F⁻¹σ(ξ)Ff(ξ), f ∈ S(R^N).

Consider the changez7→σ(ξ) into (4.6) and write it as follows:

E_α,1(σ(ξ)) =ω_α,1(−σ(ξ)) +l_α,1(−σ(ξ)). (4.8) Proposition 4.2. Let σ(ξ)∈ C^∞(R^N\{0}) be a function homogeneous of degree k >0 and such that

∂^γ

∂ξ^γ[σ(ξ)]

≤A|ξ|^k−|γ|, ξ6= 0 (4.9)

for all multi-index γ ∈ (N∪ {0})^N with |γ| ≤ [N/2] + 1. Let 1 < α < 2 and 0≤δ < k, there existsC >0such that

∂^γ

∂ξ^γ[|ξ|^δE_α,1(σ(ξ))]

≤CA|ξ|^−|γ|, ξ6= 0. (4.10) Proof. Taking (4.9) into account we obtain

∂^γ

∂ξ^γ[−σ(ξ)]^l

≤CA|ξ|^−|γ||ξ|^kl, for alll∈R. (4.11) Hence, theγ^th-order derivative of the parcel |ξ|^δω_α,1(σ(ξ)) can be estimated by

∂^γ

∂ξ^γ[|ξ|^δωα,1(−σ(ξ))]

=

∂^γ

∂ξ^γ

|ξ|^δexp(e^iπα(−σ(ξ))^1/α) +|ξ|^δexp(e^−iπα(−σ(ξ))^1/α)

≤C|ξ|^−|γ|

c0|ξ|^δ+c1|ξ|^δ+kα +· · ·+c_|γ||ξ|^δ+|γ|kα

e^{cos(πα)(−σ(ξ))1/α}

≤CA|ξ|^−|γ|.

(4.12)

To estimatel_α,1(σ(ξ)), recall that lα,1(σ(ξ)) =

Z ∞

0

Hα,1(s) exp(−s^1/α(−σ(ξ))^1/α)ds.

Using the homogeneityσ(λξ) =λ^kσ(ξ), we have

∂^γ

∂ξ^γ[|ξ|^δe^{−s1/α(−σ(ξ))1/α}]

≤C|ξ|^−|γ|

c0|ξ|^δ+c1|ξ|^δ+αks^1/α+· · ·+c_|γ||ξ|^δ+|γ|kα s^|γ|α

e^{−s1/α(−σ(ξ))1/α}

=Cs^−δk|ξ|^−|γ|

c₀|s^1kξ|^δ+c₁|s^k1ξ|^δ+kα +. . . +c_|γ||s^1kξ|^δ+|γ|kα

e^{−[−σ(s1/kξ)]1/α}

≤CAs^−δk|ξ|^−|γ|.

(4.13)

Then

∂^γ

∂ξ^γ[|ξ|^δlα,1(−σ(ξ))]

= sin(απ) απ

Z ∞

0

1

s²+ 2scos(απ) + 1

∂^γ

∂ξ^γ[|ξ|^δe^{−s1/α(−σ(ξ))1/α}] ds

≤CAsin(απ) απ

Z ∞

0

s^−δk

s²+ 2scos(απ) + 1ds

|ξ|^−|γ|

≤CA|ξ|^−|γ|,

because 0≤δ < k. These estimates prove the proposition.

In general, we obtain the following proposition for the two-parametric Mittag- Leffler function.

Proposition 4.3. Let σ(ξ)∈ C^∞(R^N\{0}) be a homogeneous function of degree k > 0 satisfying (4.9), for all multi-index γ ∈ (N∪ {0})^N with |γ| ≤ [N/2] + 1.

Then, there exists a positive constantC (independent of δandk) such that

∂^γ

∂ξ^γ[|ξ|^δE_α,β(σ(ξ))]

≤CA|ξ|^−|γ|, ξ6= 0 (4.14) provided that 1< α <2and k(^β_α−_α¹)≤δ < k.

