On the evolution equation with a dynamic Hardy-type potential

Motivated by the celebrated paper of Baras and Goldstein (Trans Am Math Soc 284:121–139, 1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, nonexistence and uniqueness of solutions and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.


Introduction and main results
We consider positive solutions of an evolution equation of the form u t (x, t) = u(x, t) + V (x, t)u(x, t), x ∈ R N \{ξ(t)}, t > 0, u(x, 0) = u 0 (x), x ∈ R N \{ξ(0)}, (1.1) where N ≥ 3 and the singular point ξ : [0, ∞) → R N is a given Hölder-continuous function. We assume that the potential V (x, t) is continuous in x ∈ R N \{ξ(t)} and t ∈ [0, ∞) and is unbounded as x → ξ(t). The initial value u 0 (x) is continuous, nonnegative and nontrivial for x = ξ(0) and is bounded for |x − ξ(0)| > 1. By a solution of (1.1), we mean a function u = u(x, t) that satisfies (1.1) in the classical sense and is bounded for |x −ξ(t)| > 1. The aim of this paper is to study the existence, nonexistence and uniqueness of solutions, and a lower and upper bound near the singular point. exists and is bounded for x ∈ R N \{ξ(t)}. In this case, the standard parabolic regularity implies that the limiting function u(x, t) is a solution of (1.1). A solution defined in this way is called a minimal solution of (1.1). We note that this does not exclude the possibility that nonminimal solutions exist.
For the existence of a minimal solution of (1.1), Baras-Goldstein [2] (see also ) studied the case where the potential function is given by It was shown in [2] that the Hardy constant is critical in the following sense: (i) If 0 < λ ≤ λ c and |x| −α 1 u 0 ∈ L 1 loc (R N ), then (1.1) has a time-global minimal solution u. Moreover, for every ε > 0 and T > 0, there exists a constant c > 0 such that (ii) If λ > λ c , then (1.1) is not well-posed, that is, there exist no positive solutions.
We note that if λ < λ c , the quadratic equation has two positive roots given by If λ = λ c , then the equation has a double root α 1 (λ c ) = α 2 (λ c ) = √ λ c . The paper [2] has attracted much attention, and many results have been obtained later for equations with singular potentials [4,9,10,12,15,16,23,24]. However, no results have been obtained when the position of the singularity depends on time. In Vol. 21 (2021) Evolution equation with a dynamic Hardy-type potential 2143 this paper, we consider the case where both the position and strength of the singularity depend on time. See [7,14,[20][21][22] for related works on such dynamic singularities. For r, R ∈ R with 0 ≤ r < R ≤ ∞, an interval I ⊂ R and x 0 ∈ R N , we introduce the following notation: Throughout this paper, we assume that for T > 0 fixed, the following conditions hold for ξ , V and u 0 : (A3) u 0 is nonnegative, nontrivial and continuous on R N \{ξ(0)} and is bounded on Note that V (x, t) may not be differentiable. In fact, we shall consider potentials such as V ( First, we give a sufficient condition for the existence of a solution and its upper bound. with some λ ∈ (0, λ c ) and R > 0. If with some k ∈ (0, α 2 (λ) + 2) and C 1 > 0, then (1.1) has a solution satisfying where ε > 0 and τ ∈ (0, T ) are arbitrary and This result with ε = 0 was proved by Baras-Goldstein [2, Theorem 2.2 (i)] when the potential V does not depend on t.
Next, we give a lower bound of solutions.
We remark that when ξ(t) does not move, the lower bound was obtained in [2, Theorem 2.1 (i)] by using the method of [17,18] (see also [5] for a related result). In this paper, we shall give a much simpler proof.
Next, we show that any two solutions of (1.1) coincide with each other if their difference is small in some sense.
with some λ ∈ (0, λ c ) and R > 0. If u 1 and u 2 are solutions of (1.1) with the same initial value such that In particular, this theorem implies that the solution given in Theorem 1.1 is unique, and hence, it must be a minimal solution. We remark that we consider solutions which are bounded outside a neighborhood of the singular point. In the context of uniqueness, we can relax this assumption for an exponential growth (see Theorem 5.1).
The following two theorems show that the conditions in Theorem 1.1 are sharp for the existence of a solution.
We remark that Theorems 1.4 and 1.5 are proved in [2] when ξ(t) does not move. In this paper, we prove these results in a totally different manner.
Proofs of the above theorems are almost self-contained and are based on new ideas using some particular solutions of the radial heat equation, precise estimate of the integral representation formula of solutions and comparison principle. Finally we remark that the method used in this paper is applicable only when γ > 1/2 in (A1). In fact, the above theorems no longer hold if γ < 1/2 (see [19]). Vol. 21 (2021) Evolution equation with a dynamic Hardy-type potential 2145 This paper is organized as follows: In Sect. 2, we give some preliminary lemmas concerning the radial heat equation. In Sect. 3, we study the existence of a minimal solution. In Sect. 4, we give a lower bound of solutions. In Sect. 5, we prove the uniqueness of a solution, namely if the difference of two solutions are small, then they coincide with each other. In Sect. 6, we show the nonexistence for large initial value. In Sect. 7, we discuss the supercritical case.

