Around a singular solution of a nonlocal nonlinear heat equation

We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.

This is a straightforward generalization of extensively studied classical nonlinear heat equation, see [37], to the case of nonlocal but linear diffusion operators defined by fractional powers of Laplacian. Study of solutions around a critical singular solution which is not smoothed out by the diffusion is, in a sense, parallel to analyses for nonlinear heat equation in [37] and, in particular, in [34,35]. This also reveals some threshold phenomena as is in the former case.
Statement of results. The form of singular steady state u ∞ of equation (1.1) is given in Proposition 2.1. Existence of global-in-time solutions for initial data in a suitable critical Morrey space is shown in Proposition 2.3. Local-in-time existence of solutions for initial data having local singularities weaker than u ∞ is proved in Proposition 2.5.
Global-in-time existence of solutions for subcritical initial data u 0 u ∞ is derived in Theorem 2.6, and this construction is based on a novel comparison principle for special solutions in Proposition 2.7. These solutions have diffusion dominated asymptotics.
Analysis of the linearization operator (nonlocal diffusion + Hardy-type potential) around the singular state in Section 3 leads to its nonlinear L 2 -asymptotic stability. Fine asymptotics of solutions in vicinity of u ∞ is given.
Finally, we interpret results on finite time blowup of solutions with "large" initial data (in [40] and recently [6]) as a kind of dichotomy, see Corollary 4.1.
Concerning the question of nonexistence of global-in-time solutions to equation (1.1), the first results have been proved in [40] with the argument based on the seminal idea of [25]. Extensions of such blowup results for more general equations with linear but nonlocal diffusion operators more general than fractional Laplacians and localized source terms are in a forthcoming paper [6] which improves some results in [1] and clarifies sufficient conditions for blowup. Interpretations of sufficient blowup conditions in [6] lead to Corollary 4.1 which shows that the discrepancy bounds between sufficient bounds for global-in-time existence and sufficient bounds for blowup (for the same quantity like the Morrey space M d(p−1)/α (R d ) norm) are well controlled for large dimensions d.
Notation. In the sequel, · p denotes the usual L p (R d ) norm, and C's are generic constants independent of t, u, ... which may, however, vary from line to line. The (homogeneous) Morrey spaces over R d modeled on L q (R d ), q 1, are defined by their norms Caution: the notation for Morrey spaces used elsewhere might be different, e.g. M s q is denoted by M q,λ with λ = dq/s in [38].
The most frequent situation is when q = 1 and we consider M s 1 ≡ M s . The spaces M d(p−1)/α (R d ) and more general M d(p−1)/α q (R d ), q > 1, are critical in the study of equation (1.1), see [38]. We refer the readers to [4,14,30] for analogous examples in chemotaxis theory.
Integrals with no integration limits are meant to be calculated over the whole space R d . The asymptotic relation f ≈ g means that lim s→∞ f (s) g(s) = 1 and f ≍ g is used whenever lim s→∞ f (s) g(s) ∈ (0, ∞).

2.
Existence of global-in-time solutions below the singular solution 2.1. Existence of the singular solution. We have the following there is a unique radial homogeneous nonnegative solution of the equation Proof. By formula (1.3) (see also [12, (3.6)]) for the fractional Laplacian, we have for Thus, a multiple of the function |x| −α/(p−1) is a solution of equation (2.1). The value of s(α, d, p) is determined from the convolution identities [12, (3.3)], see also formula (2.20).

2.2.
Small global-in-time solutions. Our purpose is to prove two global in time existence results, the first one under smallness condition of the norm of critical homogeneous Morrey space M d(p−1)/α q (R d ) with some q > 1, and the second under the assumption that the initial data u 0 is below the singular solution u ∞ (pointwisely).
