Delayed blow-up and enhanced diffusion by transport noise for systems of reaction–diffusion equations

This paper is concerned with the problem of regularization by noise of systems of reaction–diffusion equations with mass control. It is known that strong solutions to such systems of PDEs may blow-up in finite time. Moreover, for many systems of practical interest, establishing whether the blow-up occurs or not is an open question. Here we prove that a suitable multiplicative noise of transport type has a regularizing effect. More precisely, for both a sufficiently noise intensity and a high spectrum, the blow-up of strong solutions is delayed up to an arbitrary large time. Global existence is shown for the case of exponentially decreasing mass. The proofs combine and extend recent developments in regularization by noise and in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p(L^q)$$\end{document}Lp(Lq)-approach to stochastic PDEs, highlighting new connections between the two areas.


Introduction
Reaction-diffusion equations arise in many branches of applied science such as biology and chemistry (see e.g. [Rot84,Pie10] and the references therein). A major challenge in the study of such equations is the presence of commonly superlinear source terms. Even in presence of dissipation of mass, which is sufficient to show global existence in the ODE case, blow-up in finite time of strong solutions may occur, see [PS97] or [Pie10,Theorems 4.1 and 4.2]. In addition, for many problems of practical interests, such as reversible chemical reactions (see Subsection 1.2 below), existence of global unique strong solutions is still an open problem. However, this is only a first example, see also [Pie10, Section 7, Problem 1] for further comments.
In this paper we show that suitable stochastic perturbations of reaction-diffusion equations improve this situation considerably. More precisely, we show delayed blow-up and enhanced diffusion phenomena for reaction-diffusion equations with transport noise and periodic boundary condition: where i ∈ {1, . . . , } for some integer ≥ 1. Here we denote as and ν i the diffusivity of v i . Moreover, (w k,α ) k,α are complex Brownian motion on a filtered probability space and The vector fields σ k,α are smooth, divergence free and θ = (θ k ) k ∈ 2 . A precise description of the noise will be given in Subsection 3.1. The nonlinearities f i depend on v = (v i ) i=1 and are assumed to be of polynomial growth and with mass control. A prototype example is given by (1.7) which appears in the study of reversible chemical reactions, see Subsection 1.2. The term ν i ∆v i dt in (1.1) can be replaced by a general second order operator. For exposition convenience, we do not pursue this here and we refer to Remark 3.9 for comments. In this work we prove that for all T ∈ (0, ∞), there exists a choice of (θ, ν) such that, the strong solutions to (1.1) does not blow up before time T with high probability. Under additional assumptions we are also able to handle the case T = ∞. Since blow-up in finite time occurs for specific instances of (1.1) with θ ≡ 0, the presence of the noise is essential.
Transport noise is often used to study the evolution of passive scalars in turbulent flows, see e.g. [Fla11,MK99]. Such noise is often referred as Kraichnan model due to his pioneering works [Kra68,Kra94]. Roughly speaking, transport noise can be though as an idealization of the effect of an underlined turbulent transport. Heuristically, one can assume that the same type of contribution is also present in reaction-diffusion type systems lying in a turbulent flow, see Subsection 2.1. In this scenario, as experiments with chemical reactions suggest (see e.g. [GI63, LW76, STM91, MC98, KD86, ZCB20]), turbulent flows enhance the mixing of reactants. This eventually leads to an increased efficiency of the corresponding chemical reaction. In practice, the chemical reaction occurs as if the reactants have an increased diffusion compared to the one measured in standard conditions. This phenomenon is usually called enhanced diffusion. One of the aim of this paper is to provide a (possible) mathematical description of this fact by showing that, in certain norms, the solution to (1.1) is close to the solution of the corresponding deterministic problem with increased diffusivity (see Theorem 1.1 below). It seems not possible to capture this phenomenon by a deterministic model as neither the diffusivity nor the reaction-rate of the reactants are actually altered by the presence of the fluid.
1.1. Delayed blow-up and enhanced diffusion: Simplified version. To give a flavor of the results in the paper, here we state a simplified version of Theorem 3.5. To apply it, one fixes three parameters: T ∈ (0, ∞) the time horizon where one wants the solution to exist, ε ∈ (0, 1) the size of the event where the blow-up may occur and r ∈ (1, ∞) the time integrability for the norm in which we measure the enhanced diffusion.
Theorem 1.1 (Simplified version of Theorem 3.5). Let d(h−1) 2 ∨ 2 < q < ∞. Fix T ∈ (0, ∞), ε ∈ (0, 1) and r ∈ (1, ∞). Assume that f is of polynomial growth with exponent h > 1 and with mass control (see Assumption 3.1(2)-(3) below). Let v 0 ∈ L q (T d ; R ) be such that v 0 ≥ 0 (component-wise). Then there exist ν > 0 and θ ∈ 2 such that #{k : θ k = 0} < ∞ for which the unique strong solution v to (1.1) exists up to time T with high probability: P(τ ≥ T ) > 1 − ε where τ is the blow-up time of v. Finally, we have enhanced diffusion with high-probability: is the unique strong solution to the deterministic reaction-diffusion equation with enhanced diffusion on [0, T ]: In the above result one can even choose (θ, ν) uniformly with respect to v 0 such that v 0 L q ≤ N , where N ≥ 1 is fixed. In such case (θ, ν) does not depend on v 0 , but only on N . Actually, one can always enlarge ν still keeping Theorem 1.1 true. In particular, one can choose the enhanced diffusion in (1.3) as large as needed. The existence of a unique strong solution v det to (1.3) on [0, T ] is also part of the proofs. For the complete result we refer to Theorem 3.5. In Theorem 3.6 we also allow T = ∞, in the case of exponentially decreasing mass.
In light of the results in [AV22b] (recalled here in Theorem 3.3), the solution v of Theorem 1.1 is not only strong, but it is also positive and instantaneously gains regularity: a.s. for all γ 1 ∈ (0, 1 2 ) and γ 2 ∈ (0, 1). (1.5) The positivity of solutions to (1.1) is very important from an application point of view, as v i typically models concentrations. Let us stress that an additive noise would destroy the positivity of the initial data. Thus, in the context of reaction-diffusion equations, additive noise seems not appropriate to work with. Another interesting feature of transport noise is that it does not alter mass conservations, energy balance and, more generally, L q -estimates. Here we mean that, when computing v i q L q , one obtains an equality in which the noise does not contribute. Moreover, such equality is the one obtained in absence of noise, see Subsection 2.2. This shows in particular that the stochastic perturbation does not help in proving L q -bounds. The diffusivity (or mixing) behavior of the noise can only be seen in norms which are "below" the L q -energy level (e.g. L r (0, T ; L q ) with r < ∞), cf. the enhanced diffusion part of Theorem 1.1.
Compared to the deterministic theory, the strength of the results of Theorem 1.1 is that the presence of noise allows us to obtain strong unique solutions to (1.1) with arbitrary large life (at expense of enforcing the noise). Under some additional assumptions (e.g. entropy-dissipation relation), in the deterministic setting, existence of global weak solutions to (1.1) are provided in [Fis15,FHKM22,LW22]. Determining whether or not such solutions are unique and/or smooth is an open problem [Fis15,Section 4]. For the weaker notion of weak-strong uniqueness see [Fis17].
Theorem 1.1 is a regularization by noise result since solutions to the deterministic version of (1.1) blow-up in finite time for appropriate choices of f i satisfying Assumption 3.1(2)-(3) ( [PS97] or [Pie10,Theorems 4.1 and 4.2]). Regularization by noise started with the seminal work of Veretennikov [Ver81], where he proved that noise restores existence and uniqueness in ODEs. This basic result has been later extended in many directions and in particular to PDEs. It is not possible to provide a complete overview on such results and we limite ourself to the case of regularization by transport noise. In such area, a first breakthrough result has been established by Flandoli, Gubinelli and Priola [FGP10] where they prove that transport noise improves the well-posedness theory for the transport equation (see also [GM18] for scalar conservation laws). A second breakthrough has been recently obtained by Flandoli and Luo in [FL21] where they prove that a sufficiently intense noise prevents the blow-up of the Navier-Stokes equations in three dimensions and in vorticity formulation. Related results can be found in [Gal20, FGL21a, FHLN20, Lan22, FGL21b, GL22, Luo21] and in the references therein.
The results of this paper fall within this line of research providing new results and highlighting new view points on the works [FL21,FGL21a]. One of the main contribution is the connection with the theory of critical spaces for SPDEs developed in [AV21a,AV22a] which relies on the L p (L q )theory for SPDEs, pioneered by Krylov [Kry94,Kry99] and later by Van Neerven, Veraar and Weis [NVW07,NVW12]. To the best of our knowledge, the current paper is the first regularization by noise result exploiting L p (L q )-estimates. We will see that the assumption q > d(h−1) 2 in Theorem 1.1 is related to the criticality of the space L d 2 (h−1) for the SPDEs (1.1). In particular, the subcriticality of L q with q > d(h−1) 2 appears in many estimates and will be our guide in the choice of (many) parameters. We refer to Subsection 2.3 for more details.