Proof. The proof is similar the proof of Proposition 4.2. Indeed, proceeding as in (4.12), it follows that

∂^γ1

∂ξ^γ1hα,β(ξ)

:=

∂^γ1

∂ξ^γ1

h|ξ|^δexp a_α(σ(ξ)) +1−β α πi +|ξ|^δexp bα(σ(ξ))−1−β

α πii

≤CA|ξ|^−|γ1|

c0|ξ|^δ+c1|ξ|^δ+αk +· · ·+c_|γ1||ξ|^δ+|γ1α|k

ecos(π/α)(−σ(ξ))^1/α,

(4.15)

for all multi-indexγ₁. Hence, Leibniz’s rule, (4.11) and (4.15) give us

∂^γ1

∂ξ^γ[|ξ|^δωα,β(−σ(ξ))]

≤ X

γ1≤γ

γ γ1

∂^γ1

∂ξ^γ1[σ(ξ)]^1−βα

∂^γ−γ1

∂ξ^γ−γ1[hα,β(ξ)]

≤CA|ξ|^{−|γ1|−|γ−γ1|}

c0|ξ|^δ+k(α¹−^β_α) +· · ·+c|ξ|^δ+k(α¹−^β_α)^+|γ−γα^1|k

×ecos(π/α)(−σ(ξ))^1/α

≤CA|ξ|^−|γ|,

in view ofδ+k(¹_α−^β_α)≥0. Also, using (4.11) and (4.13), the Leibniz’s rule yields

∂^γ

∂ξ^γ

(−σ(ξ))^1−βα

|ξ|^δexp(−s^1/α(−σ(ξ))^1/α)

≤CA|ξ|^−|γ|s^{−kδ+βα−α1}. Hence, we estimate

∂^γ

∂ξ^γ[|ξ|^δlα,β(−σ(ξ))]

≤ Z ∞

0

|Hα,β(s)|

∂^γ

∂ξ^γ

(−σ(ξ))^1−βα |ξ|^δexp(−s^1/α(−σ(ξ))^1/α) ds

=CA(I+II)|ξ|^−|γ|

≤C|ξ|^−|γ|,

where the integralsI andII are defined by (see (4.1)) I= sin[(α−β)π]

απ

Z ∞

0

s^−δk

s²+ 2scos(απ) + 1ds, (4.16) II=−sin(βπ)

απ Z ∞

0

s^1−δk

s²+ 2scos(απ) + 1ds. (4.17) Those integrals are finite in view ofδ < k. This completes the proof of the propo-

sition.

5. Sobolev-Morrey estimates

In this section we obtain fundamental estimates which will be important to prove Theorem 3.1.

5.1. Linear estimates. Here, we present some estimates of the Mittag-Leffler op- erators{Gα,β(t)}t≥0 in Sobolev-Morrey spaces. Indeed, based on Propositions 4.2 and 4.3 with the homogeneous symbol σ(ξ) =−4π²|ξ|² of degree 2, the following lemma can be proved by proceeding as in [2, Lemma 3.1-(i)].

Lemma 5.1. Let γ1 ≤γ2 ∈ R, 1 < p1 ≤ p2 <∞, 0 ≤ µ < N, 1 < α < 2 and λ= (γ2−γ1) +^N_p^−µ

1 −^N−µ_p

2 . There is a constantC such that kGα,1(t)fk_M^γ2

p2,µ ≤Ct^−α2λkfk_M^γ1

p1,µ, ifλ <2, (5.1) kGα,2(t)fk_M^γ2

p2,µ≤Ct^−α2λkfk

M^γ1−

2 p1,µα

, ifλ+ 2

α <2, (5.2) kGα,α(t)fk_M^γ2

p2,µ≤Ct^{α−1−α2λ}kfk_M^γ1

p1,µ, if 2− 2 α

< λ <2, (5.3) for allf ∈ S⁰(R^N).

