Decay/growth rates for inhomogeneous heat equations with memory. The case of large dimensions

We study the decay/growth rates in all $L^p$ norms of solutions to an inhomogeneous nonlocal heat equation in $\mathbb{R}^N$ involving a Caputo $\alpha$-time derivative and a power $\beta$ of the Laplacian when the dimension is large, $N>4\beta$. Rates depend strongly on the space-time scale and on the time behavior of the spatial $L^1$ norm of the forcing term.


Introduction and main results
1.1. Goal. This paper is part of a project intending to give a precise description (decay/growth rates and profiles) of the large-time behavior of solutions to the Cauchy problem where u 0 and f (·, t) belong to L 1 (R N ). Here ∂ α t , α ∈ (0, 1), denotes the Caputo α-derivative, introduced in [2], defined for smooth functions by and (−∆) β , β ∈ (0, 1], is the usual β power of the Laplacian, defined for smooth functions by (−∆) s = F −1 (| · | 2s F), where F stands for Fourier transform; see for instance [16].
Fully nonlocal heat equations like (1.1), nonlocal both in space and time, are useful to model situations with long-range interactions and memory effects, and have been proposed for example to describe plasma transport [8,9]; see also [3,4,14,17] for further models that use such equations.
When the forcing term f is trivial, a complete description of the large-time behavior of (1.1) was recently given in [5,6]; see also [13]. Hence, since the problem is linear, it only remains to study the case with trivial initial datum, namely This task is by far more involved, and this paper represents a first step towards its completion. It is devoted to the obtention of (sharp) decay/growth rates of solutions to (1.2) when the forcing term satisfies (1.3) f (·, t) L 1 (R N ) ≤ C (1 + t) γ for some γ ∈ R and the spatial dimension is large, N > 4β. This already involves critical phenomena depending on the values of p and γ. If 1 ≤ N ≤ 4β, additional critical phenomena associated to the dimension appear, that make the analysis somewhat different. This case is considered in [7]. Notice that we are allowing γ to take negative values, so that f (·, t) L 1 (R N ) may grow with time.
If f (·, t) ∈ L 1 (R N ) for all t ≥ 0 and |Ff (ξ, t)| ≤ C|g(ξ)| for some function g such that (1 + | · | β )g(·) ∈ L 1 (R N ), then problem (1.2) has a unique bounded classical solution given by Duhamel's type formula where Z is the solution to (1. 2) with f ≡ 0 having a Dirac mass as initial datum; see [11,13]. If we only assume f (·, t) ∈ L 1 (R N ), the function u in (1.4) is still well defined, but it is not in general a classical solution to (1.2). Nevertheless, it is a solution in a generalized sense [12,13]. In this paper we will always deal with solutions of this kind, given by (1.4), which are denoted in the literature as mild solutions [13,15].
Notation. As is common in asymptotic analysis, g h will mean that there are constants ν, µ > 0 such that νh ≤ g ≤ µh.
1.2. The kernel Y . Critical exponents. Since the mild solution is given by the convolution in space and time of the forcing term f with the kernel Y , having good estimates for the latter will be essential for the analysis. Such estimates were obtained in [13], and are recalled next.
The kernel Y has a self-similar form, Its profile G is positive, radially symmetric and smooth outside the origin, and if N > 4β satisfies the sharp estimates In particular, we have the global bound and, since |ξ| Since the mild solution is given by a convolution of f with Y both in space and time, the threshold value that will mark the border between subcritical and supercritical behaviors will be p c , and not p * . In particular, condition (1.3) guarantees that u(·, t) ∈ L p (R N ) for p ∈ [1, p c ), but not for p ≥ p c . Hence, in order to deal with supercritical exponents p ≥ p c we need some extra assumption on the spatial behavior of the forcing term. In the present paper we will use two different such extra hypotheses, the poinwtise condition (1.12) |f (x, t)| ≤ C|x| −N (1 + t) −γ for |x| large, and the integral condition We do not claim that these conditions are optimal; but they are not too restrictive, and are easy enough to keep the proofs simple.
1.3. Precedents and statement of results. The only precedent is given in [13], where the authors study the problem in the integrable in time case γ > 1 and prove, for In particular, using (1.11) we get the sharp estimate This result is also valid for the local case, α = 1; see for instance [1,10] for the case p = 1. In this special situation Y = Z is the well-known fundamental solution of the heat equation, whose profile does not have a spatial singularity and belongs to all L p spaces.
An analogous convergence result is definitely not possible for α ∈ (0, 1) if p ≥ p * , since Y (·, t) ∈ L p (R N ) in that case, or if γ ≤ 1. Moreover, even in the subcritical range (1.14) only gives a sharp rate and a nontrivial limit profile in the diffusive scale |x| t α/(2β) ; see below. Hence we need a different approach.
As we will see, it turns out that, in contrast with the local case, and due to the effect of memory, the decay/growth rates are not the same in different space-time scales. Moreover, the scale that determines the dominant rate depends on the value of the exponent p. Our strategy will consist in tackling this difficulty directly by studying separately the rates in exterior regions, |x| ≥ νt α/2β with ν > 0, compact sets or intermediate regions |x| g(t) with g(t) → ∞ and g(t) = o(t α/2β ). We already found these phenomenon for the Cauchy problem, [5,6]. Our first result concerns exterior regions.
These estimates are sharp.
For p ∈ [1, p c ) and γ > 1 the result follows from (1.14), showing that the behavior in this regions dominates the global behavior in the subcritical case.
We now turn to the behavior in compact sets which, due to the effect of memory, will dominate the global behavior for large values of p.
Let u be the mild solution to (1.2). For every compact set K there exists a constant C such that These estimates are sharp.
Remark. Note that q c (p c ) = 1.
As expected, the rates in intermediate regions, between compact sets and exterior regions, are intermediate between the ones in such scales.
These estimates are sharp.
We also obtain results that connect the behaviors in compact sets and exterior regions, thus getting the (global) decay rate in L p (R N ).  Figure 1. Global decay/growth rates. The dotted line indicates the borderline separating decay from growth.
Notice that the borderline separating decay and growth is γ = 0 only for p ≥ p c . For p ∈ [1, p c ) the frontier is given by see the dotted line in Figure 1. For p = 1 this corresponds to γ = α. An informal explanation for this fact can be found in formula (1.4). We are integrating in time, but Y (x, t) = ∂ 1−α t Z(x, t). Hence, it is like if we were integrating α times in time. As for the behavior for the borderline γ, in general there is neither growth nor decay. The exception is the case p = p c , γ = 0, in which there is a slow logarithmic growth.
Another remarkable fact is that the rates depend on γ not only in the non-integrable case γ ≤ 1, which might have been expected, but also in part of the region γ ∈ [1, 1 + α) if p is supercritical.