The L p (L q )-setting is actually necessary to deal with (1.1) due to the polynomial growth of f . To see this, it is worth to take a closer look at the scaling limit arguments in [FL21,FGL21a] and Theorem 6.1. Following the heuristic derivation given in Subsection 2.1, where we introduce the transport noise in (1.1) as a model for small scales of the driven turbulent dynamic, one may think of Theorem 6.1 as an homogenization result for the SPDE (1.1) where the role of the scale parameter is played by the ratio θ ∞ / θ 2 (cf. also Subsection 2.2). The point is that, although the vector fields σ k,α are smooth, in the scaling limit they are fast oscillating and they average as the scale parameter goes to zero. The limit contribution of the fast oscillating noise is the additional diffusive term ν∆v i appearing in (1.3). Looking at Theorem 6.1 in this perspective, Theorem 1.1 (and the main results of the paper) can be seen as a large scale regularity result (in the homogenization sense, see e.g. [She18, AKM19, GNO20]) for the SPDEs (1.1). The homogenization view-point is also interesting for mathematical reasons. Indeed, as it is standard in homogenization theory, even in presence of smooth diffusive matrix, one cannot prove estimates uniformly in the scale parameter. In practice, one cannot use further information on the diffusivity matrix besides ellipticity and boundedness. Therefore one is forced to use tools from PDEs with (rough) L ∞ -coefficients, such as Moser iterations and DeGiorgi-Nash-Moser estimates. A similar situation appears here, where, to run the scaling limit argument of Theorem 6.1, one need an estimate coming from a Moser type iteration, see Theorem 4.1(2).
1.2. Reversible chemical reactions. In this subsection we apply Theorem 1.1 to a model for reversible chemical reactions. For an integer ≥ 1 and two collections of nonnegative integers (q i ) i=1 , (p i ) i=1 (note that either α i = 0 or β i = 0 for some i is allowed), consider the chemical reaction: (1.6) q 1 V 1 + · · · + q V where R ± are the reaction rates and (V i ) i=1 are the reactants. Let v i be the concentration of the reactant V i with diffusivity ν i > 0. Finally R ± > 0 are the reaction rates. The law of mass action postulates that the concentration v i satisfies the deterministic version of (1.1) with The conservation of the reactants mass is equivalent to ask for a collection of positive constants (α i ) i=1 such that 1≤i≤1 α i (q i − p i ) = 0 (below referred as mass conservation condition). The following result is a special case of Theorem 1.1.
s. for all γ ∈ [0, 1 2 ). Item (2) follows from (1.4). Item (3) is stronger than (1.5) and still follows from the result of [AV22b] where one also uses the fact that f i are smooth (see Remark 3.4(b)). Interestingly, item (3) shows that v is not only a strong solution to (1.1) but it is also classical in space.
As before, we remark that the transport noise does not interact with the mass, energy and L q -balances. For instance, under the mass conservation condition, by integrating (1.1) with (1.7), one can show the pathwise conservation of mass: In absence of noise, existence for large time of unique strong solutions to (1.1) with (1.7) it is not known even for the (apparently) simple situation of (1.6) with = 2 (cf. [Pie10, Remark 3.2]). Let us mention that, already the case q 1 = p 2 = 1 and q 2 = p 1 = 2 appears problematic. Indeed, in the latter situation, (1.1) with (1.7) has cubic nonlinearities with non triangular structure (cf. [Pie10, Theorem 3.5]). For this system of PDEs, existence for large time of strong solutions is not known. However, if ≥ 1 and max 1≤i≤ q i ∨ p i ≤ 1, then the existence of global strong solutions has been proven recently in [FMT20] (see also [CGV19] for the case = 4 and p i , q i ≤ 1).
1.3. Further comments on the literature. We collect here further references to the related literature. To the best of our knowledge, in the deterministic case, the investigation of the effect of a velocity field on the dynamics of passive scalars was first studied by Constantin, Kiselev, Ryzhik and Zlatož [CKRZ08]. For some results in a nonlinear deterministic L 2 -setting see [IXZ21]. The case of a linear dynamic in a turbulent fluid, modeled by a transport noise, was also studied by Gess and Yaroslavtsev in [GY21]. There the authors proved stabilization and enhanced dissipation by noise for passive scalar and they provide an interesting overview of previous results. A somehow intermediate situation has been studied by Bedrossian, Blumenthal and Punshon-Smith where the dynamics in the (deterministic) passive scalar equation is given by the solution of a stochastic Navier-Stokes type equations, see [BBPS21,BBPS22b,BBPS22a]. However, in the latter results, one is limited to consider situations where Navier-Stokes type equations are known to be globally well-posed, e.g. d = 2 for the Navier-Stokes equations.
1.4. Notation. Here we collect the notation which will be used throughout the paper. We write A P1,...,P N B (resp. A P1,...,P N B) whenever there exists a positive constant C depending only on the parameters P 1 , . . . , P N such that A ≤ CB (resp. A ≥ CB). Similarly, we write C(P 1 , . . . , P N ) > 0 if the constant C depends only on P 1 , . . . , P N . Moreover, R is the set of real numbers, R + = (0, ∞), Z is the set of integers, . We also employ the notation a ∨ b = max{a, b} and a ∧ b = min{a, b}. In the following, for an integer ≥ 1, we denote by H s,q (T d ; R ), B s q,p (T d ; R ) the set of -valued maps in the Bessel potential and in the Besov classes, respectively (see e.g. [Gra14,Tri95,Saw18]). Often, below we write L q , H s,q etc. instead of L q (T d ; R ), H s,q (T d ; R ) etc., if no confusion seems likely. With (·, ·) θ,p and [·, ·] θ we denote the real and the complex interpolation functor, respectively. We refer to [BL76,HNVW16,Tri95] for definitions and basic properties. Below we collect some further notation which may be non standard. In the following X is a Banach space and I = (a, b) is an open interval.
• L p (a, b, w κ ; X) is the set of all strongly measurable maps f : I → X satisfying If κ = 0, then we write L p (a, b; X) instead of L p (a, b, w 0 ; X). • W 1,p (a, b, w κ ; X) or W 1,p (I, w κ ; X) denotes the space of all f ∈ L p (a, b, w κ ; X) such that f ∈ L p (a, b, w κ ; X) endowed with the natural norm.
• For γ 1 , γ 2 > 0, C γ1,γ2 ((a, b) × T d ) denotes the set of all bounded maps u such that • For a function space A, we sometimes write A(I, w κ ; X) instead of A(a, b, w κ ; X). Moreover, we write f ∈ A loc (O, w κ ; X) provided f ∈ A(O , w κ ; X) for all compact set O ⊆ O. Finally we collect the probabilistic notation. Further notation will be fixed in Subsection 3.1. Throughout the paper, (Ω, A, (F t ) t≥0 , P) denotes a filtered probability space. A measurable map τ : Ω → [0, ∞] is a stopping time if {τ ≤ t} ∈ F t for all t ≥ 0. For a stopping time τ , F τ denotes the σ-algebra of the τ -past, i.e. A ∈ F τ provided A ∩ {τ ≤ t} ∈ F t for all t ≥ 0 (see e.g. [Kal02]). For a Banach space X, a stochastic process φ : Finally, we write k,α instead of k∈Z d 0 1≤α≤d−1 , if no confusion seems likely.
Acknowledgements. The author thanks Mark Veraar for helpful comments and acknowledges Caterina Balzotti for her support in creating the pictures.

Derivation, enhanced diffusion and criticality
In this section we illustrate some basic ideas leading to the proof of our main results. However, before going into the mathematical details, we first provide an heuristic derivation of (1.1).
2.1. Heuristic derivation. Inspired by [FL21, Subsection 1.2], we motivate transport noise by the idea of separating large and small scales and to model the small scale by noise. This corresponds to some intuition of turbulence. With this in mind, we formally derive (1.1) by considering its deterministic version in which v i is transported by a velocity field u of a fluid in which v i lies: Following [FL21], we decompose u as u L + u S , where u L and u S denote the large and the small scale part, respectively. Roughly speaking, in a turbulent regime, the u S varies very rapidly in time compared to u L . In this case, one may replace u S by an approximation of white noise, i.e. − k,α θ k (σ k,α · ∇)v i •ẇ k,α t , and therefore (2.1) coincide (1.1) with an additional deterministic transport noise. The deterministic transport term (u L · ∇)v i does not play any role in the analysis, and therefore we drop it from the results below (see Remark 3.9 for some comments). In case of 2D fluid dynamic models, the approximation of small scales by a transport term can be made rigorous, see [FP22]. Let us also remark that the noise (1.1) is in the Stratonovich formulation, which, from a modeling point of view, seems the correct one due to its connections with Wong-Zakai type results. Moreover, as we will see in Subsection 2.2 below, the Stratonovich noise does not alter the mass and energy balances. This is consistent with the intuition of the stochastic perturbation in (1.1) as a transport term.
