Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over R^d and in L^p-spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous second-order elliptic operator with unbounded coefficients defined in IxR^d, (I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem.


Introduction
This paper is devoted to continuing the analysis started in [5]. We consider a family of linear second-order differential operators A(t) acting on smooth functions ζ as where I is either an open right halfline or the whole ℝ. Then, given T > s ∈ I, we are interested in studying the nonlinear Cauchy problem where ψ u (t, x) = ψ (t, x, u(t, x), ∇ x u(t, x)). We assume that the coefficients q ij and b i (i, j = 1, . . . , d), possibly unbounded, are smooth enough, the diffusion matrix Q = [q ij ] i,j=1,...,d is uniformly elliptic and there exists a Lyapunov function φ for A(t) (see Hypothesis 2.1 (iii)). These assumptions yield that the linear part A(t) generates a linear evolution operator {G(t, s) : t ≥ s ∈ I} in C b (ℝ d ). More precisely, for every f ∈ C b (ℝ d ) and s ∈ I, the function G( ⋅ , s)f belongs to C b ([s, +∞) × ℝ d ) ∩ C 1,2 ((s, +∞) × ℝ d ), it is the unique bounded (being equivalent to the restriction of the Lebesgue measure to the Borel σ-algebra in ℝ d ), the corresponding L p -spaces differ. Formula (1.5) and the density of C b (ℝ d ) in L p (ℝ d , μ s ) allow to extend G(t, s) to a contraction from L p (ℝ d , μ s ) to L p (ℝ d , μ t ) for any t > s and any p ∈ [1, +∞) and to prove very nice properties of G(t, s) in these spaces.
In view of these facts, it is significant to extend N(t, s) to an operator from L p (ℝ d , μ s ) to L p (ℝ d , μ t ) for any I ∋ s < t. This can be done if p ≥ p 0 (see Hypothesis 2.1 (v)), ψ(t, x, ⋅ , ⋅ ) is Lipschitz continuous in ℝ d+1 uniformly with respect to (t, x) ∈ (s, T] × ℝ d and, in addition, sup t∈(s,T] √ t − s‖ψ(t, ⋅ , 0, 0)‖ L p (ℝ d ,μ t ) < +∞. In particular, each operator N(t, s) is continuous from L p (ℝ d , μ s ) to W 1,p (ℝ d , μ t ). We stress that the first condition on ψ may seem too restrictive, but in fact it is not. Indeed, the Sobolev embedding theorems fail to hold, in general, when the Lebesgue measure is replaced by any of the measures μ t . This can be easily seen in the particular case of the one-dimensional Ornstein-Uhlenbeck operator, where the evolution system of measures is replaced by a time-independent measure μ (the so-called invariant measure), which is the Gaussian centered at zero with covariance 1/2. For any ε > 0, the function x → exp(2(2p + ε) −1 |x| 2 ) belongs to W k,p (ℝ, μ) for any k ∈ ℕ but it does not belong to L p+ε (ℝ d , μ). Under the previous assumptions, for any f ∈ L p (ℝ d , μ s ), N( ⋅ , s) can be identified with the unique mild solution to problem (1.2) which belongs to L p ((s, T) × ℝ d , μ) ∩ W Since, as it has been stressed, in this context the Sobolev embedding theorems fail to hold in general, the summability improving properties of the nonlinear evolution operator N(t, s) are not immediate and true in all the cases. For this reason in Section 4 we investigate properties such as hypercontractivity, supercontractivity, ultraboundedness of the evolution operator N(t, s) and its spatial gradient. Differently from [5], where ψ = ψ(t, u) and the hypercontractivity of N(t, s) is proved assuming ψ(t, 0) = 0 for any t > s, here we consider a more general case. More precisely we assume that there exist ξ 0 ≥ 0 and ξ 1 , ξ 2 ∈ ℝ such that uψ(t, x, u, v) ≤ ξ 0 |u| + ξ 1 u 2 + ξ 2 |u||v| for any t ≥ s, x, v ∈ ℝ d , u ∈ ℝ. Under some other technical assumptions on the growth of the coefficients q ij and b i (i, j = 1, . . . , d) as |x| → +∞, we show that as in the linear case, (see [3,4]), the hypercontractivity and the supercontractivity of N(t, s) and ∇ x N(t, s) are related to some logarithmic Sobolev inequalities with respect to the tight system {μ t : t ∈ I}. These estimates are the natural counterpart of the Sobolev embedding theorems in the context of invariant measures and evolution systems of measures.