Radial heat equation
We start from a simple equation If the initial value is radially symmetric, then this equation is reduced to to obtain the radial heat equation where d = N − 2α 1 (λ) > 2 corresponds to the spatial dimension. See [1,3,11,13] for the analysis of the radial heat equation. We first consider (2.2) with a nonnegative and nontrivial initial value w 0 (r ), Lemma 2.1. If d > 2, then any nonnegative and nontrivial solution w of (2.2) satisfies w(0, t) > 0 for t > 0.
Next, let us consider forward self-similar solutions of (2.2) of the form where l > 0 is a constant and σ (l) is given in Lemma 2.2(iii). Substituting this in (2.2), we see that ϕ must satisfy For d > 2 and l > 0, there exists a unique solution of (2.4) subject to the initial condition ϕ(0) = 1 and ϕ ρ (0) = 0, and the solution has the following properties: Proof. The existence and uniqueness of a solution of (2.4) with ϕ(0) = 1 and ϕ ρ (0) = 0 can be proved in the same manner as [6, Lemma 3.1].
On the other hand, using g(ρ) ≤ ρ −l , multiplying (2.5) by ρ l−d , integrating it on (0,ρ) forρ > θ and applying (i), we obtain converges to a positive constant as ρ → ∞. (iv) Assume that d − δ < l < d for some small δ > 0. Then we can take θ and γ in (2.6) to be independent of l. For any ε > 0, we take ρ 0 > θ such that Then by (ii) and the continuity of ϕ with respect to l, there exists δ > 0 such that is decreasing, we obtain σ (l) ≤ 2ε. Since ε > 0 is arbitrary, the proof is complete.
. Moreover, the function v has the following properties:

Existence in the subcritical case
In this section, we prove Theorem 1.1 by using a uniform upper estimate of the solution u n for the cut-off problem. We use solutions of a simple equation Proof of Theorem 1.1. For n ≥ 1, set V n := min{V, n}. Let u n be a unique solution of the approximate problem (1.2). Note that u n satisfies For a while, we fix n, and setũ(x, t) Let ε > 0 be arbitrarily given. For k < α 2 (λ) + 2, we take constants δ = δ(ε) and 0 < T δ ≤ T independent of n such that where C ξ is a constant given by (A1). We claim that there exist w n and w + on t ∈ [0, T δ ] such that w n = [w n ] and w n ≤ w + for t ∈ [0, T δ ]. Once we prove this claim, by the uniqueness of solutions of (1.2), we obtain where Q (0,∞),[0,T δ ] is given in (1.3). We prove the claim by using solutions of a simple problem for which we denote a solution by v = v(x, t; k). Note that the existence of a solution v follows from k < α 2 (c 0 λ) + 2 and Lemma 2.3. By (A2), (A3) and the assumption on V , there exists a constant M > 0 independent of n such that We will see that Direct computations show that w + is a solution of the following problem: which is equivalent to (3.6) These computations together with (3.2) and (3.4) yield for t ∈ (0, T δ ]. This proves (3.5).

Lower bound of solutions
In this section, assuming the existence of a minimal solution of (1.1), we shall derive a lower bound of solutions by using a solution of a simple equation v t = v+λ |x| −2 v (0 < λ < λ c ).

Uniqueness of solutions
We prove the uniqueness of solutions. The proof is based on nontrivial modifications of the method of Marchi [16, pp. 1075-1079]. In that paper, the position of the singularity does not move in time, while in our paper, the singularity moves in time and its motion may not be smooth.
Note that, in the definition of solutions of (1.1), we assume the boundedness of solutions for |x − ξ(t)| ≥ 1. However, for uniqueness, this can be relaxed as follows, and so we prove the following result which is stronger than Theorem 1.3.
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