2.2. Enhanced diffusion and the homogenization view-point. The issue of globale existence for parabolic PDEs is usually addressed by showing energy estimates. In practice, one derives a-priori bounds on suitable L q -norms of the solutions to the corresponding PDEs. Blow-up criteria for SPDEs (c.f. Theorem 3.3(3)) shows that an a-priori pathwise L ∞ t (L q x )-estimate with q > d(h−1) 2 ∨ 2 is sufficient to prove global existence for system of reaction diffusion equations like (1.1). Thus, one is tempted to apply the Itô formula to compute v i q L q and deriving a-priori bounds. However, due to the divergence free of σ k,α , one has In particular, the martingale part in the Itô formula vanishes and one obtains, a.s. for all t ∈ [0, τ ), The above equality coincides with the L q -balance in absence of stochastic perturbation in (1.1). Therefore it is clear that the noise cannot help to improve such estimates. To capture the enhanced diffusion (or mixing) induced by the transport noise one has to look at weaker norms compared to the one appearing in the energy-type balance (2.2), e.g. L r (0, T, L q ) with r ∈ (1, ∞). From a mathematical perspective, the key step to understand the enhanced diffusive effect of the noise is the scaling limit result of Theorem 6.1. In that result, we consider a sequence of (θ (n) ) n≥1 and the sequence of corresponding solutions (v (n) ) n≥1 to (1.1) and we show convergence (in "relatively weak norms") of the solutions to a deterministic system of reaction-diffusion equations with enhanced dissipation provided lim n→∞ θ (n) ∞ / θ (n) 2 = 0. Here we exploit the fact that the vector fields σ k,α are objects with high oscillations and in the limit as n → ∞ they average. The limiting contribution of the noise is the diffusive term ν∆v i in (1.3). Now Theorem 1.1 follows by choosing n so large that the solution to (1.1) is not far from (1.3). Since the transport noise in (1.1) models small scale effects, the above argument share the same philosophy of large scale regularity theory in the theory of homogenization, see e.g. [She18, AKM19, GNO20]. As commented in the Susbection 1.1, this interpretation naturally bring us to the use of tools from the theory of PDEs with L ∞ -coefficients, such as Moser iterations.
In a way, this view-point allows us to give an heuristic motivation for the failure of the scaling limit argument in [FL21] for the full advective noise (see [FL21,Appendix 2]). Recall that the vorticity formulation in [FL21] is obtained by applying ∇× to the Navier-Stokes equations with transport noise. Due to Leibniz rule, this creates a (lower order) term which cannot be controlled via the L ∞ -norm of the coefficients itself and therefore the scaling argument is doomed to fail.
2.3. The role of criticality. Several choices of the spaces done in this paper are motivated by the (local) invariance of the SPDEs (1.1) under parabolic scaling. Recall that f is of polynomial growth with exponent h > 1 (see Assumption 3.1(2)). As discussed in [AV22b, Subsection 1.2], the Lebesgue space L d 2 (h−1) is critical for (1.1). Here we do not discuss the case of critical Besov spaces, as the Lebesgue ones are the natural to deal with when working with L ∞ -coefficients. With an eye towards the main scaling argument of Theorem 6.1, where one needs to use compactness, we work within the subcritical regime L q with q > d(h−1) 2 . Indeed, within this range, one can lose regularity to obtain compactness, still being in a spaces where (1.1) is well-posed. The subcriticality also plays an important role in the main estimates. Indeed, a fairly straightforward consequence of it is the existence of ε > 0 such that (cf. Lemma 4.3) The criticality of the L q is equivalent to ask for which q the inequality (2.3) holds with ε = 0. The sub-criticality gives us the play parameter ε > 0 which can be used to show a-priori estimates via a simple buckling argument. Indeed, the Young inequality shows that, for all p ∈ (2, ∞), Choosing δ > 0 small enough, one can use maximal L p -regularity estimates to close a bound for v L p (0,T ;H 1,q ) in terms of v L r (0,T ;L q ) . However, there is no general way estimate the latter term. Following [FL21,FGL21a], we introduce a cut-off in the equation (1.1). We design the cut-off φ R,r (·, v) in a way that φ R,r (·, v)v L r (0,T ;L q ) R 1, see (4.2) below. Thus, for the cut-off version of (1.1), the inequality (2.4) readily proves an estimate, cf. Theorem 4.1(1). The cut-off can later be removed by using the enhanced diffusive effect of the noise.
The same sort of argument also enters in the Moser type iteration used in Theorem 4.1(2). More precisely, looking at the L q -balance of (2.2), the condition q > d(h−1) 2 yields the existence of β ∈ (0, 1) such that the RHS(2.2) can be estimated as (cf. Lemma 4.5) Again, by balancing the contribution of v L r (0,T ;L q ) with the cut-off φ R,r , one sees that the energy term max 1≤i≤ T 0 T d |v i | q−2 |∇v i | 2 dxds can be absorbed on the LHS(2.2) with the same buckling argument via Young inequality.

Statement of the main results
In this section we state our main result concerning reaction diffusion equations (1.1). Here we actually consider the following generalization of (1.1) where we also include a conservative term: As above, i ∈ {1, . . . , } for some integer ≥ 1. As before c d def = d d−1 and ν, ν i > 0. The unexplained parameters appearing in the stochastic perturbation of (3.1) will be described in Subsection 3.1.
The nonlinearities (f, F ) will be assumed to be of polynomial growth, see Assumption 3.1 for the precise conditions. This section is organized as follows. In Subsection 3.1 we describe the noise and its basic properties, in Subsection 3.2 we collect the main assumptions, definition and a local existence result taken from [AV22b]. Finally in Subsection 3.3 we state our main results whose proofs will be commented in Subsection 3.4.
3.1. Description of the noise. Here we specify the quantities (θ k , σ k,α , w k,α ) appearing in the stochastic perturbation in (3.1). Here we follow [FL21,FGL21a] . Moreover, we assume that θ is normalized and it is radially symmetric, i.e.
Let us remark that the stochastic integration on the RHS(3.4) is understood in the Itô-sense. In the paper we will always understood the Stratonovich noise on the LHS(3.4) as the RHS(3.4), namely an Itô noise plus a diffusion term. However, note that the diffusion term ν∆v i does not provide any additional diffusion, as in the usual energy estimates, it is balanced by the Itô correction coming from the Itô-noise. In particular (3.4) is consistent with Subsection 2.2.
3.2. Main assumptions, definitions and local existence. In this subsection we collect our main definitions and assumptions. The following will be in force throughout this paper.

Conditions
(2)-(4) in Assumption 3.1 are typically employed in the study of reaction-diffusion equations, see e.g. [Pie10] and the references therein. The growth of the nonlinearities (F, f ) in (2) is chosen so that the mapping v → f (·, v) and v → div(F (·, v)) has the same (local) scaling (see [AV22b, Subsection 1.2]). As shown in [AV22b], the above conditions ensure the existence of solution to (3.1), with certain properties, under mildly regularity assumption on v 0 . For the reader's convenience, we summarize the one needed in this paper in Theorem 3.3 below.
To introduce the definition of solutions we use the interpretation (3.4) of the Stratonovich noise. Recall that the family (w k,α ) k,α induces an 2 -cylindrical Brownian motion W 2 given by where k ∈ Z d 0 and α ∈ {1, . . . , d − 1}. Note that W 2 is real valued due to (3.3). Definition 3.2. Assume that Assumption 3.1 holds for some h > 1. Suppose that θ satisfies be a stochastic process.
a.s. for all j ≥ 1 the following holds for all t ∈ [0, τ j ]: A sequence of stopping times (τ j ) j≥1 satisfying the above is called a localizing sequence.
Next we recall the following results of [AV22b] which will be needed below.
Before going further let us discuss the role of δ in Theorem 3.3 (in practice, one chooses δ close to 1). Note that the case δ = 1 is not included in the result as it would lead to a weight w κ p,δ ∈ A p/2 as κ p,δ = κ p,1 = p 2 −1 (here A r denotes the r-th Muckenhoupt class, see e.g. [Gra08]). Recall that the A p/2 -setting are the natural one for SPDEs, see e.g. [AV20, Section 7] or [LV21]. Finally, we note that the choice of the value κ p,δ is optimal. Indeed the (space-time) Sobolev index of the path space H γ,p (0, T, w κ p,δ ; H 2−δ−2γ,q ) is equal to the one of the space of initial data L q .
For later use we collect some further observations in the following In an attempt to make this work as independent as possible from [AV22b], we use Theorem 3.3(1)-(3) only to prove Theorem 3.6, while Theorem 3.5 only uses the local well-posedness of (3.1). A careful inspection of the proof of Theorem 3.5 shows that (3.9) is not used (however, it will be needed for solutions for its deterministic version, see Proposition 5.1). Finally, Remark 3.4(a) (resp. (b)) is used in Theorem 1.2 (resp. Proposition 4.2 below).
3.3. Main results. In this subsection we state the main results of this paper. To this end, let us introduce the following deterministic version of (3.1) with enhanced diffusion: where ν > 0 is as in (3.1). The notion of (p, κ, δ, q)-solution to (3.12) is as in Definition 3.2. Compared to Definition 3.2, for (3.12), we can use the full positive A p -range κ ∈ [0, p − 1) as the problem (3.12) is deterministic. To economize the notation we say that v is a (p, q)-solution to (3.12) in case is a (p, κ, δ, q)-solution to such problem with δ = 1 and κ = κ p,δ = κ p,1 .
To apply the next result one needs to fix five parameters (N, T, ε, ν 0 , r). Roughly speaking, N bounds the size of the initial data v 0 , T is the time horizon where our solutions lives, ε bounds the size of the event where the solution v might explode, ν 0 is the lower bound for the enhanced diffusion and r is the time integrability exponent in which we measure the convergence of (3.1) to the deterministic problem (3.12) with enhanced diffusion.