α > 1, and the initial condition u 0 is sufficiently small in the sense of the norm of the homogeneous Morrey space M (for a number q > 1) is small enough, then a solution of problem (1.1)-(1.2) is global in time, see [38, Proposition 6.1] (as well as a counterpart for the chemotaxis system cf. [4,Th. 1]). Similarly as the former result, it can be proved directly for α ∈ (0, 2) (while the proof in [38] for α = 2 was by contradiction) using the Picard iterations of the mapping For useful estimates of the heat semigroup in Morrey spaces we refer the reader to [27] and [41,Th. 3.8]. They extend immediately to the estimate for 1 < p 1 < p 2 < ∞ When either p 1 = 1 or p 2 = ∞, the norm are . 1 and . ∞ , resp. The crucial estimate for the convergence of the Picard iterations is , and this follows since They are convergent in the norm ||| . ||| for max{p, q} < r < pq, since Then, usual bootstrapping arguments like in [37, proof of Th. 15.2, p. 81] apply and u(t) ∈ L ∞ follows for each t > 0. The author is indebted to Philippe Souplet for this argument showing regularity. Note that this existence result remains valid for nonlinearities with behavior |F (u)| ∼ |u| p , u → 0.
Remark 2.4. Let us recall that for α = 2 the number p F = 1 + 2 d , called the Fujita exponent, borders the case of a finite-time blowup for all positive solutions (for p p F ) and the case of existence of some global in time bounded positive solutions (if p > p F ) (see for example [37] and references therein). The exponent 1 + α d is a counterpart of the Fujita exponent in the case α ∈ (0, 2), see [40].
In the next proposition u 0 is supposed to have asymptotics for large |x| like u ∞ (x) ≍ 1 |x| 2 but local singularities strictly weaker since u 0 ∈ Ms(R d ) with somes > s is assumed.
, then there exists T > 0 and a local in time solution Proof. We follow the approach and notations in the proof of Proposition 2.3. We will estimate the nonlinear operator N (u) in the norms of the spaces M s (R d ), Ms(R d ) and These bounds (2.5)-(2.7) lead in a standard way to the convergence of the Picard iterations for initial data in M s (R d ) ∩ Ms(R d ) of arbitrary size and for sufficiently small t > 0.

Large global-in-time solutions. The main result in this Section is
as well as Then u can be continued to a global-in-time solution which still satisfies the bound Thus,s < s can be chosen as close to d(p−1) α as we wish. Therefore, Proposition 2.5 on local-in-time solutions applies to those data.
The result in Theorem 2.6 is based on the following restricted comparison principle, see [12, Th. 4.1, Th. 5.1] for analogous albeit more complicated constructions for radially symmetric solutions of chemotaxis systems. Proposition 2.7. For each δ ∈ (0, 1) and each K > 0 there exist γ 0 ∈ 0, α p−1 (independent of K and sufficiently close to α p−1 ) such that every solution u ∈ C 1 (R d × (0, T ]) with the properties and the initial data satisfying satisfies the estimate Once the comparison principle in Proposition 2.7 is proved, the local-in-time solution constructed in Proposition 2.5 can be continued onto some interval [T, T + h], and further, step-by-step with the same h > 0, onto the whole half- Proof of Theorem 2.6. Approximating u 0 in Theorem 2.6 by initial data we obtain a global-in-time solution via a monotonicity argument. This procedure is an adaptation of the monotone approximation argument for the classical nonlinear heat equation in [26]. The first step is to show the property that u k increase with k. This can be done in standard way writing the equation for the difference w of two approximating solutions w = u l − u k , l > k, as This is a linear equation of the type in the next Section. The associated semigroup conserves positivity of the initial data w 0 . Then, the pointwise monotone limit of u k 's exists and satisfies equation (1.1) in the weak sense (using the Lebesgue dominated convergence theorem).
We sketch the proof of the comparison principle.