Exterior region
In this section we prove Theorem 1.1, which gives the behavior of all L p norms of the mild solution u to (1.2) in exterior regions, {(x, t) ∈ Q : |x| ≥ νt α/(2β) }, ν > 0.

Compact regions
In this section we prove Theorem 1.2, which gives the large-time behavior of the L p norms of the mild solution to (1.3) in compact sets K.
Proof of Theorem 1.2. Let t ≥ 1. We have |u| ≤ I + II, where Using the global bound (1.9) for Y we get |f (y, s)| dyds.
Let q = 1 if p ∈ [1, p c ), q > q c (p) as in (1.13) if p ≥ p c . Let r satisfy 1 + 1 p = 1 q + 1 r . Then r ∈ [1, p c ), and in particular r ∈ [1, p * ). Thus, for all t ≥ 2 we have In order to bound II we take r ∈ [1, p c ) as before. Then, using (1.11) we get which combined with the estimate for I yields the result.
In order to prove that estimate (1.16) is sharp we consider f (x, t) = (1 + t) −γ χ K+B 1 (x), where K is any compact set with measure different from 0. We have If x ∈ K and |x − y| < 1, then y ∈ K + B 1 . Notice that |x − y| < 1 and s < t − 1 imply that |x − y|(t − s) −α/(2β) ≤ 1. Therefore, using the self-similar form (1.5) of Y and the bound from below (1.6) for the profile G, for all x ∈ K we have for some constant C > 0. Thus, no matter the value of γ, for all x ∈ K and t large enough,

Intermediate scales
In this section we study the large-time behavior of the L p norms of the mild solution to (1.3) in regions where |x| g(t) with g(t) → ∞ such that g(t) = o(t α/(2β) ), which is the content of Theorem 1.3.

Estimates in R N
In this section we establish the behavior of the global L p (R N ) norms of the mild solution to (1.2), Theorem 1.4.
Proof of Theorem 1.4. Due to the results of theorems 1.1 and 1.2, it is enough to show that the estimates are true in some region of the form {R ≤ |x| ≤ δt α/(2β) } with R, δ > 0.