(1) (Delayed blow-up) The solution v exists up to time T with high probability: (2) (Enhanced diffusion) There exists a (unique) (p, q)-solution v det to (3.12) on [0, T ], and the solutions v and v det are close in the following sense: It is interesting to note that the parameters (ν, θ) are independent of v 0 satisfying (3.13) (however, they may depend on N ). The choice of (ν, θ) is not unique. Indeed, as the proof of Theorem 3.5 shows, one can always enlarge ν still keeping the assertions (1)-(2) true. The same is also valid for Theorem 3.6 below. Other possible choices of θ will be given in Remark 3.8 below. Finally, let us remark that v det in Theorem 3.5(2) is actually a (p, q)-solution to (3.12) given by Proposition In case of exponentially decreasing mass we can allow T = ∞ in Theorem 3.5. By Theorem 3.3(1), exponentially decreasing mass happens if Assumption 3.1(4) holds with a 0 = 0 and a 1 < 0. To apply the following result one five parameters (N, ε, ν 0 , r, q 0 ). Compared to Theorem 3.5, the time horizon is T = ∞ and we have an additional parameter q 0 < q for the space integrability in the enhanced diffusion assertion.
(1) (Global existence) The solution v is global in time with high-probability: (2) (Enhanced diffusion) There exists a (unique) (p, q)-solution v det to (3.12) on [0, ∞) and The parameters (ν, θ) in Theorem 3.6 are independent of v 0 satisfying (3.14). Moreover, we remark that v det in item (2) is as in Lemma 5.2 and therefore v det ∈ L ζ (R + ; L q0 (T d ; R )) for all ζ < ∞. Let us conclude this subsection with several remarks.
Remark 3.8 (On the choice of θ). The proof of Theorems 3.5 and 3.6 also reveals other possible choices of θ. Indeed, for each sequence (θ (n) ) n≥1 ⊆ 2 (Z d 0 ) satisfying (3.2) for all n ≥ 1 and (3.15) lim n→∞ θ (n) ∞ (Z d 0 ) = 0, there exists n * > 0 sufficiently large such that the assertions of Theorems 3.5-3.6 hold for all θ = θ (n) with n ≥ n * (cf. Proposition 6.1 below). As in [FL21], an example is given by The above example also satisfies #{k : θ (n) k = 0} < ∞ for all n. Interestingly, the sequence (3.16) satisfies supp θ (n) ⊆ {n ≤ k ≤ 2n} and therefore it only acts on high Fourier modes. Moreover, as we may enlarge n, such frequencies can be chosen as large as needed. We will employ such sequence later on, but of course other choices are possible, see e.g. [FL21, Remark 5.7].
Remark 3.9 (Inhomogeneous diffusion/deterministic transport). The operator ν i ∆v in (3.1) can be replaced by a general second order operator div(a i · ∇v i ) are α-Hölder continuous with α > 0 and a i is a bounded elliptic matrix with ellipticity constant ν i > 0. In such a case, the results of Theorems 3.5-3.6 still hold if δ < 1 + α (this restriction comes from the application of [AV21b] in Theorem 4.1 below).
Remark 3.10 (The case of constant mass). The assumptions a 0 = 0 and a 1 < 0 in Theorem 3.6 cannot be removed in general. However, in case of constant mass (i.e. a 0 = a 1 = 0 in Assumption 3.1(4)), we expect that Theorem 3.6 still holds. Indeed, it is often true that solutions to the deterministic version of (3.1) converges exponentially to a steady state v ∞ , see e.g. [AMT00, DF06, DFM08, DFT17, DJT20] for some examples. In this scenario, Theorem 3.6 concerns the case v ∞ = 0. However, compared to the references before, here we do not assume any global existence a-priori and in particular any assumption on h. It would be interesting to see if entropy methods, as used in the above references, can allow us to extend Theorem 3.6 in case a 0 = a 1 = 0.
Remark 3.11 (Navier-Stokes equations). It is natural to ask for similar results for Navier-Stokes equations perturbed by transport noise. Note that the equations considered in [FL21] are not equivalent to those (see [FL21, Subsection 1.2 and Appendix 2]). Although the L p (L q )-setting for the Navier-Stokes equations with transport noise has been developed in [AV21c], at the moment an extension of Theorems 3.5-3.6 to such problem seems out of reach. Among others, one of the main issue seems the extension of Theorem 4.1(2) below. To prove the latter we exploit the fact that the nonlinearities (f (·, v), div(F (·, v))) and the transport noise are local in v. The latter fact is not true for the Navier-Stokes equations due to the Helmholtz projection.
3.4. Strategy of the proofs. In this subsection we summarize the strategy in the proof of our main results. It consists of three main steps: (1) Global existence and a-priori estimates for (3.1) with cut-off.
(2) Global existence for the deterministic version of (3.1) for high diffusivity.
Roughly, the strategy follows the one of [FL21,FGL21a]. However, as commented in Subsection 2.3, to handle the arbitrary large growth of the nonlinearities in (3.1), in (1)-(2) we exploit the full strength of maximal L p (L q )-techniques.
(1): In Section 4, we consider (1.1) with cut-off on T d : Here, for R ≥ 1 and suitably parameters q, r ∈ (1, ∞), φ R,r is a cut-off given by φ R, where φ is a bump function satisfying φ| [0,1] = 1. As we have seen in Subsection 2.3 the choice of the cut-off is related to the subcriticality of L q with q > d(h−1) 2 . In Theorem 4.1 we prove global existence of unique strong solutions to (3.17) and a-priori estimates with constants independent of θ (recall that we are assuming θ 2 = 1). The latter estimates are obtained by mimicking a Moser-type iteration. Recall that, as commented in Subsection 2.2, we cannot use the spatial smoothness of the noise to obtain estimates with constants independent of θ. In the proof of Theorem 4.1 the subcriticality of L q plays a key role.
(2): In Section 5 we show that the deterministic reaction-diffusion equations has a unique strong solutions on [0, T ], for any given T < ∞, provided µ i (T ) 0; see Proposition 5.1. Moreover, we investigate certain weak-strong uniqueness result for a class of weak solutions appearing in the scaling limit argument of (3), see Corollary 5.5.
(3): For all n ≥ 1, considers the solution v (n) to (3.17) with θ = θ (n) where θ (n) is as in (3.16). Then, using the θ-independence of the a-priori estimate in (1) and a compactness argument, up to a subsequence, we have that v (n) → v det in probability in L r (0, T ; L q ) where v det solves (3.18) with µ i = ν + ν i (enhanced diffusion). Here ν is as in the stochastic perturbation of (3.17). Theorems 3.5-3.6 now follow by choosing ν very large so that (2) applies with v det L r (0,T ;L q ) ≤ R − 1 and choose n * large enough so that v (n) − v det L r (0,T ;L q ) ≤ 1 for all n ≥ n * with high probability. Thus, for all n ≥ n * , we have φ R,r (·, v (n) ) = 1 and therefore v solves (3.1) on [0, T ] with θ = θ (n) .
Due to technical problems related to anisotropic spaces (cf. the discussion below Theorem 3.3), the above argument works only if v 0 has positive smoothness in a Besov scale, see Proposition 6.5. To show Theorem 3.5 we need an approximation argument which requires to study (3.1) with a stronger cut-off compared to the one used in Section 4, see Lemma 6.6. Finally, to prove Theorem 3.6, we exploit that the mass is exponentially decreasing due to Theorem 3.3(1) with a 0 = 0 and a 1 < 0. See [FGL21a, Theorem 1.5] and [FL21, Theorem 1.6] for similar situations.
The proof of Theorem 4.1 shows that r 0 ∈ (1, ∞) depends only on (p, q, κ, h, d). Recall that where (i) follows from 2 1+κ p < 1 and (ii) from elementary embeddings (see e.g. [Saw18, Proposition 2.1]). Hence v 0 ∈ L q and the RHS in the estimate of (2) is finite. The crucial point in Theorem Thus Theorem 4.1(2) and Sobolev embeddings yield, for all T ∈ (0, ∞), and the implicit constant is independent of (θ, v 0 ). The proof of Theorem 4.1 is spread over this section. More precisely, the proof of Theorem 4.1(1) and (2) are given in Subsections 4.2 and 4.3, respectively. In Subsection 4.1 we investigate local existence for (4.1) which is an important preparatory step for the proof of Theorem 4.1(1).

4.1.
Local existence for reaction-diffusion equations with cut-off. In this subsection we begin our analysis of the problem (4.1) with cut-off. Here we prove the existence of local unique solutions to (4.1). Moreover, we provide a general blow-up criterium for the local solution to (4.1) which will be used in Subsection 4.2 to prove that such solutions are actually global.
Proposition 4.2 (Local existence and blow-up criterium with cut-off). Let the assumptions of Theorem 4.1 be satisfied. Then there exists r 0 (p, q, κ, h, d) ∈ (1, ∞) for which the following hold for all r ∈ [r 0 , ∞).