For what concerns the ultraboundedness of N(t, s) and ∇ x N(t, s) we first prove an Harnack-type estimate which establishes a pointwise estimate of |N(t, s)f| p in terms of G(t, s)|f| p for any f ∈ C b (ℝ d ), p > p 0 and t > s. This estimate, together with the evolution law and the ultraboundedness of G(t, s), allow us to conclude that, for any f ∈ L p (ℝ d , μ s ) and any t > s, the function N(t, s)f belongs to W 1,∞ (ℝ d , μ t ) and to prove an estimate of ‖N(t, s)f‖ W 1,∞ (ℝ d ,μ t ) in terms of ‖f‖ L p (ℝ d ,μ s ) . Finally, assuming that ψ(t, x, 0, 0) = 0 for every t ∈ (s, +∞) and x ∈ ℝ d , we prove that the trivial solution to the Cauchy problem (1.2) is exponentially stable both in W 1,p (ℝ d , μ t ) and in C 1 b (ℝ d ). This means that ‖u f (t, ⋅ )‖ X ≤ C X e −ω X t as t → +∞ for some constants C X > 0 and ω X < 0, both when X = W 1,p (ℝ d , μ t ) and In the first case, the space X depends itself on t. We stress that, under sufficient conditions on the coefficients of the operators A(t), which include their convergence at infinity, in [2,11] it has been proved that the measure μ t weakly * converges to a measure μ, which turns out the invariant measure of the operator A ∞ , whose coefficients are the limit as t → +∞ of the coefficients of the operator A(t). This gives more information on the convergence to zero of ‖u f (t, ⋅ )‖ W 1,p (ℝ d ,μ t ) at infinity. We refer the reader also to [12] for the case of T-time periodic coefficients.
To get the exponential stability of the trivial solution in C b (ℝ d ), differently from [5] where a nonautonomous version of the principle of linearized stability is used and more restrictive assumptions on ψ are required, we let p tend to +∞ in the decay estimate of ‖u f (t, ⋅ )‖ W 1,p (ℝ d ,μ t ) , since all the constants appearing in this estimate admit finite limit as p tends to +∞. In particular, we stress that we do not need any additional assumptions on the differentiability of ψ but, on the other hand, we require that the mild solution u f of (1.2) is actually classical.

Notations
For k ≥ 0, by C k b (ℝ d ) we mean the space of the functions in C k (ℝ d ) which are bounded together with all their derivatives up to the [k]-th order.
where [k] denotes the integer part of k. When k ∉ ℕ, we use the subscript "loc" to denote the space of all ; Lip(ℝ d+1 )) and C α/2,α (J × ℝ d ) (α ∈ (0, 1)), respectively, the set of all functions f : , uniformly with respect to (t, x) ∈ J × ℝ d , and the usual parabolic Hölder space. The subscript "loc" has the same meaning as above.
We use the symbols D t f , D i f and D ij f to denote respectively the time derivative ∂f ∂t and the spatial derivatives ∂f ∂x i and ∂ 2 f ∂x i ∂x j for any i, j = 1, . . . , d. The open ball in ℝ d centered at 0 with radius r > 0 and its closure are denoted by B r and B r , respectively. For any measurable set A, contained in ℝ or in ℝ d , we denote by A the characteristic function of A. Finally, we write A ⋐ B when A is compactly contained in B.