We finish this subsection by noticing that{∂tGα,1(t)}t≥0and{∂tGα,2(t)}t≥0are bounded in Morrey spaces. Indeed, a straightforward computation gives us

d

dtEα,1(−4π²|ξ|²t^α) =−4π²|ξ|²

t^α−1Eα,α(−4π²|ξ|²t^α) , fort >0 andξ6= 0. It follows from Lemma 5.1-(iii) that

k∂tGα,1(t)fk_Mp

2,µ≤CkGα,α(t)fk_M2

p2,µ ≤Ct^−α2

“_N−µ

p1 −^N−µ_p

2

”−1

kfk_Mp

1,µ. (5.4) Using

tEα,2(−4π²|ξ|²t^α) = Z t

0

Eα,1(−4π²|ξ|²s^α)ds, (5.5) Lemma 5.1-(i) yields

k∂tG_α,2(t)fkMp2,µ=kGα,1(t)fkMp2,µ ≤Ct^−α2

“_N−µ

p1 −^N−µ_p

2

”

kfkMp1,µ. (5.6)

5.2. Nonlinear estimates. This subsection is devoted to estimate the nonlinear term Nα(u) on the functional space Xβ. Firstly, let us denote B(ν, η) byspecial beta function B(ν, η) = R1

0(1−t)^ν−1t^η−1dt which is finite, for all η, ν > 0. Let k₁, k₂, k₃<1, fort >0 ands >0 the changes of variableτ 7→τ sand s7→stgive us

I(t) = Z t

0

(t−s)^−k1 Z s

0

(s−τ)^−k2τ^−k3dτ ds

=B(1−k2,1−k3) Z t

0

(t−s)^−k1s^−k2−k3+1ds

=B(1−k2,1−k3)B(1−k1,2−k2−k3)t^{2−k1−k2−k3}.

(5.7)

We freely use (5.7) in the next proof.

Lemma 5.2. Under assumptions of Theorem 3.1, there is a positive constantK= K(κ1, κ2)such that

kNα(u)− Nα(v)kX_β ≤Kku−vkX_β kuk^ρ−1_X

β +kvk^ρ−1_X

β +kuk^q−1_X

β +kvk^q−1_X

β

. (5.8) Proof. RecallN_α(u) and rewrite it as follows:

Nα(u)(t) = Z t

0

Gα,1(t−s) Z s

0

rα(s−τ) κ2|u|^ρ−1u+κ1|∇xu|^q dτ ds

=:N_α¹(u)(t) +N_α²(u)(t).

(5.9) The proof is divided in three steps.

First step: Estimates for N_α¹(u). In (5.1), let (γ1, γ2, p1, p2) = (0,1, r/ρ, r) and 1< ρ < rto obtain

kN_α¹(u)(t)− N_α¹(v)(t)k_M1 r,µ

≤C Z t

0

(t−s)^−λ1 Z s

0

rα(s−τ)kf(u)−f(v)k_Mr/ρ,µdτ ds, wheref(u)(τ) =κ1|u(τ)|^ρ−1u(τ) andλ1= ^α₂ +^α_{2 N−µ}

r/ρ −^N−µ_r

. Using that

| |a|^ρ−1a− |b|^ρ−1b| ≤C|a−b| |a|^ρ−1+|b|^ρ−1

, for allρ >1 (5.10) and ^ρ_r = ¹_r+^ρ−1_r , the H¨older inequality (2.5) yields

kN_α¹(u)(t)− N_α¹(v)(t)k_M¹_r,µ ≤C|κ2| Z t

0

(t−s)^−λ1θ(s)ds, (5.11) whereθ(s) is given by

θ(s) = Z s

0

(s−τ)^α−2ku(τ)−v(τ)kM_r,µ ku(τ)k^ρ−1_M

r,µ+kv(τ)k^ρ−1_M

r,µ

dτ

≤C Z s

0

(s−τ)^α−2τ^−ρβτ^βku(τ)−v(τ)kMr,µ×

×τ^β(ρ−1) ku(τ)k^ρ−1_M

r,µ+kv(τ)k^ρ−1_M

r,µ

dτ

≤C Z s

0

(s−τ)^α−2τ^−ρβdτku−vkX_β kuk^ρ−1_X

β +kvk^ρ−1_X

β

.

(5.12)

Notice thatα(ρ−1)^N−µ_2r =α−(ρ−1)β yields

−λ₁+α−ρβ=−α

2 + (ρ−1)β−α+α−ρβ=−α 2 −β.