(1) (Local existence and regularity) There exists a (unique) Proposition 4.2 does not follow directly from the results of [AV21a,AV22a] as the setting used there does not allow for the non-local (in time) operator v → φ R,r (·, v). However, the methods of [AV21a,AV22a] are still applicable with minor modifications. Below we give some indications how to extend the proofs of [AV21a,AV22a] to the present situation.
Proof of Proposition 4.2 -Sketch. We split the proof into three steps.
To establish the blow-up criterium of (2) we follow the arguments in [AV22a, Subsection 5.2] which was devoted to the proof of [AV22a, Theorem 4.10(2)] that is closely related to (2). The result of Step 2 should be compared with [AV22a, Lemma 5.4].
The last claim follows as in [AV22a, Remark 5.6] once (4.6) is proven. To prove (4.6), we argue by contradiction with the maximality of (v, τ ). Hence, by contradiction, assume that Thus there exist M, η > 0 and a set V ∈ F τ such that P(V) > 0, and a.s. on V, one has τ > η and Let φ be as below (4.2). For all u ∈ L r (τ, T ; L q ) we set Consider the following version of (4.1) with modified cut-off: ) by (4.7). One can check that the proof of [AV22a, Proposition 5.1] extends to the present setting (more precisely, the estimates below [AV22a, (5.9)] also hold). Thus, reasoning as in [AV22a, Proposition 5.1], one sees that there exists a (p, 0, δ, q)-solution (u, λ) to (4.8) such that λ > τ ∨ η a.s. (note that we use the trivial weight at time λ ≥ η). Set This contradicts the maximality of (v, τ ). Hence the claim of Step 2 follows.
Step 3: (2) holds. The claim of this step follows verbatim from the proof of Theorem 4.10(1) in [AV22a, Subsection 5.2] (here we are using that the SPDEs (4.1) are semilinear).

4.2.
Proof of Theorem 4.1(1). We begin with the following interpolation inequalities involving the nonlinearities in (3.1). Here the subcritical nature of the spaces considered comes into play.
The key point is that the RHS(4.9)-(4.10) grows sub-linearly in u H 1,q .
Proof of Lemma 4.3. We split the proof into two steps.
Step 1: (4.9) holds. Recall that q > 2 and d ≥ 2 by assumption. By Sobolev embeddings, Therefore, using Assumption 3.1(2), we have Without loss of generality we assume that hζ > q, otherwise the previous inequality already gives (4.9). If hζ > q, then by Sobolev embeddings we have H ϕ,q → L hζ for some ϕ > 0 such that Step 1, it remains to note that the condition ϕh < 1 follows from q > d 2 (h − 1).
Step 1: where the implicit constants are independent of (t, u). Below we only prove the first estimate as the second one follows similarly.
Since C 1 is independent of (v 0 , n) and lim n→∞ γ n = τ ∧ T a.s., the claim of Step 2 with c 0 = 2C 1 follows by letting n → ∞ in the above estimate.
Step 3: Conclusion. By Step 2 we know that v ∈ L p (0, τ ∧ T, w κ ; H 1,q ) a.s. for all T < ∞. From Steps 1 we deduce that, for all T < ∞, Therefore, by Proposition 4.2(2), Hence τ ≥ T a.s. for all T < ∞ and therefore τ = ∞ a.s. . This strategy has two basic advantages. Firstly, the role of the sub-criticality is clear from the estimates of Lemma 4.3 which in combination of a (relatively) soft argument gives global existence for (4.1). Secondly, in an L p (L q )-setting, pathwise uniqueness is more difficult to achieve as it often difficult to estimate differences like f (·, v (1) ) − f (·, v (2) ). Indeed, such estimate seems possible only if one enforces the cut-off, cf. Lemma 6.6 below.
Proof of Theorem 4.1(2). Fix T ∈ (0, ∞). Without loss of generality we may assume that r 0 ≥ r * where r * is as in Lemma 4.5. To prove the claim of Step 1, we compute T d |v i | q dx and we estimate the nonlinearities by employing Lemma 4.5. As in [DG15, Lemma 2], we need an approximation argument. For N ≥ 1, set One can check that there exists c ≥ 1 independent of N ≥ 1 and y ∈ R such that Moreover, for all y ∈ R, as N → ∞ we have where we used that the martingale part cancels since = 0 a.s.
Here (i) follows from the chain rule and (ii) from integrating by parts as well as div σ k,α = 0. For the reader's convenience, we split the remaining proof into several steps.
By taking N → ∞ in (4.18) and using (4.16) we have, a.s. for all t ∈ [0, T ], It remains to discuss the legitimacy of using the Lebesgue domination theorem to take N → ∞ in (4.18). Firstly, recall that, by Theorem 4.1(1), κ < p 2 − 1 and (4.3), Hence (4.15) shows |v i | q−2 |∇v i | 2 ∈ L 1 ((0, T ) × T d ) a.s. Moreover, by Assumption 3.1(2), Combining the above with (4.20), (4.15) and Lemma 4.5 we get For the F -term we argue similarly. By Assumption 3.1(2) and the Cauchy-Schwartz inequality, Thus (4.19) is proved. To conclude the proof of Step 1, it is enough to note that Step 2: Let ν 0 def = min 1≤i≤ ν i . Then there exists K > 0, independent of (θ, v 0 ), such that In this step we use that r 0 ≥ r * where r * is as in Lemma 4.5. We first estimate the f -term. Let e v be the first exit time of t → v L r (0,t;L q ) from the interval [0, 2R], i.e.
By (4.2) and Assumption 3.1(2) we have where the implicit constants depend only on (q, h, , f (·, 0) L ∞ (T d ;R ) ). By Lemma 4.5, for some β ∈ (0, 1), where in (i) we used v L r (0,ev;L q ) ≤ 2R by definition of e v . Hence we proved that Similarly we estimate the F -term. By Cauchy-Schwartz inequality and Assumption 3.1(2), Since |v| ≤ 1≤i≤ |v i |, the last integral can be estimated as in (4.21). Putting together the above estimates, one obtains the claim of Step 2.
Step 3: Conclusion. Summing over i ∈ {1, . . . , } the estimate of Step 1 and using the estimate of Step 2, one gets (4.23) sup where C T is independent of (θ, v 0 ). We remark that T 0 T d |v i | q−2 |∇v i | 2 dxds < ∞ a.s. due to (4.15). Therefore the term q(q − 1) 3ν0 4 1≤i≤ T 0 T d |v i | q−2 |∇v i | 2 dxds obtained by summing the estimate of Step 1 can be absorbed on the LHS of the corresponding estimate.
To conclude the proof of Theorem 4.1(2), it remains to show where C T is independent of (θ, v 0 ). By Step 1 with q = 2, it remains to show that To this end, recall that q ≥ 2 and 0 ≤ φ R,r (·, v) ≤ 1. Thus, by Assumption 3.1(2), where the last inequality follows from (4.21) and (4.23). With a similar argument one can show Thus we have proved (4.25).
Proof of Lemma 4.5. As above, we break the proof into steps. Below we set 1/0 def = ∞.
From the proof of Lemma 4.5 we can extract the following result which will be used later on.

Deterministic reaction-diffusion equations with high diffusivity
In this section we investigate deterministic reaction-diffusion equations: where i ∈ {1, . . . , } for some integer ≥ 1, µ i > 0 and (F, f ) are as in Assumption 3.1. The results of this section will be used in combination with Theorem 6.1 below to prove the results stated in Subsection 3.3. This section is organized as follows. In Subsection 5.1 we show the existence global unique solutions to (5.1) provided the diffusivities µ i are sufficiently large. Finally, in Subsection 5.2 we prove an uniqueness result for a class of weak solutions to (5.1) which naturally appears when dealing with certain compactness arguments, see the proof of Theorem 6.1.

Reaction-diffusion equations with high diffusivity.
Here we show the existence on large time intervals of solutions to (5.1) with µ i 0. Recall that (p, q)-solutions to (5.1) have been defined below (3.12) and that (p, q)-solutions are unique by definition.
Then there exists µ 0 (N, q, p, d, T, h, a i ) > 0 for which the following assertions holds provided (3) For some C 0 (T, N, q, p, d, h) > 0, the (p, q)-solution v of (1) satisfies Note that (µ 0 , C 0 ) are independent of v 0 satisfying the conditions in (5.2). Before going into the proof of the above result, we collect some observations. To apply L p (L q )-techniques, it is convenient to use that v 0 ∈ B 0 q,p since L q (p≥q) → B 0 q,p . Moreover, (1) and the trace embedding of anisotropic maps yield (see e.g. [ ).
The previous and p > 2 imply that v(t) ∈ B 1−2/p q,p ⊆ L q for all t > 0 (cf. (4.3) for the inclusion). In particular, the term sup t∈(0,T ] v(t) q L q in (3) is well-defined. However, since B 0 q,p → L q , it is a part of the proof of Proposition 5.1 to show its finiteness for t small. A similar remark holds for the second term estimated in (3) since the argument in (4.15) holds only on the interval [s, T ] with s > 0. Finally, as in (4.4)-(4.5), Proposition 5.1(3) and Sobolev embeddings yield In the following we need another interpolation inequality. For all t ∈ R + and u ∈ L ∞ (0, t; L 2 (T d ))∩ L 2 (0, t; H 1 (T d )) such that T d u dx = 0 a.e. on [0, t], where γ = d d + 2 and the implicit constant is independent of (t, u). The estimate (5.5) follows from the Poincaré inequality, interpolation and the Sobolev embedding H γ (T d ) → L 2/γ (T d ).