Proof of Proposition 2.7. Let u be a solution of problem (1.1)-(1.2) for an initial data satisfying (2.12). The proof of inequality (2.13) is by contradiction. Suppose that there exists t 0 ∈ (0, T ] which is the first moment when u(x, t) hits the barrier b(x) defined in (2.12). By a priori C 1 regularity of u(x, t) and by property (2.11) the value of t 0 is well defined. Moreover, there exists x 0 ∈ R d satisfying u(x 0 , t 0 ) = b(x 0 ). Define the number (2.14) where two parts of the graph of the barrier b meet. Let us consider the auxiliary function With this choiceũ hits a constant part of the graph of the modified barrier |x| γ b(x). Let us compute according to formula (1.3) for the fractional Laplacian The passage from (2.16) to (2.17) is obvious, see an analogous reasoning in [12, (4.19)]. Formula (2.17) follows by (1.3) above. Formula [12, (3.6)] obtained for α + γ < d from (1.3) and convolution identities for powers |x| −γ , etc., [12, (3.3)] gives for γ = α p−1 (so that α + γ < d by assumption p > 1 + α d−α ) We used in this passing from (2.18) to (2.19). In the first case when γ = α p−1 , we recall formula (2.2) for the constant s(α, d, p) which leads tõ The passage from (2.18) to (2.19) in the middle term follows sinceũ(., t 0 ) assumes its maximal value at x 0 and (2.19) equals The inequality ∂ ∂tũ (x 0 , t) t=t 0 < 0 contradicts the assumption thatũ hits for the first time the constant level δs(α, d, p) at t = t 0 .
Remark 2.8. This kind of result in Theorem 2.6 is known for the classical nonlinear heat equation (cf. [29, Th. A]) but the proof in [5] seems be somewhat novel. Similar pointwise arguments are powerful tools and, as such, they have been used in different contexts as e.g. fluid dynamics and chemotaxis theory: [17,20,19,28], and free boundary problems. If u 0 is radially symmetric and u 0 (x) < δu ∞ (x) for some δ < 1, then the solution of (1.

Asymptotic behavior of solutions below the singular solution
We consider in this section behavior of solutions close to the singular solution u ∞ but lying below it. We use the same approach as in [34] for classical nonlinear heat equation.
Introducing a new variable w(x, t) = u ∞ (x) − u(x, t), where u = u(x, t) is a solution to (1.1)-(1.2) the considered problem takes the form where the constant s(α, d, p) is defined in formula (2.2). Note that the last term on the right-hand side of equation (3.1) is nonpositive, namely with the operator Hw = (−∆) α /2 w − s(α, d, p)p|x| −α w. Consequently, using the condition 0 u 0 (x) u ∞ (x) and the just mentioned comparison principle we can write First, we concentrate on existence and properties of solutions to the linear initial value problem w(x, 0) = w 0 (x).
appearing in a fractional version of the Hardy inequality. Following those arguments, we define the weights ϕ σ (x, t) ∈ C(R d \ {0}) as where σ ∈ (0, d − α) satisfies the following equality Moreover, there exists a positive constant C such that for all t > 0 and all x, y ∈ R d \ {0} where the functions ϕ σ are defined in (3.8) and G α (x − y, t) is the fractional heat kernel.
Remark 3.2. Another important exponent for α = 2 is the Joseph-Lundgren exponent which plays a crucial role in the study of stability of solutions to the classical nonlinear heat equation (see [34] and references therein). The analogue of this exponent for α ∈ (0, 2) is the critical value of p for which the assumption s(α, d, p) p−1 p The following theorem is the consequence of the estimates stated in formulas (3.10).
holds for a constant C > 0 and all t 1.