Assumptions and preliminary results
Let {A(t) : t ∈ I} be the family of linear second-order differential operators defined by (1.1). Hypotheses 2.1. Our standing assumptions on the coefficients of the operators A(t) are as follows.
x)] ij is symmetric and there exists a function with positive infimum κ 0 , such that for any (t, x) ∈ I × ℝ d and any ξ ∈ ℝ d . (iii) There exists a nonnegative function φ ∈ C 2 (ℝ d ), diverging to +∞ as |x| → +∞, such that for any (t, x) ∈ I × ℝ d and some positive constants a and c. (iv) There exists a locally bounded function ρ : ).
(2.1) Under Hypotheses 2.1 (i)-(iii) (actually even under weaker assumptions) it is possible to associate an evolution operator {G(t, s) : t ≥ s ∈ I} to the operator A(t) in C b (ℝ d ), as described in the Introduction. The function where p(t, s, x, dy) are probability measures for any I ∋ s < t, x, y ∈ ℝ d . This implies that for any I ∋ s < t, f ∈ C b (ℝ d ) and p ≥ 1. Moreover, Hypotheses 2.1 (iv) and (v) yield the pointwise gradient estimates , ε > 0 and some positive constants C 0 and C ε , where σ p is given by (2.1), with p instead of p 0 . We stress that the pointwise estimates (2.3) and (2.4) have been proved with the constants C 0 and C ε also depending of p. Actually, these constants may be taken independent of p. Indeed, consider for instance estimate (2.4). If p ≥ p 0 , then using the representation formula (2.2) we can Under Hypotheses 2.1 we can also associate an evolution system of measures {μ t : t ∈ I} with the operators A(t). Such a family of measures is tight, namely for every ε > 0 there exists r > 0 such that μ s (ℝ d \ B r ) < ε for any s ∈ I. The invariance property (1.5) and the density of for any t > s. As it has been stressed in the Introduction, in general evolution systems of measures are infinitely many, but, under suitable assumptions, there exists a unique tight evolution system of measures. This is, for instance, the case when Hypotheses 2.1 are satisfied as well as the following two conditions: (i) q ij and b i belong to C α/2,1+α loc ([a, +∞) × ℝ d ) for any i, j = 1, . . . , d and some a ∈ I. Moreover, q ij belongs for any (t, x) ∈ [a, +∞) × ℝ d , or the diffusion coefficients are bounded in [a, +∞) × ℝ d . For more details and the proofs of the results that we have mentioned, we refer the reader to [9][10][11]13].

The semilinear problem in a bounded time interval
Given I ∋ s < T, we are interested in studying the Cauchy problem (1.2) both in the case when f ∈ C b (ℝ d ) and in the case when f ∈ L p (ℝ d , μ s ). Hypotheses 3.1. Our standing assumptions on ψ are as follows.
(i) The function ψ : [s, T] × ℝ d × ℝ × ℝ d → ℝ is continuous. Moreover, there exists β ∈ [0, 1) such that for any R > 0 and some constant L R > 0 where C 0 is the constant in (2.4). Moreover, for any R > 0, θ ∈ (0, 1) and t ∈ (s, s+δ], u f (t, ⋅ ) belongs to C 1+θ (B R ) and there exists a positive constant C R,T−s such that Proof. Even if the proof is quite standard, for the reader's convenience we provide some details.
