Self-similar asymptotics of solutions to heat equation with inverse square potential

We show that the large-time behavior of solutions to the Cauchy problem for the linear heat equation with the inverse square potential is described by explicit self-similar solutions.


Introduction
In this paper, we study properties of weak solutions to the linear and singular initial value problem u(x, 0) = u 0 (x), (1.2) where n 3, the parameter λ ∈ R is given and assumptions on the initial condition u 0 are stated below. The initial value problem (1.1)-(1.2) was popularized by Baras and Goldstein [1] who discovered the "instantaneous blow up" of solutions, namely, the fact that Cauchy problem (1.1)-(1.2) has no positive local in time solutions if λ > (n−2) 2 4 (see also [7] for a simple proof via the Harnack inequality). Moreover, for 0 < λ (n−2) 2 4 , the authors of [1] found necessary and sufficient conditions for u 0 so that a nonnegative solution exists. Note that the number (n−2) 2 4 appears as the best constant in the Hardy inequality R n |∇u(x)| 2 dx (n − 2) 2 4

Main results
First, let us briefly review an asymptotic result for the initial value problem for the classical heat equation u(x, 0) = u 0 (x), (2.4) which was the main motivation for our paper. If we assume that u 0 ∈ L 1 (R n ), we obtain the following convergence where u = u(x, t) is the solution of problem (2.4) given as the convolution u(t) = G(t) * u 0 , the quantity M = R n u 0 (x) dx = R n u(x, t) dx is constant in time, and G(x, t) = 4π t −n /2 exp( −|x| 2 /4t) is the heat kernel. To prove the relation in (2.5), it suffices to use the explicit formula for solutions to problem (2.4). In the first step, one should assume that u 0 is a smooth, compactly supported function and use the Taylor expansion of the heat kernel G(x − y, t) (see, for example, [6, Theorem in Sect. 1.1.4]). Next, to complete the proof of (2.5) for any u 0 ∈ L 1 (R n ), it suffices to approximate this initial datum by a sequence of smooth, compactly supported functions and to use the well-known inequality G(t) * u 0 p u 0 p for every p ∈ [1, ∞]. More Vol. 13 (2013) Self-similar asymptotics of solutions 71 information about the asymptotic expansion of solutions to the heat equation can be found in the paper by Duoandikoetxea and Zuazua [4]. The goal of this note is to show an analogous result for initial value problem (1.1)-(1.2) and, due to the nonexistence result by Baras and Goldstein, we have to assume that λ < (n−2) 2 4 . By a technical obstacle, our method does not work in the critical case λ = (n−2) 2 4 , see Remark 2.4, below. The fundamental role in this paper is played by the parameter σ = σ (λ) defined by the formula which is a smaller root of a quadratic equation Moreover, the number σ = σ (λ) satisfies and σ (λ) < 0 for λ < 0. (2.8) To state our result, we recall a weight function, which is systematically used in the works [8][9][10]: Moreover, we connect with this weight time-dependent norms defined as follows: In other words, f q,ϕ σ (t) = ϕ where · q denotes the standard norm in the Lebesgue space L q (R n ). Note that for q = 2, the norm · 2,ϕ σ (t) agrees with the usual L 2 -norm on R n . Below, in Proposition 3.3, we show that for every λ ∈ − ∞, (n−2) 2 4 and each u 0 ∈ L 2 (R n ), problem (1.1)-(1.2) has a unique global-in-time weak solution in a standard energy space. Moreover, the operator −H = u + λ |x| 2 generates a semigroup of linear operators on L 2 (R n ) (see Proposition 3.4 below) and the solution agrees with a semigroup solution. In addition, due to the results from [8,9] (see also [10,14,15]), this solution has the integral representation where the kernel K satisfies estimates recalled in Theorem 4.1 below. This integral formulation allows us to study solutions of problem (1.1)-(1.2) with initial conditions Our main result on the large-time asymptotics of solutions to problem (1.1)-(1.2) is stated in the following theorem.
and the function ϕ is the self-similar solution of (1.1).
Observe that by (2.6) for λ = 0, we have σ = 0 and the relation (2.12) reduces to (2.5) since the value of the denominator of A is equal to (4π) n /2 . REMARK 2.2. Note that, analogously as the constant M = R n u(x, t) dx in the case of the heat Eq. (2.4), the following quantity A = R n |x| −σ u(x, t) dx is also constant in time for sufficiently regular solutions to the initial value problem (1.1)-(1.2). Indeed, if we multiply Eq. (1.1) by the function |x| −σ and integrate over R n , we get, for a sufficiently regular solution that decays sufficiently fast as |x| → ∞, the following equality