Proof. First, for every fixed x ∈ R d , we apply the estimate of the kernel e −tH in Theorem 3.1 in the following way Next, we split the integral on the right-hand side into three parts I 1 (x, t), I 2 (x, t) and I 3 (x, t) according to the definition of the weights ϕ σ and the assumptions on the function w 0 . Let us begin with I 1 (x, t) We use the same argument to deal with
Theorem 3.4. Assume that | · | σ w 0 ∈ L ∞ (R d ) and lim |x|→∞ |x| σ w 0 (x) = 0, where σ ∈ (0, d − α) satisfies equation (3.9). Then Proof. For every fixed x ∈ R d we use the estimate from Theorem 3.1 as follows We decompose the integral on the right-hand side according to the definition of ϕ σ and we estimate each term separately. Substituting y = zt 1 /α and using the fact that ϕ σ (x, t) 2t σ /α |x| −σ if |x| t 1 /α we obtain Hence, t σ /α sup follows by the Lebesgue dominated convergence theorem, because G x t 1 /α − z, 1 is bounded and the function |z| −2σ is integrable for |z| 1. By the assumption imposed on w 0 , given ε > 0 we may choose t so large that sup |y| √ t |y| σ w 0 (y) < ε.
Now, using the inequality 1 t 1 /α |y| σ for |y| t 1 /α , we obtain Since R d G(x − y, t) dy = 1 for all t > 0, x ∈ R d and since ε > 0 is arbitrary, we get Let us define the weighted L q (R d )-norm as follows Note, that in particular for q = 2, the norm · 2,ϕσ(t) coincides with the usual L 2 -norm on R d .
Proof. Using estimates of the kernel e −tH we get Observe that the weights are bounded by We split the integral into two terms and applying (3.15) we get using estimates of the semigroup generated by the fractional Laplacian. Moreover, applying the Young inequality and estimates of this semigroup we arrive at  2) constructed in Theorem 2.6 with exponent p satisfying assumption (3.11) and σ ∈ (0, d − α) fulfill equation (3.9). Assume that there exist constants b > 0 and ℓ ∈ σ, d − σ such that for all |x| 1. Then hold for a constant C > 0 and all t 1. Moreover, Fila and Winkler [22] showed the uniform convergence of solutions to the singular one on R d \ B r (0), where B r (0) is the ball centered at the origin with the radius r.
Proof. It suffices to use inequality (3.3) and to estimate its right-hand side by Theorem 3.3.
We can improve Theorem 3.6 for the limit exponent ℓ = σ as follows.
Theorem 3.8. Let u = u(x, t) be a solution to problem (1.1)-(1.2) constructed in Theorem 2.6 with exponent p satisfying assumption (3.11) and σ ∈ (0, d − α) fulfill equation (3.9). Suppose that there exists a constant b > 0 such that and, moreover, lim Then the relations Remark 3.9. A similar result for the classical case, namely for α = 2, can be found in [23,21,22,24], where authors proved estimates from below of the L ∞ -norm of solutions using matched asymptotics.
Proof. As in the proof of Theorem 3.6, it is sufficient to use (3.3) together with Theorem 3.4, substituting w 0 (x) = u ∞ (x) − u 0 (x). Proof. Since we have inequality (3.4), it suffices to prove that (3.12) in Theorem 3.3 enables us to write for all x ∈ R d \ {0} and t > 0. Next, using the explicit form of the weights ϕ σ , we define the function An easy computation shows that the function F attains its maximum at and this is equal to max for some constant C(b) 0. Hence, we get (3.18).
To obtain (3.19), we use the result from Theorem 3.4. It follows from (3.13) that for every ε > 0 there exists T > 0 such that for all x ∈ R d \ {0} and t > T . Hence, by (3.4), we have Now, using again the explicit form of the weights ϕ σ , we consider the function Elementary computations give us that the function G attains its maximum at , we see that the maximum of the function G diverges to infinity if ε tends to zero. This completes the proof of (3.19).
Our next goal is to prove the asymptotic stability of the singular solution u ∞ in the Lebesgue space L 2 (R d ).