Step 1. We prove that there exists δ > 0 such that, for any f ∈ C b (ℝ d ) satisfying the condition there exists a mild solution to problem (1.2) defined in the time interval [s, s + δ]. For this purpose, we consider the operator Γ, defined by the right-hand side of (1.4) for any u ∈ B Y δ (R 0 ) (the ball of Y δ centered at zero with radius R 0 ). Clearly, the function ψ u is continuous in (s, s + δ] × ℝ d and ψ u (t, ⋅ ) is bounded in ℝ d for any t ∈ (s, s + δ]. Moreover, estimating |ψ u (t, x)| ≤ |ψ u (t, x) − ψ(t, x, 0, 0)| + |ψ(t, x, 0, 0)| and taking (3.1) into account, we can easily show that the function t → (t − s) β ‖ψ u (t, ⋅ )‖ ∞ is bounded in (s, s + δ). Hence, Proposition A.1 and estimates (1.3) and (2.4) show that Γ(u) ∈ Y δ for any t ∈ (s, s + δ] and u ∈ B Y δ (R 0 ). To show that, for a suitable δ ∈ (0, 1], Γ is a 1/2-contraction in B Y δ (R 0 ), we observe that, using again (3.1), it follows that , where c 1 , as the forthcoming constants, is independent of δ and u, if not otherwise specified. Hence, choosing δ properly, we can make Γ a 1/2-contraction in B Y δ (R 0 ). It is also straightforward to see that Γ maps B Y δ (R 0 ) into itself, up to replacing δ with a smaller value if needed. It suffices to split Γ(u) = (Γ(u) − Γ(0)) + Γ(0), use the previous result and estimate As a consequence, Γ has a unique fixed point in B Y δ (R 0 ), which is a mild solution of (1.2) and satisfies (3.2).
Step 2. We prove the uniqueness of the mild solution u f . For this purpose, let u 1 , u 2 ∈ Y δ be two mild solutions. By Lemma A.2, the function r → h(r) Using (3.5), we estimate the last two integral terms in the right-hand side of (3.6), which we denote by I(t) and J(t). Replacing (3.5), with j = 0, in I(t), we get The same arguments show that J(t) can be estimated pointwise in [s, s + δ] by the right-hand side of (3.7), with c 3 (M) being possibly replaced by a larger constant c 4 (M). Summing up, we have proved that The generalized Gronwall lemma (see [7]) yields h(t) ≡ 0 for any t ∈ (s, s + δ), i.e., u Step 3. We prove (3.2) and (3.3).
can be proved in the same way.
Step 4. We prove that for some constant c 7 , independent of f . For this purpose, we observe that the results in the previous steps show that the function ψ u satisfies the estimate (t − s) β ‖ψ u (t, ⋅ )‖ ∞ ≤ c 8 ‖f‖ ∞ for any t ∈ (s, s + δ], the constant c 8 being independent of f . Applying Proposition A.1 and estimate (A.5), we complete the proof. Corollary 3.3. In addition to the assumption of Theorem 3.2 suppose that there exist β ∈ [0, 1) and γ ∈ (0, 1) such that 2β + γ < 2 and for any t ∈ (s, T], x, y, v ∈ B R , u ∈ [−R, R], any R > 0 and some positive constant C R . Then, for any f ∈ C b (ℝ d ), the mild solution u f to problem (1.2) belongs to C 1,2 ((s, s + δ] × ℝ d ) and it is a classical solution to (1.2). and . From these estimates, adding and subtracting ψ(t, y, and some positive constantC, depending on R and u. Now, using Proposition A.1, we conclude Then, the proof of the previous theorem can be repeated verbatim with , endowed with the natural norm, and we can show that the mild solution to problem (1.2) belongs to C We now provide some sufficient conditions for the mild solution to problem (1.2) to exist in the large. Such conditions will be crucial to define the nonlinear evolution operator associated with the Cauchy problem (1.2).

Hypotheses 3.5.
We introduce the following assumptions.
(i) For any R > 0 there exists a positive constant L R such that For any τ > s ∈ I there exist positive constants k 0 , k 1 and a, and a functionφ ∈ C 2 (ℝ d ) with nonnegative values and blowing up at infinity such that In the rest of this section, for any p ∈ [p 0 , +∞) and T > s we denote by Proof. We split the proof into two steps.