Now, integration by parts gives us
because σ is assumed to satisfy Eq. (2.7). However, in our reasoning below, we have never used the fact that the quantity A = R n |x| −σ u(x, t) dx is independent of time.
REMARK 2.3. The denominator of the constant A defined in (2.13) was chosen in such a way to have R n |x| −σ u 0 (x) − AV (x, 1) dx = 0 (see Theorem 5.1 below). Notice also that A is equal to the norm of the function V (x, 1) in the weighted space L 2 (R n , e |x| 2 /4 dx) (see next remark).  [13,Theorem 10.3], who proved that for λ = (n−2) 2 4 and for any u 0 ∈ L 2 (R n , e |x| 2 /4 dx) Cauchy problem (1.1)-(1.2) admits a solution u satisfying In the proof, they worked in similarity variables and used an improved version of the Hardy inequality (1.3). Here, our reasoning is different and allows us to deal with a large class of initial conditions due to the optimal estimates of the fundamental solution of Eq. (1.1). However, we are not able to handle the critical value of the parameter λ = (n−2) 2 4 . REMARK 2.5. It is worth pointing out that, by the assumption from Theorem 2.1, we have σ > 0, hence, we consider the initial datum u 0 which is not too singular at zero. On the other hand, for λ < 0, we have σ < 0, hence, the initial condition u 0 has to decay at infinity sufficiently fast.
REMARK 2.6. The decay rate in (2.12) is optimal in the following sense. Using the (1) .

Applying this property of the self-similar solution, we may write
In the next section, we discuss questions on the existence of solutions to initial value problem (1.1)-(1.2).

Existence of solutions
We begin our study of properties of solutions to (1.1)-(1.2) by deriving explicit solutions to Eq. (1.1). and
Proof of Proposition 3.1. One can check by a direct calculation that these functions satisfy Eq. (1.1) for all x ∈ R n \ {0} and t > 0. Here, however, we sketch a reasoning which allowed us to find these formulas. We look for a solution of Eq. (1.1) in the radial and self-similar form for some α > 0. Hence, it follows from direct calculations, that the function U = U (|x|) satisfies the equation Next, by a sequence of substitutions, we reduce Eq. (3.4) to the equation for some m ∈ R with the explicit solution e − r 2 8 . Indeed, first, we substitute the function Next, denoting W (r ) = r b g(r ) for some b ∈ R to be chosen later, we obtain Here, we assume that the parameter b satisfies the quadratic equation (cf. Eq. (2.7)) which, for λ < (n−2) 2 4 , has two roots Vol. 13 (2013) Self-similar asymptotics of solutions 75 Choosing these numbers as b in (3.6) and substituting α = n+b 2 , it is easy to see that Eq. (3.6) takes the form (3.5) with m = n + 2b. Therefore, recalling the explicit solution g(r ) = e − r 2 8 of (3.6) and inverting our substitutions, we obtain the solution (3.1) for b = b 1 and the solution (3.2) for b = b 2 . Now, we recall a classical result on the existence of weak solutions to problem Proof. If λ 0, solutions of (1.1)-(1.2) satisfy the following a priori estimate , the Hardy inequality (1.3) implies the following energy inequality Thus, by the Galerkin method and a priori estimates (3.8) and (3.9), we obtain immediately following, for example, [5, Sect. 7.1.2] the existence and the uniqueness of a weak solution to problem (1.1)-(1.2).
However, we can also study solutions of problem (1.1)-(1.2) via the semigroup theory. In this approach, we consider a sesquilinear form Assumptions (1) and (2) from the above proof imply, in fact, that {e −t H } t 0 is an analytic semigroup of linear operators on L 2 (R n ). However, in the following, we do not need this additional property of {e −t H } t 0 . REMARK 3.5. It follows from Proposition 3.4 that the weak solution from Proposition 3.3 can be written in the form u(x, t) = e −t H u 0 (x).

