A Remark on a Nonlocal-in-Time Heat Equation

Schauder's fixed point theorem is used to derive the existence of solutions to a semilinear heat equation. The equation features a nonlinear term that depends on the time-integral of the unknown on the whole, a priori given, interval of existence.


INTRODUCTION
This note is dedicated to the nonlocal problem on a bounded smooth domain Ω ⊂ R n with d ∈ C 1 (Ω) such that d(x) ≥ d 0 > 0 for x ∈ Ω. The potential ϕ, the weight a, the initial datum u 0 , and the right-hand side f are suitable given functions. Equation (1.1) is used in the modeling of a biological nanosensor in the chaotic dynamics of a polymer chain in an aqueous solution and has been introduced and considered in [4][5][6][7]. It can also be seen as a toy model for equilibrium states in age-structured diffusive populations (with t referring to the age of individuals), see [8,9] for instance.
Note that the unknown weighted time-integral u = ∞ 0 a(s) u(s) ds depends on the whole, a priori given interval of existence (0, ∞) (the case of a bounded time interval (0, T ) is included in (1.1), of course, by taking a with bounded support). Hence, only global solutions are of interest. Moreover, (1.1) is no usual evolution problem satisfying a Volterra property since solutions at a time instant depend also on future time instants. For the homogeneous version of (1.1) on a bounded time interval (0, T ) with vanishing right-hand side f and without weight a, existence of weak solutions was derived in [5,6] and strong solutions in [10]. The non-homogeneous problem (1.1) on a bounded interval (0, T ) was investigated in [7] where, for unbounded potentials ϕ, a truncation approach and weak compactness methods were used to prove the existence of weak solutions under fairly general conditions. The purpose of this work is to propose an alternative approach to (1.1) for deriving the existence of mild and strong solutions under slightly different conditions. This approach has been used in [10] and may also be a template for other nonlocal problems. More precisely, we shall use the fact that solutions to (1.1) may be written as mild solutions in the form where (e tA(ū) ) t≥0 is the contraction semigroup on L p (Ω) generated by the operator subject to Dirichlet boundary conditions (see below for details). Integrating the representation (1.2) yields the equivalent fixed point equation forū. We then shall focus on this fixed point equation and prove, in particular, that the right-hand side of (1.3) enjoys suitable compactness properties with respect toū that allow us to apply Schauder's theorem leading to the following existence result: for some p ∈ (max{1, n/2}, ∞) and let u 0 ∈ L ∞ (Ω). Then there is a mild solution u ∈ C R + , L p (Ω) whereṘ + := (0, ∞).
In Section 2 we prove Theorem 1.1. The crucial compactness properties of the integral terms appearing on the right-hand side of (1.3) are postponed to Section 3. The proofs there are inspired by the works [2,3] and may be extended to more general frameworks than the one considered herein for (1.1), e.g. to other semilinear and possibly quasilinear equations (see Remark 3.3 in this regard). Also some of the assumptions in Theorem 1.1 may be weaken, e.g. for linearly bounded ϕ or smoother a.

PROOF OF THEOREM 1.1
Notation and Preliminaries. We use the notation Since ϕ is uniformly continuous and bounded on bounded sets, it follows that (considered as Nemytskii the closed ball in L ∞ (Ω) of radius R 0 centered at the origin. Recall that p ∈ (max{n/2, 1}, ∞) and note that, given anyū ∈ X, the mapping ϕ(ū) : Setting A(ū)w := div d∇w − ϕ(ū)w , w ∈ W 2 p,D (Ω) , it then follows from standard perturbation results that In fact, since ϕ is nonnegative, (e tA(ū) ) t≥0 is a positive contraction semigroup on each L q (Ω) for q ∈ (1, ∞] (which, however, is not strongly continuous for q = ∞), hence Moreover, we have s(A(ū)) ≤ s 0 < 0 for its spectral bound with s 0 denoting the spectral bound of the operator w → div d∇w . It then follows from [1, II.Lemma 5.1.3] that there is ν > 0 and, given 2θ In the following we fix 2θ ∈ (n/p, 2) and note the compact embedding Let us also observe that, given t > 0 andū,v ∈ X, we have We then use (2.3) and (2.4) to get We are now in a position to provide the proof of Theorem 1.1.

COMPACTNESS PROPERTIES
We provide the compactness results used in the proof of Theorem 1.1. This section relies on the papers [2,3] and adapts these ideas to our setting.
We first consider the non-homogeneous part.
Proof. Given T > 0 introduce and It suffices to prove that F ∈ C (X, X T ) is compact for every T > 0 since the assertion then follows by a diagonal sequence argument and the assumption a ∈ L 1 (R + , L ∞ (Ω)).
(iii) Next, we claim that Let δ ∈ (0, T ). Using (2.2) we have, for 0 ≤ t ≤ δ, On the other hand, for δ ≤ t ≤ T , we use (2.2) to get For the first term on the right-hand side we use (2.2) to estimate while we use (3.4) and (3.2) for the second term to obtain Gathering these estimates we derive, for δ ≤ t ≤ T , Since f ∈ L 1 (R + , L ∞ (Ω)) we may first choose δ > 0 small enough and then let λ tend to zero to conclude from (3.7) and (3.8) that (3.6) indeed holds true.
We prove a compactness result for the part involving the initial condition: Then the set {G(ū) ;ū ∈ X} is precompact in L ∞ (Ω).