Step 1. We prove that, for any Once this is proved, we can use Hypotheses 3.5 (i) to deduce, adding and subtracting ψ(t, Applying the same arguments as in Step 2 of the proof of Theorem 3.2, we can show that also the function t . This is enough to infer that u f can be extended beyond τ f , contradicting the maximality of the interval [s, τ f ). To x n ) ≥ 0, so that, multiplying both the sides of (3.12) by v n (t n , x n ) + n −1φ (x n ) > 0 and using Hypotheses 3.5 (ii) we get Repeating the same arguments with u f being replaced by −u f , we conclude that u f is bounded also from below by a positive constant independent of b. Since b is arbitrary, it follows that Step 2. Fix f, g ∈ C b (ℝ d ), p ≥ p 0 and let ||| ⋅ ||| p be the norm defined by on smooth functions v, where ω is a positive constant to be chosen later on and to fix the ideas we assume that p < +∞. From Hypothesis 3.5 (i), where L R is replaced by a constant L, it follows that for any r ∈ (s, T]. Hence, recalling that each operator G(t, r) is a contraction from L p (ℝ d , μ r ) to L p (ℝ d , μ t ) and using the second pointwise gradient estimate in (2.4) and the invariance property of the family {μ t : t ∈ I}, we conclude that To estimate the integral terms in the last side of (3.13), we fix δ > 0 and observe that Hence, minimizing over δ > 0, we conclude that the left-hand side of estimate (3.14) is bounded from above by √ 8ω −1/2 . Splitting √ t − s ≤ √ t − r + √r − s and arguing as above, also the last term in square brackets in the last side of (3.13) can be estimated by ( √ 8 + √π)ω −1/2 . It thus follows that and estimate (3.11) follows at once. Estimate (3.10) can be proved likewise. Hence, the details are omitted.
As a consequence of Theorem 3.6 we prove the existence of a mild solution to problem (1.2) in the time domain , μ t ) for almost every t ∈ (s, T] and, for such values of t, the equality where A t is negligible with respect to the measure μ t (or, equivalently, with respect to the restriction of the Lebesgue measure to the Borel σ-algebra in ℝ d ). Proof. Fix f ∈ L p (ℝ d , μ s ) and let (f n ) ⊂ C b (ℝ d ) be a sequence converging to f in L p (ℝ d , μ s ). By (3.11), (u f n (t, ⋅ )) is a Cauchy sequence in W 1,p (ℝ d , μ t ) for any t ∈ (s, T]. Hence, there exists a function v such that u f n (t, ⋅ ) converges to v(t, ⋅ ) in W 1,p (ℝ d , μ t ) for any t ∈ (s, T]. Moreover, writing (3.10), with f being replaced by f n , and letting n tend to +∞ we deduce that v satisfies (3.10) as well.
Next, using (3.11) we can estimate and for any q ∈ [1, 2) if p ≥ 2 and for p = q otherwise. Hence, recalling that for almost every t ∈ (s, T). Letting n tend to +∞ in formula (1.4), with f n replacing f , we deduce that u f is a mild solution to problem (1.2). The uniqueness follows, arguing as in the proof of Theorem 3.2 with the obvious changes.