Proof of Proposition 5.1. Through the proof, we fix T ∈ (0, ∞) and v 0 ∈ L q ⊆ B 0 q,p . To economize the notation, here we denote by c T a constant which may change from line to line and depends only on (N, q, p, d, T, h, a i ), where (h, a i ) is as in Assumption 3.1. Next we collect some useful facts. By [PSW18, Theorem 1.2], there exists a (p, q)-solution (v, τ ) to (5.1) such that ). Moreover, [PSW18] also shows the existence of positive constants (T 0 (v 0 ), ε 0 (v 0 )) for which the following holds: For all u 0 ∈ L q such that v 0 − u 0 ≤ ε 0 there exists a (p, q)-solution (u, γ) to (5.1) with initial data u 0 satisfying γ > T 0 and (5.7) v − u W 1,p (0,T0,wκ p ;W −1,q )∩L p (0,T0,wκ p ;W 1,q ) v 0 − u 0 L q where the implicit constant is independent of u 0 (but depends on v 0 ). Combining a linearization argument and the maximum principle for the heat equation, one can check that Assumption 3.1(3) and v 0 ≥ 0 a.e. on T d yield (see e.g. [Pie10] and [AV22b, Appendix B] for the conservative term div(F (·, v))) Arguing as for Theorem 3.3(1), by Assumption 3.1(4) and (5.8), we have (5.9) sup Below, w.l.o.g., we assume that µ i ≥ 1 for all i ∈ {1, . . . , }. Finally, we set Now we break the proof into several steps. In Steps 1-3 we prove the estimate in Proposition 5.1(3) with T replaced by τ ∧ T . In Step 4 we prove that τ ≥ T and therefore Proposition 5.1(1) and (3) follows from Steps 1-4. Finally, in Step 5 we Proposition 5.1(2). This will complete the proof of Proposition 5.1.
As we remarked below the statement of Proposition 5.1, the case s = 0 of (5.10) is not immediate as sup s∈(0,t] v(s) q L q and t To overcome this difficulty, we use an approximation argument. To this end, we first prove (5.10) on an interval separated from t = 0. Namely, for all 0 < s < t < τ ∧ T , we claim that Here it is important that c T on the RHS(5.11) does not depend on s but only on (N, q, p, d, T, h).
To see (5.11), we can argue as follows. Firstly, as the weight w κ acts only at t = 0, we have ; W 1,q ) for all s > 0. In particular, the terms on the LHS are finite, cf. (4.15). Similarly, one can also show that RHS(4.15) is finite (see (5.14) below for the more involved weighted case). Now, since q > 2, the proof of (5.10) for s > 0 follows as in the proof of Theorem 4.1(2) in Subsection 4.3 by computing ∂ t v i (t) q L q for a fixed i ∈ {1, . . . , } and them summing up over i. Compared to Subsection 4.3, the term where in (i) we used Assumption 3.1(2) and in (ii) q > 2. The F -term can be estimate similarly also using the Cauchy-Schwartz inequality, cf. (4.22) for a similar situation. Now we would like to take the limit as s ↓ 0 in (5.11). To this end, let us first prove that In particular, the last term on RHS(5.11) is finite also if s = 0. To prove (5.13), due to (5.6), it suffices to show that, for all t ∈ (0, ∞), (5.14) W 1,p (0, t; w κp ; W −1,q ) ∩ L p (0, t; w κp ; W 1,q ) → L q+h−1 (0, t; L q+h−1 ).
By Sobolev embeddings with power weights (see e.g. [AV21a, Proposition 2.7] or [MV12, Corollary 1.4]), the above holds provided we find θ ∈ (0, 1) such that In case p > q + h − 1, then in the first condition in (5.16) the equality is not allowed and in that case one also needs to use [AV22a, Proposition 2.1(3)] in combination with the above mentioned Sobolev embeddings with power weights. To check (5.16), we can argue as follows. The second condition in (5.16) is satisfied with θ = 1 2 1 − d(h−1) q(q+h−1) . Note that θ ∈ (0, 1) since h > 1, q ≥ 2 and q > d(h−1) 2 ≥ d(h−1) q+h−1 by assumption. With the above choice of θ, one can check that also the first condition in (5.16) is satisfied since q > d 2 (h − 1). Therefore (5.14) holds. It remains to show (5.10). Due to (5.11), to prove (5.10) it suffices to show (5.10) for t ∈ (0, T 0 ] where T 0 > 0 is as before (5.7) and c T independent of T 0 . The advantage is that, for t ∈ (0, T 0 ], we can use the continuous dependence on the initial data (5.7) and prove the claimed estimate by approximation.
Thus, by Fatou's lemma, (5.10) for t ∈ (0, T 0 ] and c T independent of T 0 follows by letting n → ∞ in the corresponding estimate for v (n) using also that inf n≥1 τ (n) ≥ T 0 .
To prove the last assertion in Step 1, note that, if a 0 = 0 and a 1 < 0, then (5.9) yields v i L 1 (0,τ ;L 1 ) v 0 L 1 and therefore all the constants in the starting estimate (5.11) can be chosen independently of T .
In this step we use the interpolation inequality (5.5) in a similar way as we did in the proof of Lemma 4.5 with (4.26). However, here we need an explicit dependence on the diffusivity µ and therefore we use the homogeneous estimate (5.5). Let us fix i ∈ {1, . . . , }. Since q > d(h−1) 2 , we have q + h − 1 < q γ . By interpolation, we have, for all 0 ≤ t < τ ∧ T , Next we estimate the second term on the RHS(5.18). Note that, for all 0 ≤ t < τ ∧ T , I2,i def = Next we estimate I 1,i and I 2,i , separately. We begin with I 1,i . By (5.5) with u = |v i | q/2 , where in (i) we used that µ ≥ 1. Next we look at I 2,i . Recall that q > 2 and let ϕ(q, h, d) ∈ (0, 1) be such that 1 − ϕ + ϕγ q = 2 q . Again, by interpolation, L q/γ (0,t;L q/γ ) . Using the estimates for I i,1 and I i,2 in (5.19), we have, for all 0 ≤ t < τ ∧ T , where we have absorbed the term 2 −1 v i q L q/γ (0,t;L q/γ ) appearing in the estimate of I i,2 in the LHS of (5.20). This is possible since v i L q/γ (0,t;L q/γ ) < ∞ for all 0 ≤ t < τ ∧ T , as it follows from the estimates of I 1,i , the fact that I 2,i T sup r∈(0,t) v(r) q L q < ∞ for 0 ≤ t < τ ∧ T , (5.10) and (5.14).
The last assertion in Step 2 follows by using that c T in (5.10) can be chosen to be independent of T and, in the estimate of I 2,i , that v i ∈ L q/γ (R + ; L 1 ) by (5.9) with a 0 = 0 and a 1 < 0.
Step 3: Fix N ≥ 1 and let v 0 be as in (5.2). Then there exist µ 0 , K 0 > 0 depending only on (N, q, p, d, T, h, a i ) such that the (p, q)-solution (v, τ ) to (5.1) satisfies The proof requires some preparation. Recall that v 0 L q ≤ N and v ∈ L q+h−1 loc ([0, τ ); L q+h−1 ) by (5.2) and (5.14) respectively. Thus the estimate of Step 2 implies: T are independent of (t, v 0 , µ). It is routine to check that ψ µ,R has a unique maximum on [0, ∞) and The idea is to choose µ 0 (R, θ, d) large enough so that M µ0,R > 1 + N q , cf. Figure 1(a). This eventually leads to a contradiction with (5.22). To this end, let us begin by noticing that [0, ∞) µ → M µ,R is increasing. Thus there exists µ 0 (N, q, p, d, T, h) > 0 such that Now suppose that µ ≥ µ 0 . Assume by contradiction that (5.25) sup Next, note that the mapping is continuous, non-decreasing and satisfies X(0) = 0. Thus (5.25) implies the existence of τ 0 > 0 such that X(τ 0 ) = m µ,R < ∞. Note that (5.24) imply ψ µ,R (X(τ 0 )) = M µ,R > 1 + N q . This leads to a contradiction with (5.22). By continuity, the same argument also yields where m µ,R is as in (5.23) and we used that ψ µ,R restricted on [0, m µ,R ] is invertible by construction. Standard considerations show that (cf. Figure 1(b)) Whence the estimate (5.21) with K 0 def = K µ0,R follows from the previous and the Fatou lemma.
Step 4: Let µ 0 be as in Step 3 and assume that µ ≥ µ 0 . Then τ ≥ T . Combining the estimates of Steps 3 and 4 we have N, q, p, d, h).