Proof of Theorem 3.11 i). According to estimates (3.3) it is enough to estimate the L 2norm of the expression e −tH w 0 for every w 0 satisfying two conditions: w 0 ∈ L 1 (R d ) and | · | −σ w 0 ∈ L 1 (R d ). Applying (3.14) with q = 2, r = 1, and using the definition of the functions ϕ σ (x, t), we may write This establishes formula (3.20).
Since the second term on the right-hand side converges to zero as t → ∞ by the first part of Theorem 3.11, we get lim sup t→∞ e −tH w 0 2 Cε.
This completes the proof of Theorem 3.11 ii), because ε > 0 can be arbitrarily small.

Decay of solutions.
We prove an asymptotic result for solutions considered in Theorem 2.6 Theorem 3.12. Let u be a solution of problem (1.1)-(1.2) with u 0 ∈ L 1 ϕσ(t) (R d ) satisfying the assumptions of Theorem 2.6. Then lim t→∞ u(t) q,ϕσ(t) = 0 for each 1 q ∞ holds.

Complements and comments
A sufficient condition for blowup of solutions of equation (1.1) with p > 1 + α [40] has been interpreted in [6] as for some m α,d,p > 0. Indeed, we have equivalence  and/or suitable pointwise estimates comparing the initial condition u 0 with the singular solution u ∞ . It is of interest to compare these constants c(α, d, p) and C(α, d, p) with the Morrey space norm | |u ∞ | | M d(p−1)/α of the singular stationary solution u ∞ > 0 of (1.1) in (2.1)-(2.2). Sufficient conditions for local-in-time existence of solutions are intimately connected with the problem of initial traces, i.e. a characterization of u 0 such that u(t) tends to u 0 weakly as t → 0 for a given nonnegative local solution of equation (1.1). Conditions on such u 0 's, roughly speaking, mean that local singularities are weaker than a multiple of |x| −α/(p−1) (or lim x→x 0 |x−x 0 | α/(p−1) u 0 (x) J(α, d, p) for a universal constant J(α, d, p) > 0) so that the size of u 0 in M d(p−1)/α loc (R d ) is universally bounded.
Remark 4.2 (initial traces). General results on the existence of initial traces (i.e. u 0 's) for arbitrary nonnegative weak solutions of equation (1.1) (u = u(t) defined on (0, T )) can be inferred from [2,3] using local moments like those in [12] with the weight functions (1 − |x|) If in both (i) and (ii) of Corollary 4.1 on dichotomy there were a single functional norm ℓ instead of those of M d(p−1)/α q (R d ) with some q > 1 and M d(p−1)/α (R d ) (both spaces are critical) this will be unique, up to equivalence (a personal communication of Philippe Souplet). Such a norm is called the dichotomy norm. Note that, however, problem (1.1)-(1.2) is not well posed in the critical space M d(p−1)/α (R d ), similarly to the case of radial solutions of the parabolic-elliptic Keller-Segel system studied in [30,12] with M d/2 (R d ) data as well as for the fractional Keller-Segel system with M d/α (R d ) data in [7]. Namely, there is no continuity of solutions of the Cauchy problem (1.1)-(1.2) with respect to the initial data in this norm, see the following Remark 4.3. In view of the result in [18], one can barely expect that thisl would be the norm in M d(p−1)/α (R d ) based solely on L 1 loc properties of functions.
Remark 4.3 (solutions may depend discontinuously on the initial data in M d(p−1)/α ). If the condition lim sup R→0 R α/(p−1)−d {|y−x|<R} |u 0 (y)| dy > K(α, d, p) is satisfied for some x ∈ R d and a constant K(α, d, p) C(α, d, p) > 0, then solutions are not continuous with respect to the initial data at u 0 ; in fact, the existence times of approximating solutions tend to 0 when the initial data are cut: 1 I {|y−x|>Rn} u 0 , R n → 0 as n → ∞. This can be inferred from the sufficient condition for blowup and the estimate of the existence time for solutions, see analogous arguments in [12,14].