Let us now prove the last part of the statement. We again use an approximation argument. Fix t > s ∈ I and R > 0. At a first step, we estimate the norm of the operator first for any f ∈ C b (ℝ d ), and then, by density, for any f ∈ L p (ℝ d , μ r ). Since, for θ ∈ (0, 1), with equivalence of the corresponding norms, by an interpolation argument and (3.16) we deduce that for any s < r < t < T. Hence, if for any n ∈ ℕ we consider the function z n , which is the integral term in (1.4), with u being replaced by u f n , and use (3.11) and the fact that for any n ∈ ℕ. We have so proved that, for any θ ∈ (0, 1) and almost every t ∈ (s, T], the function u f (t, ⋅ ) belongs to W 1+θ (B R+1 ) and Similarly, Using these estimates, we can now show that ψ u f (r, ⋅ ) ∈ W θ,p (B R+1 ), for any θ < γ. For this purpose, we add and subtract ψ(t, y, u f n (t, x), ∇ x u f n (t, x)), use condition (3.15) and the Lipschitz continuity of ψ with respect to the last two variables to infer that for any t ∈ (s, T), x, y ∈ ℝ d and m, n ∈ ℕ. Hence, using (3.17) we obtain and, using (3.18), From these two estimates we conclude that for any (t, x) ∈ (s, T) × ℝ d , any β ∈ (0, 1) and any m, n ∈ ℕ. Hence, for any θ < γ and β, such that (0, 1) ∋ θ = θ/β + d(1 − β)/(pβ), a long but straightforward computation reveals that and, consequently, for any t ∈ (s, T). We are almost done. Indeed, by interpolation from Proposition A.3 we deduce that . From this and the previous estimate we conclude that for any t ∈ (s, T] and β > θ , so that for any m, n ∈ ℕ, thanks to (3.16). From this estimate it is easy to deduce that (u f n ) is a Cauchy sequence in W 0,2 p,loc ((s, T) × ℝ d ). Since u f n is a classical solution to problem (1.2), we conclude that (D t u f n ) is a Cauchy sequence in L p loc ((s, T) × ℝ d ). It thus follows that u f ∈ W 1,2 p,loc ((s, T) × ℝ d ) and it solves the equation The arguments in the proof of Theorem 3.6 and Corollary 3.7 allow us to prove the following result. Further, u f satisfies (3.10) and (3.11), with the supremum being replaced by the essential supremum.
Proof. To prove property (i), it suffices to apply the Banach fixed point theorem in the space of all the func- Step 2 of the proof of Theorem 3.6, with p = +∞. The uniqueness of the so obtained solution follows from the condition Lip(ℝ d+1 ) < +∞, in a standard way. To prove property (ii), one can argue by approximation. We fix f ∈ L p (ℝ d , μ s ), approximate it by a sequence (f n ) ⊂ C b (ℝ d ), converging to f in L p (ℝ d , μ s ), and introducing a standard sequence (ϑ n ) of cut-off functions. If we set ψ n = ϑ n ψ for any n ∈ ℕ, then each function ψ n satisfies the assumptions in property (i) and

The evolution operator and its summability improving properties
Suppose that, besides Hypotheses 2.1, the assumptions on ψ in Theorem 3.6 hold true for any I ∋ s < T or ψ ∈ C(I × ℝ d × ℝ × ℝ d ) ∩ B(J × ℝ d ; Lip(ℝ d+1 )) for each J ⋐ I and ψ( ⋅ , ⋅ , 0, 0) ∈ C b (ℝ d+1 ). Then, for any f ∈ C b (ℝ d ) and s ∈ I the mild solution to problem (1.2) exists in the whole of [s, +∞). Hence, we can set N(t, s)f = u f (t, ⋅ ) for any t > s. Each operator N(t, s) maps Moreover, the uniqueness of the solution to problem (1.2) yields the evolution law N(t, s)f = N(t, r)N(r, s)f for any r ∈ (s, t) and f ∈ C b (ℝ d ). Hence {N(t, s) : I ∋ s < t} is a nonlinear evolution operator in C b (ℝ d ). It can be extended to the L p -setting, for any p ≥ p 0 , using the same arguments as in the first part of the proof of Corollary 3.7. Clearly, if ψ(t, x, ⋅ , ⋅ ) is Lipschitz continuous in ℝ d+1 , uniformly with respect to (t, x) × J × ℝ d , for any J ⋐ I, then by density, we still deduce that N(t, s) satisfies the evolution law and, moreover, each operator N(t, s) is bounded from L p (ℝ d , μ s ) to W 1,p (ℝ d , μ t ) and