To conclude the proof it remains to show that τ ≥ T . To this end, we apply the blow-up criterium of [PSW18, Corollary 2.3(ii)], which ensures that Here we also used that B 0 q,p = (W −1,q , W 1,q ) 1− 1+κp p ,p and κ p = p 2 − 1. Let us note that even if [PSW18] deals with bilinear nonlinearities, the blow-up criterium of [PSW18, Corollary 2.3(ii)]   where we recall that the weight κ = p 2 − 1 is allowed in case of deterministic equations. We prove τ ≥ T by contradiction. Assume that τ < T . Then (5.27) and the embedding Step 5: (2) holds. Recall that τ ≥ T by Step 4. Let where the limit is taken in the L q -norm, u is the solution of (5.1) with initial data u 0 ∈ L q and MR q,p (t) def = W 1,p (0, t, w κp ; W −1,q ) ∩ L p (0, t, w κp ; W 1,q ) for all t ∈ R + . Note that T * > 0 by (5.7). To prove (2) it is enough to show that T = T * and that the supremum in (5.29) is reached. To this end, one can argue by contradiction, we leave the details to the reader. In the argument it is convenient to use that v([s, T ]) ⊆ B (see e.g. [PSW18, Theorem 1.2]).
In the proof of Theorem 3.6 we need uniform estimates on the half-line [0, ∞). In case of exponentially decreasing mass, we obtain them by slightly modifying the proof of Proposition 5.1.
then the following assertion holds: For each v 0 ∈ L q (T d ; R ) such that v 0 ≥ 0 a.e. on T d and v 0 L q ≤ N , there exists a (unique) (p, q)-solution v to (5.1) on [0, ∞) such that, for all q 0 ∈ [1, q), where c 0 > 0 depends only on (a i , q 0 , q).
To prove (5.32), one can repeat the arguments in Step 3 of Proposition 5.1. Indeed, due to Step 2 of the same proof, the constant c T in (5.17) can be made independent of T since we are assuming a 0 = 0 and a 1 < 0. 5.2. Uniqueness for weak solutions to reaction-diffusion equations. In this subsection we prove uniqueness results for weak solutions to deterministic reaction-diffusion equations. Such results will be needed in the proof of Theorem 3.5. In particular, the class of maps considered in the following result is the one used in Lemma 6.2 below. We begin by proving the following uniqueness result for (5.1).
The result of Proposition 5.3 is not really surprising since X ⊆ L ∞ (0, T ; L q ) ∩ L q (0, T ; L ξ ) and therefore the class of solutions considered there are somehow close to the strong ones.
It will prove convenient later to see that (p, q)-solutions of Proposition 5.1 belongs to X . In particular, they are in the class of weak solutions considered in Proposition 5.3.
Before going into the next step we collect some facts.
Step 1 shows that v (j) solves (5.49) in its differential form where the equality is understood in H −1 . For exposition convenience, in Step 2 we prove the claim of Proposition 5.3 assuming that (in case d = 2 we choose ξ 0 large enough) where ψ depends only on (h, d, q).
Step 3 is devoted to the proof of (5.44).
Step 2: v (1) ≡ v (2) . By a standard iteration argument, to prove the claim of Step 2 it suffices to show the existence of δ * > 0 such that, for all s ∈ [0, T ], Note that the evaluation at s in first condition of (5.45) is well defined since v (j) ∈ C([0, T ]; L 2 ) by Step 1. The remaining part of this step is devoted to the proof of (5.45). Let ε > 0 be fixed later.
We estimate I F in a similar way. To begin, note that for all t ∈ [s, s + δ], Again, by Assumption 3.1(2), where the last inequality follows by noticing that the the second line in the above estimate coincides with the LHS in the first line in the estimate of I f . Using the above estimates, we get .
Corollary 5.5 (Weak-strong uniqueness for (5.34)). Let Assumption 3.1(2)-(4) be satisfied. Let ∞) and v 0 ∈ L q (T d ; R ). Let ξ and X be as in Proposition 5.3. Assume that there exists a solution v (1) ∈ X of (5.1) in the weak formulation of (5.34) satisfying Let v (2) ∈ X be a weak solution to (5.49) in the following sense: Then Due to (5.50), v (1) is also a weak solution to the problem (5.49) with cut-off. In the proof of Proposition 6.5, we check (5.50) by taking the (strong) (p, q)-solution to (5.1). Hence, to some extend, Corollary 5.5 shows that weak solutions coincide with strong ones to (5.49) (if there are any) and that (5.50) is a regularity assumption. This explains the name of Corollary 5.5.
Proof. The idea is to reduce to the case analyzed in Proposition 5.3 by mimicking a stopping time argument. To this end, let us set It remains to prove that e = T . Indeed, if the latter holds, then φ R,r (·, v (2) ) ≡ 1 and therefore v (2) is also a weak solution to (5.1) (i.e. it satisfies (5.34) for all η ∈ C ∞ (T d ; R )). Hence, applying Proposition 5.3, we eventually have v (1) ≡ v (2) .
6. Proofs of Theorems 3.5 and 3.6 In this section we prove Theorems 3.5 and 3.6. To prove both results we can now argue as in [FL21,FGL21a]. In particular, as a central step we prove a scaling limit result for stochastic reaction-diffusion equations with cut-off (4.1), see Subsection 6.1. Theorems 3.5 and 3.6 will be proved in Subsections 6.2 and 6.3, respectively.
6.1. The scaling limit for reaction-diffusion equations with cut-off. In this subsection we continue our investigation of reaction-diffusions with cut-off initiated in Section 4. Recall that the cut-off equation reads as follows: where φ R,r is as in (4.2) for R > 0, r ∈ [r 0 , ∞) and r 0 is as in Theorem 4.1. The aim of this subsection is to prove the following scaling limit result. It can be seen as a version of [FL21, Theorem 1.4] or [FGL21a, Proposition 3.7] in our setting and it is of independent interest. Recall that weak solutions to (6.1) are understood as in Corollary 5.5.
(3) For some γ ∈ (0, 1), there existence a unique weak solution to the following deterministic system of reaction-diffusion equation with cut-off: Denote by v (n) the (p, κ, 1, q)-strong solution to (6.1) with data v (n) 0 (see Theorem 4.1) and let v be as in (3). Then Eq. (6.2) shows the enhanced diffusive effect of the transport noise in (6.1). Note that the enhanced diffusivity depends on the strength of the noise through the parameter ν. The proof of Theorem 6.1 actually gives a stronger result. More precisely, we show that (6.2) also holds in case the L r (0, T ; L q )-norm is replaced by L 2 (0, T ; H 1−γ ) ∩ C([0, T ]; H −γ ) ∩ L r (0, T ; L q ) where γ > 0 is arbitrary (this is needed to obtain the assertions of Remark 3.7).
The proof of Theorem 6.1 requires some preparation and it will be given at the end of this subsection. We begin with a compactness result.
Proof. For notational convenience, we fix i ∈ {1, . . . , } and we drop it from the notation if no confusion seems likely. The proof of (6.6) follows almost the one of [FGL21a], see p. 1779. Since the argument exploits several basic properties of the noise, we include some details.
We are ready to prove Theorem 6.1. Here we follow [FGL21a, Proposition 3.7].
Recall that weak solution to (6.9) in the class X K are defined in Corollary 5.5. Fix π ∈ C ∞ (T d ; R ). Let J π : X K → C([0, T ]) be given by where u ∈ X K , t ∈ [0, T ] and ·, · denotes the pairing in the duality (H γ , H −γ ). In the following we prove the continuity of the map J π,f : The remaining terms in J π can be treated analogously using also that X K ⊆ L 2 (0, T ; H 1−γ ). By Lebesgue domination theorem, we have, for all u (1) , u (2) ∈ X K , where we used Assumption 3.1(2) and that φ is bounded and Lipschitz continuous. By Remark 4.6 and q ≥ 2 we have, for some α, β > 0 and all u ∈ X , . Thus the continuity of J π,f on X K follows from by combining the above estimates and using that u (1) L q (0,T ;L ξ ) , u (2) L q (0,T ;L ξ ) ≤ K a.s. Since J π is continuous, we may define the pushforward measures of µ (n) and µ under the map J π , respectively: Observe that µ (n) π,# µ π,# as µ (n) µ and that µ (n) ψ,# is the law with J π v (n) . Moreover J π v (n) satisfies (6.11) By (6.6) in Lemma 6.3 and θ (n) ∞ → 0 (see assumption (2)) we have, for all a ∈ (1, ∞), Using the above and assumption (1) in (6.11), one can check that supp µ π,# = {0}. The conclusion follows from the separability of H −γ and the density of the embedding C ∞ → H −γ (cf. the last part of the proof of [FGL21a, Proposition 3.7]).
Step 2: Let v be as in (3). Then v (n) → v in probability in X . In particular (6.2) holds. It suffices to show that where δ v is the Dirac measure at v ∈ X . To see this recall that v is independent of ω ∈ Ω. Hence, δ v due to (6.12). It remains to prove (6.12). By (3), v is the unique weak solution in X ⊆ X K to the reactiondiffusion equation with cut-off (6.9) and therefore (6.13) u = (u i ) i=1 ∈ X K : u is a weak solution to (6.9) = {v}.
The arguments of Theorem 6.1 also yield a suitable continuity of weak solutions for system of deterministic reaction-diffusion equations with cut-off: As it will be needed in the proof of Theorem 3.5, we formulate it in the next result. Recall that weak solutions to (6.14) in X are defined in Corollary 5.5.