Properties of fundamental solution
We recall two-sided estimates of the fundamental solution of Eq. (1.1), which was found independently by Liskevich and Sobol [8, Remarks at the end of Sec. 1] and Milman and Semenov [9, Theorem 1] (see also [10,14,15]).
Moreover, there exist positive constants C 1 , C 2 > 0 and c 1 , c 2 > 1, such that for all t > 0 and all x, y ∈ R n \ {0}
Vol. 13 (2013) Self-similar asymptotics of solutions 77 REMARK 4.2. In fact, the authors of [8,9] used more regular weight functions t with a suitable constant C and such that σ is sufficiently regular for √ t |x| 2 √ t. It can be checked directly that there exist positive constants C 1 and C 2 for which the inequalities hold true, where ϕ σ is defined by (2.9). By this reason, we are allowed to use the weights ϕ σ instead of σ in estimates (4.2).
Estimates (4.2) of the fundamental solution allow us to consider a class of initial conditions, which is different than L 2 (R n ). In particular, by direct calculations involving the estimates of the kernel K from Theorem 4.1 and the definition of the weight functions ϕ σ , we obtain Next, applying the Young inequality for the convolution, the scaling property of the heat kernel G = G(x, t), and the fact that 2σ < n, we get the following inequality that is valid for every q ∈ [1, ∞], all t > 0, each measurable function f such that f 1,ϕ σ (t) is finite and a constant C = C(q, 1) independent of t and f . Let us emphasize, that this kind of inequalities have been systematically used in [9] to derive the kernel estimates (4.2).
Next, it is worth pointing out that the operator e −t H is symmetric, thus, its kernel K (x, y, t) is also symmetric with respect to x and y. Moreover, the function K (x, y, t) in representation (4.1) is unique. Now, let us discuss its continuity.  K (x, y, t) The Moser iteration technique applied to Eq. (4.5) allowed the authors of [10] to show the Hölder continuity of solutions to Eq. (4.5) (see [10,Theorem 3.8] for more details). This implies thatS(x, y, t) is a continuous function with respect to x for each y ∈ R n and by symmetry, a continuous function with respect to y for each x ∈ R n . The same reasoning can also be directly applied in the case λ 0 following [10] (see also [2]). Now, we show the continuity ofS =S(x, y, t) with respect to both variables (x, y) and, for simplicity of the exposition, we consider (x, y) = (0, 0). The proof for (x, y) = (0, 0) is completely analogous. First, let us notice that the functionS satisfies the usual Chapman-Kolmogorov equality, because it is a fundamental solution to Eq. (4.5) with φ(x) = |x| −σ . We fix ε > 0 and take x, y ∈ R n such that |x| < δ and |y| < δ for some δ > 0 and we apply the Chapman-Kolmogorov equality, we get Since the functionsS(x, z, t /2) andS(z, 0, t /2) have the Gaussian estimates (cf. Theorem 4.1), using the elementary inequality |z − x| 2 |z| 2 /2 − δ 2 , we obtain Now, the continuity of the functionS(z, y, t /2) with respect to y for every z ∈ R n and S(x, z, t /2) with respect to x for every z ∈ R n combining with the Lebesgue dominated convergence theorem, allows us to find δ such that This completes the proof of Lemma 4.3.
THEOREM 4.4. The kernel of the operator e −t H provided by Theorem 4.2 has the self-similar form, namely, Vol. 13 (2013) Self-similar asymptotics of solutions 79 for all t > 0 and x, y ∈ R n \ {0}.
Proof. Let us note that if the function u = u(x, t) is the solution of the problem (1.1)-(1.2), then the function u α = u α (x, t) = u(αx, α 2 t) is also a solution of (1.1) for any α > 0 with the initial datum u α 0 (x) = u 0 (αx). However, by Theorem 4.1, every semigroup solution of problem (1.1)-(1.2) has the form Substituting αy =ȳ and recalling the definition of u α , we get Introducing the new notationx = αx andt = α 2 t, we obtain for allx ∈ R n \ {0} and t > 0. From the uniqueness of the kernel K (x, y, t) and its continuity for x = 0 and y = (by Lemma 4.3), we infer the equality for all x, y ∈ R n \ {0}, t > 0 and α > 0. Substituting α = √ t, we complete the proof of (4.6).
Using the self-similar form of K (x, y, t) and its continuity stated in Lemma 4.3, we prove two technical lemmas, which will be needed to obtain our main result. LEMMA 4.5. For every x, y ∈ R n and t > 0, define where K (x, y, t) is the fundamental solution from Theorem 4.1 and ϕ σ (x, t) is the weight function stated in (2.9). Then, for every ε > 0 there exists δ > 0 such that for all t > 0. Proof. By Lemma 4.3 and the explicit form of the function ϕ σ , we immediately obtain that S ∞ (x, y, t) can be extended to a continuous function (also denoted by S) for all x, y ∈ R n and t > 0.
Let ε > 0. Using the self-similar form of kernel K (x, y, t) and of the weight which is possible due to the Gaussian estimates of the function S ∞ (x, y, 1), cf. the definition of S ∞ (x, y, t) and inequalities (4.2).
Next, for fixed R > 1, the uniform continuity of function S ∞ (x, y, 1) for |x| R and |y| δ (cf. Lemma 4.3) allows us to find δ such that This completes the proof of inequality (4.7). LEMMA 4.6. Let us define where S ∞ (x, y, t) is the function from Lemma 4.5 and ϕ σ (x, t) is the weight stated in (2.9). Then for every ε > 0 there exists δ > 0 such that for all t > 0. Vol. 13 (2013) Self-similar asymptotics of solutions 81 Proof. We fix ε > 0. In view of the self-similar form of the kernel K (x, y, t), the function S 1 (x, y, t) has the same scaling property. Hence, we obtain First, we can choose R > 1 so large that since, by Theorem 4.1, we have the Gaussian estimates of the function S ∞ (z, w, 1) and the function ϕ σ (z, 1) is equal to 1 for |z| R > 1.
We conclude this section by an estimate involving the weighted L 1 -norm. 2 4 , then ii) if λ < 0, then for all t > 0.