Continuity properties of the nonlinear evolution operator
In the following theorem, assuming the above conditions on ψ, we prove an interesting continuity property of the operator N(t, s). Proof. Let (f n ) and f be as in the statement. To ease the notation, we write u f n and u f for N( ⋅ , s)f n and N( ⋅ , s)f , respectively. Moreover, we set h n (r, ⋅ ) = G(t, r)(|u f n (r, ⋅ ) − u f (r, ⋅ )| p + |∇ x (u f n (r, ⋅ ) − u f (r, ⋅ ))| p ) for any n ∈ ℕ, t > s and r ∈ (s, t], and we denote by L R,T any constant such that and T > 0. As a first step, formula (3.2) shows that, for any T > 0, there exists a positive constant M T such that ‖u f ‖ ∞ + ‖u f n ‖ ∞ ≤ M T . Fix p ∈ (1, 2). Using formula (1.4), we can estimate x G(t, r)(ψ u fn (r, ⋅ ) − ψ u f (r, ⋅ ))(x) dr p for any (t, x) ∈ (s, +∞) × ℝ d and j = 0, 1. By the representation formula (2.2), Hölder inequality, estimates (2.4) and (4.2), we deduce that for any t ∈ (s, s + T) and some positive constant c T . Hence, the function h n ( ⋅ , x) satisfies the differential inequality for any t ∈ (s, s + T) and x ∈ ℝ d . Since h n ( ⋅ , x) is continuous in (s, t] and h n (r, x) ≤C T (r − s) −p/2 for some positive constantC T , independent of n, and any r ∈ (s, t), we can apply [8, Lemma 7.1] and conclude that for any t ∈ (s, s + T). Hence, for any R > 0. By [9, Proposition 3.1 (i)], ‖G(r, s)|f n − f| p ‖ C b (B R ) vanishes as n → +∞ for any r > s. Hence, by dominated convergence, ‖h n (t, ⋅ )‖ C b (B R ) vanishes as n → +∞ for any t ∈ (s, s + T), which means that, for any t ∈ (s, s + T), u f n (t, ⋅ ) and ∇ x u f n (t, ⋅ ) converge uniformly in B R to u f (t, ⋅ ) and ∇ x u f n (t, ⋅ ), respectively. The arbitrariness of R and T yields the assertion.

Hypercontractivity
Throughout this and the forthcoming subsections we set for any smooth enough function ζ . To begin with, we recall the following crucial result.

Hypotheses 4.3.
We introduce the following assumptions.
, for any T > s ∈ I and some constant which may depend also on s and T, and there exist two constants ξ 0 ≥ 0 and ξ 1 such that uψ(t, x, u, v) ≤ ξ 0 |u| + ξ 1 u 2 + ξ 2 |u||v| for any t ≥ s, x, v ∈ ℝ d and u ∈ ℝ. (ii) There exists a nonnegative functionφ : ℝ d → ℝ, blowing up at infinity such that Aφ + k 1 |∇φ| ≤ aφ in ℝ d for some locally bounded functions a, k 1 . (iii) There exist locally bounded functions C 0 , C 1 , C 2 : for any t ∈ I and any x ∈ ℝ d . (iv) There exists a positive constant K such that for any t > s, f ∈ C 1 b (ℝ d ) and q ∈ (1, +∞).
We can now prove the main result of this subsection. Proof. To begin with, we observe that it suffices to prove (4.4) and (4.5) for functions f ∈ C 1 b (ℝ d ). Indeed, in the general case, the assertion follows approximating f with a sequence (f n ) ⊂ C 1 b (ℝ d ) which converges to f in L p (ℝ d , μ s ). By (3.10), N(t, s)f n converges to N(t, s)f in W 1,p (ℝ d , μ t ) for almost every t > s. Hence, writing (4.4) and (4.5) with f being replaced by f n and letting n tend to +∞, the assertion follows at once by applying Fatou lemma.