(2) Suppose that there exists a unique weak solution v ∈ X to (6.14) such that, for some γ 0 , Moreover, for all n ≥ 1, there exists a weak solution v (n) det ∈ X to (6.14) with initial data v In applications (2) will be checked using Proposition 5.1 and Corollary 5.5.
Proof. It is enough to show that for each subsequence of (v (n) det ) n≥1 , we may find a subsequence such that v (n) det → v det in X . As above, to economize the notation, we do not relabel subsequences. By Lemma 6.2 and the bound in (2), there exists in u ∈ X such that v (n) det → u in X . By (1) and arguing as in the Step 1 of Theorem 6.1 we may pass to the limit in the weak formulation of (6.14) (cf. Corollary 5.5). Hence u ∈ X is a weak solution to (6.14). The uniqueness of v det (see assumption (2)) forces u = v det . 6.2. Proof of Theorem 3.5. As a preparatory step for Theorem 3.5, we prove the following version of it with sufficiently smooth initial data v 0 where (θ, ν) depend only the L q -norm of v 0 . Once this is proved, Theorem 3.5 follows from such result and a standard density argument. Recall that the existence and uniqueness for (3.1) is ensured by Theorem 3.3.
The above result can be proven following the proof of [FGL21a, Theorem 1.4]. As our setting (slightly) differs from the one of [FGL21a], we include some details.
Proof of Proposition 6.5. Throughout this proof we let (N, ε, T, ν 0 , r) be as in the statement of Proposition 6.5. Without loss of generality we assume r ≥ r 0 where r 0 is as in Theorem 4.1. Moreover, to make the argument below more transparent, we display the dependence on the initial data for the equation considered. For instance, the (p, κ, 1, q)-solution to (3.1) with data v 0 will be denoted by (v(v 0 ), τ (v 0 )).
We begin by collecting some useful facts. Set Note that B N ⊆ L N . Proposition 5.1 ensures the existence of positive constants ν ≥ ν 0 and R > 1, both independent of v 0 ∈ L N , for which the deterministic reaction-diffusion equations (5.1) with Due to (6.20) and (4.2), v det (v 0 ) is a (p, q)-solution on [0, T ] to the deterministic problem with cut-off (6.14) where µ i = ν i + ν, R as above and initial data v 0 ∈ L N . Finally, Let (θ (n) ) n≥1 be the sequence defined in (3.16). For any n ≥ 1, Theorem 4.1 provides a unique strong solution v (n) cut (v 0 ) to the reaction-diffusion equations with cut-off (4.1) for all initial data v 0 ∈ B N , R is as in (6.20) and θ = θ (n) .
The key idea now is to prove that, for all ε ∈ (0, 1), We break the proof of (6.21) in several steps. The proof of (6.21) is postponed to Step 4. In Step 1 we prove that (6.21) implies the assertions (1)-(2) of Theorem 3.5 and (6.21). In Steps 2 we prove additional interpolation estimates, which complements the one in Lemma 4.3, and leads to the proof of (6.17) given in Step 3.
Here in (i) we used that 1+κ p,δ p = 1− δ 2 and that the space for the initial data is (H −δ,q H 2−δ,q ) δ 2 ,p = B 0 q,p and in (ii) that L q → B 0 q,p as p ≥ q. The implicit constants in (6.28) depends on (θ, ν) which has been fixed so that (6.16) holds. In particular they are independent of v 0 ∈ B N . Hence, the estimate (6.17) follows from (6.28), the Chebyshev inequality and the fact that P(γ = T ) > 1 − ε.
lim sup Thus there exists a (not-relabeled) subsequence of data (v Moreover, up to extract a further subsequence, we can assume that, as n → ∞, ≥ 0 on T d for all n ≥ 1, see (6.18)-(6.19). The choice of ν and the comments below (6.20) show that there exists a (p, q)-solution v where Y is as in Proposition 6.4. Recall that, due to (6.20), v det (v (n) 0 ) are actually (p, q)-solutions to (5.1) with µ i = ν i + ν provided by Proposition 5.1 and therefore in the class considered in Proposition 6.4. By Corollary 5.5 and (6.20), we also have that v det (v 0 ) ∈ X is also unique in the class of weak solutions, where X is as in Proposition 6.4. Hence, the latter result ensures that The above and (6.29) yield (6.31) lim sup Next we derive a contradiction with Theorem 6.1. To this end we first check its assumptions (1)-(3) of Theorem 6.1. Note that (1) follows from (6.30) and v (n) 0 ∈ B N for all n ≥ 1.
(2) follows from the above choice of θ (n) as in (3.16). Finally, (3) follows from v 0 ∈ L N and the comments below (6.20). Let us stress that the uniqueness part of the assumption (3) in Theorem 6.1 follows from Corollary 5.5 and (6.20). Hence Theorem 6.1 is applicable and it yields (6.2) with v (n) = v and v = v det (v 0 ). The latter gives a contradiction with (6.31) and completes Step 4.
To prove Theorem 3.5 we use a density argument and the fact that the conditions in Proposition 6.5 are uniformly w.r.t. v 0 L q . To set up a convenient density argument we need an additional estimate for stochastic reaction diffusion equations with a modified cut-off. The choice of the cut-off is now inspired by the estimates (6.16)-(6.17).
Fix K > 0 and δ, η > 0. As in Proposition 6.5, we set κ p,δ = p(1 − δ 2 ) − 1. Let φ ∈ C ∞ (R) be such that φ| [0,1] = 1 and φ| [2,∞) = 0. Finally, set Consider the following stochastic reaction equations with (a modified) cut-off: The notion of (p, κ, δ, q)-solutions to (6.33) can be given as in Definition 3.2. The main difference of (6.33) compared to (4.1) analyzed in Section 4 is that the action of the cut-off Φ K,r,δ,η (·, v) is stronger than the one used in (4.1), i.e. (4.2). Let us note that the truncation chosen in (6.33) is too strong to run the arguments of Section 4. On the other hand, the one in (4.2) seems not enough to obtain the stability estimate of Lemma 6.7 below (cf. Remark 4.4). Such estimate is the last ingredient in the proof of Theorem 3.5. To this end, we need the following estimates.
and therefore The latter condition holds as it is equivalent to β < 1 − h h−1 1+κ p which holds by construction.
The next result is the last ingredient we need to prove Theorem 3.5.
6.3. Proof of Theorem 3.6. Following the arguments of [FL21,FGL21a] we deduce Theorem 3.6 from Theorem 3.5 and Lemma 5.2. As in [FL21] we need that the stochastic problem (3.1) is globally well-posed for small initial data, see assumption b) in [FGL21a, Theorem 1.5]. This will be the content of the following result.
Proposition 6.8 ensures the absence blow-up with high probability provided v(γ) L q 1 is small with high probability as well. The smallness of the norm v(γ) L q 0 is not surprising as the mass is exponentially decreasing by Theorem 3.3(1) with a 0 = 0 and a 1 < 0.
Next we prove Proposition 6.8 and afterwards Theorem 3.6.
Proof of Proposition 6.8. The proof follows as the one of Lemma 5.2 (see also Proposition 5.1).
Here we use the smallness of η instead of the one of µ −1 . Let (r, q 0 , q 1 , ε) be as in the statement of Proposition 6.8. For η, S > 0 and a stopping time γ : Ω → [S, ∞), set (6.54) Below we also write V instead of V γ,η if no confusion seems likely. Below we assume that (6.55) P(V) > 1 − ε. As in Lemma 5.2, below, we frequently use that the exponential decay of the mass: (6.56) T d |v| dx N e −|a1|t a.s. for all t ∈ [0, τ ).
To prove the claim, by Step 3 and the fact that P(V) > 1 − ε, it is enough to show that (6.61) v L r (γ,∞;L q 0 ) ≤ ε a.s. on V.
The proof of Theorem 3.6 follows by combining the Theorem 3.5, Proposition 6.8 and the exponential decay of solution to (5.1) shown in Lemma 5.2. For the reader's convenience, before going into the proof, we summarize the main argument. By Theorem 3.5 and Lemma 5.2, we know that v(γ) − v det (γ) and v det (γ) are small provided γ ≥ S is big enough (here v and v det is the solution to (3.1) and (5.1), respectively). Thus v(γ) is small as well. Hence Theorem 3.6 follows from the previous observation and Proposition 6.8.
Proof of Theorem 3.6. Let (N, ε, ν 0 , r, q 0 ) be as in the statement of Theorem 3.6. Recall that q 0 < q and without loss of generality we may assume that q 0 > d(h−1) 2 ∨ d d−δ . Finally fix q 1 ∈ (q 0 , q). Next we collect some further parameters which are independent of v 0 satisfying (3.14). Let µ 0 > 0 be as in Lemma 5.2 and let (S, η) be as in Proposition 6.8 with ε replaced by ε 2 . Lemma 5.2 ensures the existence of T > 0, independent of v 0 satisfying (3.14), for which the following assertion is satisfied provided ν ≥ µ 0 : For all v 0 as in (3.14), there exists a (p, q)-solution v det on [0, ∞) to the deterministic problem (3.12) satisfying v det L r (T,∞;L q 1 ) + sup t≥T v det (t) L q 1 ≤ ε ∧ η 4 . (6.62) Without loss of generality we may assume S ≤ T and ν 0 ≤ µ 0 .