Self-similar asymptotics-proof of Theorem 2.1
Our main result stated in Theorem 2.1 on the large-time asymptotics of solutions to (1.1)-(1.2) will be a direct consequence of the following property of the semigroup e −t H of linear operators constructed in Proposition 3.4.
where for q = ∞ the expression 1 q is understood as 0. Proof. For simplicity of the exposition, we divide this proof into a series of steps. First, we consider a compactly supported function ψ such that | · | −σ ψ ∈ L 1 (R n ) and Step 1. Convergence in weighted L ∞ .
If y ∈ supp ψ, we assume that ϕ σ (y, t) = √ t |y| σ for sufficiently large t > 0. Hence, using the definition of weight function ϕ σ and norm · ∞,ϕ σ (t) for large t > 0, we have Applying (5.3), we get Now, we fix ε > 0 and we observe that for sufficiently large t, since the function ψ has a compact support, we may write Step 2. Convergence in weighted L 1 . Now, let q = 1. Using the definitions of · 1,ϕ σ (t) and the function S 1 from Lemma 4.6, we arrive at J (t) ≡ t − σ 2 e −t H ψ 1,ϕ σ (t) = R n R n S 1 (x, y, t)ψ(y)|y| −σ dy dx.
We fix δ > 0 and we notice that for sufficiently large t, since the function ψ has a compact support. We deal with the term on the right-hand side in the most direct way R n |y| −σ |ψ(y)| dy.
Step 3. Convergence in weighted L q . To show the L q -estimates for 1 < q < ∞, we use the definition of the norm · q,ϕ σ (t) as follows: Factors on the right-hand side converges to zero as t → ∞ by Step 1 and Step 2 of this proof.
Step 4. General initial datum. Let us complete the proof of (5.2) for every w 0 satisfying the assumptions of Theorem 5.1. Notice that for every R > 0 there exists a constant c R ∈ R such that the compactly supported function ψ(x) =