μ t ) and satisfies the estimates
We split the rest of the proof into two steps. In the first one we prove (4.4) and in the latter one (4.5).

Supercontractivity
In the next theorem we prove a stronger result than Theorem 4.5, i.e., we prove that the nonlinear evolution operator N(t, s) satisfies a supercontractivity property. For this purpose, we introduce the following additional assumption. Hypothesis 4.7. There exists a decreasing function ν : (0, +∞) → ℝ + blowing up as σ tends to 0 + such that for any r ∈ I, σ > 0 and f ∈ C 1 b (ℝ d ).
Remark 4.8. Sufficient conditions for (4.12) to hold are given in [3]. In particular, it holds true when (2.3) is satisfied with p = 1 (see Remark 2.2) and there exist K > 0 and R > 1 such that ⟨b(t, x), x⟩ ≤ −K|x| 2 log |x| for any t ∈ I and |x| ≥ R. Theorem 4.9. Let Hypotheses 2.1, 4.3 (i)-(iii) and 4.7 be satisfied. Then, for any t > s ∈ I, p 0 ≤ p < q < +∞ and any f ∈ L p (ℝ d , μ s ), N(t, s)f belongs to W 1,q (ℝ d , μ t ) and (4.14) Here, c 2 , c 3 , c 4 : (0, +∞) → ℝ + are continuous functions such that lim r→0 + c k (r) = +∞ (k = 2, 3, 4). Proof. The proof of this result follows the same lines of the proof of Theorem 4.5. For this reason we use the notation therein introduced and we limit ourselves to sketching it in the case when f ∈ C 1 b (ℝ d ).
Step 2. Fix q > p. By Step 1, N((t + s)/2, s)f belongs to L q (ℝ d , μ (t+s)/2 ) and The same arguments used in Step 2 of the proof of Theorem 4.5 show that N(t, s)f ∈ W 1,q (ℝ d , μ τ ) for any τ > t+s 2 and Estimate (4.14) follows with c

Stability of the null solution
In this section we study the stability of the null solution to problem (1.2) both in the C b -and L p -settings. For this reason, we assume that ψ( ⋅ , ⋅ , 0, 0) = 0. Proof. (i) Estimate (5.1) can be obtained arguing as in the proof of Theorem 4.5, where now p(t) = p for any t ≥ s. As far as the gradient of N(t, s)f is concerned, we fix t > s + 1 and observe that N(t, s)f = N(t, t − 1)N(t − 1, s)f .

Examples
Here, we exhibit some classes of nonautonomous elliptic operators and some classes of nonlinear functions ψ which satisfy the assumptions of this paper.
‖g(t, ⋅ )‖ ∞ for any t ∈ J. Clearly, for any fixed t ∈ J, the sequence (z n (t)) is increasing and is bounded from above by ‖g(t, ⋅ )‖ ∞ . To prove that (z n (t)) converges to ‖g(t, ⋅ )‖ ∞ , we fix a sequence (x n ) ⊂ ℝ d such that |g(t, x n )| tends to ‖g(t, ⋅ )‖ ∞ as n → +∞. For any n ∈ ℕ, let k n ∈ ℕ be such that x n ∈ B k n . Without loss of generality, we can assume that the sequence (k n ) is increasing. Then z k n (t) = ‖g(t, ⋅ )‖ C(B kn ) ≥ |g(t, x n )| for any n ∈ ℕ. Hence, the sequence (z k n (t)) converges to ‖g(t, ⋅ )‖ ∞ and this is enough to conclude that the whole sequence (z n (t)) converges to ‖g(t, ⋅ )‖ ∞ as n → +∞.

Funding:
The authors are members of GNAMPA of the Italian Istituto Nazionale di Alta Matematica. This work has been supported by the INdAM-GNAMPA Project 2016 "Equazioni e sistemi di equazioni ellittiche e paraboliche associate ad operatori con coefficienti illimitati e discontinui".