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 in the pointwise sense 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.


Introduction and main results
Nonlinear evolution problems involving fractional Laplacian describing the anomalous diffusion (or the α-stable Lévy diffusion) have been extensively studied in the mathematical and physical literature, see [43] for the Cauchy problem (1.1)-(1.2), and [9][10][11][12] for other examples of problems and for extensive list of references. One of these models is the following initial value problem for the reaction-diffusion equation with the anomalous diffusion

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 (1.5) 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 [41]. 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 [41]. We refer the readers to [5,15,33] 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, ∞).

Existence of global-in-time solutions below the singular solution 2.1. Existence of the singular solution
We have the following Proposition 2.1. For p > 1 + α d−α , there is a unique radial homogeneous nonnegative solution of the equation with the constant (

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 ∞ in the pointwise sense.
We will use the technique of Picard iterations of constructing mild solutions, i.e. those satisfying the Duhamel formula A systematic approach to the existence of mild solutions via iterations of nonlinear mappings, as those on the right-hand side of formula (2.3), for various evolution equations (motivated mainly by hydrodynamics and chemotaxis theory) is presented in full detail in, e.g., [32,33]. Similarly as the former result, it can be proved directly for α ∈ (0, 2) (while the proof in [41] 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 [29] and [44,Th. 3.8]. They extend immediately to the estimate for 1 < p 1 When either p 1 = 1 or p 2 = ∞, the norms 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  [40] and references therein). The exponent 1 + α d is a counterpart of the Fujita exponent in the case α ∈ (0, 2), see [43].
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 The first estimate is in the M s (R d ) norm These bounds (2.6)-(2.8) 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 [13,Th. 4.1, Th. 5.1] for analogous albeit more complicated constructions for radially symmetric solutions of chemotaxis systems. (2.12) and the initial data satisfying (2.14) Once the comparison principle in Proposition 2.7 is proved, the localin-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 . By Proposition 2.5, the solution u is locally in time in L ∞ (R d ), hence smooth by standard arguments.
Proof of Theorem 2.6. Approximating u 0 in Theorem 2.6 by initial data u 0k (x) = min{u 0 (x), (1 − 1 /k)u ∞ (x)} satisfying (2.13) with δ = 1 − 1 /k, k = 2, 3, 4, . . . , 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 [28]. 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 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.
where two parts of the graph of the barrier b meet. Let us consider the auxiliary functioñ With this choiceũ hits a constant part of the graph of the modified barrier |x| γ b(x). Thus,ũ(x, t 0 ) attains its local maximum at x 0 . Indeed, the existence of an x 1 = x 0 such thatũ(x 1 , t 0 ) >ũ(x 0 , t 0 ) would contradict the choice of t 0 as the first hitting point of the barrier. Clearly, ∇ũ(x 0 , t 0 ) = 0 and ∂ ∂tũ (x 0 , t 0 ) 0 hold. We will show that nevertheless ∂ ∂tũ (x 0 , t) t=t0 < 0, a contradiction. Now, let us compute according to formula (1.3) for the fractional Lapla- The passage from (2.17) gives for γ = α p−1 (so that α + γ < d by assumption p > 1 + α d−α ) We use 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 .
In the second case when γ < α p−1 but close to α by the definition of the barrier (2.13). Since the coefficient in formula (2.20) is continuous when γ α p−1 , so (p − 1)γ α, the right-hand side of inequality (2.19) is strictly negative for γ close enough to α p−1 . This happens for γ ∈ (γ 0 , α p−1 ) with γ 0 = γ 0 (δ, α p−1 ) independent of other parameters.  better decay rate under stronger assumptions on the regularity of the initial data even for somewhat bigger size of data than in the proof of Theorem 2.6.

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 [37] for classical nonlinear heat equation. 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 or, equivalently, First, we concentrate on existence and properties of solutions to the linear initial value problem 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 . The semigroup e −tH of the linear operators generated by H can be written as the integral operator with a kernel denoted by e −tH (x, y), namely Moreover, there exists a positive constant C such that for all t > 0 and all 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 d−1 which plays a crucial role in the study of stability of solutions to the classical nonlinear heat equation (see [37] 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 (2π) α cα is fulfilled. Observe that the assumption is satisfied for certain d (large) and p (close to d d−α > 1). This follows from the asymptotics of the expression s(α, d, p) The following theorem is the consequence of the estimates stated in formulas (3.10).
holds for a constant C = C(b) > 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) Here, the constant C(b) depends on b. We use the same argument to deal with Finally, we estimate  (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, 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  (3.11), and σ ∈ (0, d−α) fulfill equation (3.9). Suppose that there exists a constant b > 0 such that and, moreover, Then the relations

Remark 3.9.
A similar result for the classical case, namely for α = 2, can be found in [23][24][25][26], 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 for a constant C > 0 and all t 1, and 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 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 for some constant C 0. Since σ > α (p−1) , 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 2 -norm 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