Ultracontractivity and Gaussian bounds for evolution families associated with non-autonomous forms

We develop a variational approach in order to study qualitative properties of non-autonomous parabolic equations. Based on the method of product integrals, we discuss invariance properties and ultracontractivity of evolution families in Hilbert space. Our main results give sufficient conditions for the heat kernel of the evolution family to satisfy Gaussian-type bounds. Along the way, we study examples of non-autonomous equations on graphs, metric graphs, and domains.


Introduction
Non-autonomous evolution equations are partial differential equations in which relevant coefficients of the differential operator and/or in the boundary conditions are time-dependent, thus allowing for underlying models that are variable over time.
In the autonomous case (i.e., evolution equations with time-independent coefficients), well-posedness is known to be equivalent to generation of a semigroup in a suitable Banach space; in comparison, the theory for well-posedness of non-autonomous problems on general Banach spaces is more rudimentary. If the coefficients of a non-autonomous equation are piecewise constant, then one may find a solution by following the orbit of the semigroup governing a given problem as long as the coefficient stays constant; then "freeze" the system; use the final state as an initial condition for a new evolution equation with new (constant) coefficient, and so on: this boils down to consider the composition of a finite numbers of semigroups.
A theory originally developed by J.-L. Lions shows that well-posedness in Hilbert space can be proved under much weaker assumptions, most notably mere measurability of the time dependence, provided the problem has a nice variational structure: this is typically the case if the differential equation is parabolic. By adapting the setting of (time-independent) bounded elliptic forms it is thus possible to show that the equation has a solution that is, in particular, continuous in time, cf. Theorem 2.1 for a precise statement. This motivates the study of non-autonomous forms, a topic which has received much attention in the last decade: we mention among others [ADLO14,AD17,ADF17,Ouh15,Fac17]. In all these articles, the focus lies on properties of solutions of partial differential equations, with a focus on maximal regularity issues and hence allowing for inhomogeneous terms.
Our main aim in this paper is to develop a more abstract theory with an operator theoretical flavor. Indeed, Lions' result paves the way to the possibility of defining an evolution family (or evolution system, or propagator ), i.e., a family of operators U (·, s) mapping each initial data u(s) = x ∈ H to the orbit of the solutionu (t) + A(t)u(t) = 0 a.e. on [s, T ] . Because the initial condition may well be imposed at instants s = 0, this actually define a twoparameter family U := (U (t, s)) (t,s)∈∆ of bounded linear operators on H by U (t, s)x := u(t), where ∆ := {(t, s) ∈ (0, T ) 2 : s < t}.
Some good compendia on such evolution families are [Tan79,Chapt. 7], [Paz83,Chapt. 5], [Fat83,Chapt. 7], [EN00, Section VI.9], or the monograph [CL99]. The tumultuous development of Hilbert space methods, and especially the theory of Dirichlet forms, have been fruitful also in the non-autonomous environment: a theory of non-autonomous Dirichlet forms has been recently introduced in [ADO14]. If A(t) ≡ A, the above abstract Cauchy problem is autonomous and its solution is simply given by hence the findings in [ADO14] can be regarded as a strict generalization of the classical theory of Markovian operators and Dirichlet forms represented e.g. in [FOT10]. Our goal is to complement these results, thus setting up a non-autonomous variational program analogous to the autononomous one outlined in classical monographs like [Ouh05,Are06]: to this aim we study further operator theoretical properties, including extrapolation to L p -spaces, ultracontractivity, or Gaussian-type bounds on integral kernels of evolution families. It should be mentioned that ultracontractivity and kernel estimates have been observed already in [Aro68,Dan00] for specific instances of parabolic non-autonomous equations; in particular, Aronson observed in [Aro67] that the fundamental solution (t, s; x, y) → Γ(t, s; x, y) of a certain class of non-autonomous diffusion equations in (domains of) R d satisfies (1.1) Γ(t, s; x, y) ≤ K G(t − s; x − y) where (t, x) → G(t, x) is the Gaussian kernel that yields the fundamental solution of the (autonomous) heat equation on R d . Analogous Gaussian bounds have ever since been proved for integral kernels of semigroups generated by large classes of second-order elliptic operators, possibly with complex coefficients [Ouh04]; in the non-autonomous case, Aronson's original findings have been extended to operators on domains under Dirichlet or Robin boundary conditions in [Aro68,Dan00].
In this paper we are going to introduce a general approach, based on the so-called Davies' Trick, to prove Gaussian bounds for heat kernels of evolution families that govern non-autonomous parabolic equations. Inspired by some techniques introduced in [Dan00,Ouh04], we show the applicability of our methods by showing that a large class of elliptic operators with complex-valued, bounded measurable (both in time and space) coefficients are associated with evolution families that satisfy Gaussian bounds, thus extending the main results in [Ouh04] to the non-autonomous setting.
Our approach will heavily rely upon the method of product integrals, whose historical evolution is thoroughly discussed in [Sla07], [Fat83,§ 7.10] and whose scope has been extended to nonautonomous forms with measurable dependence on time in [EL16,SL15]. We adapt it to our present setting, thus deriving in Theorem 2.5 a version that we will use over and over again in different contexts throughout this paper.
The present paper is organized as follows. After describing our mathematical framework in Section 2, in Section 3 we present sufficient conditions that enforce qualitative properties based on the lattice structure of L 2 -spaces, including stochasticity or domination. Also, we are able to discuss cases where evolution families on L 2 -spaces extrapolate to further L p -spaces. This is a key feature of the theory of Dirichlet forms in the autonomous case and shows the flexibility of the Hilbert space approach in the non-autonomous context, too. Even evolution equations on structures that change over time can be studied by means of our theory.
Gaussian-type bounds are shown to depend on ultracontractivity properties of certain operator families related to U . This approach requires, in turn, suitable common bounds in L p -norm, uniformly on all compact subsets of ∆. Inspired by similar criteria in the autonomous setting we show that efficient conditions based on Sobolev-type inequalities can enforce such bounds. In Section 4 we develop a theory of ultracontractive evolution families: a technical difficulty we face is related to the failure of self-adjointness of evolution families, a phenomenon that typically occurs even when all operators A(t) are self-adjoint. We take over an idea from [Dan00] and circumvent this problem by studying some non-autonomous form associated with a tightly related backward evolution equation.
It has been known since [Dav87] that ultracontractivity is an important ingredient to prove Gaussian bounds for semigroups. Expanding the scope of Davies' trick, in Section 5 we are going to present different sufficient conditions under which an evolution family satisfies Gaussian bounds.
In particular, our approach allows us to show Gaussian bounds for the evolution family associated with a large class of elliptic operators, thus generalizing the pioneering results in [Aro68,Dan00].
Several applications of increasing complexity are reviewed in Section 6: we discuss well-posedness and qualitative properties of dynamical systems on undirected graphs tightly related to the theory of dynamic (positive) graphs discussed in [Šil08] as well as models of Black-Scholes-types equations with time-dependent volatility [Hes93]; we extend the kernel estimates in [Mug07] to more general non-autonomous diffusion equations on possibly infinite networks; and finally, we prove Gaussian bounds for the heat kernel for a large class of elliptic operators with time-dependent, possibly complex coefficients, thus deducing the main results in [Dan00,Ouh04] as special cases.

Evolution families: Notations and preliminary results
Throughout this paper H is a separable, complex Hilbert space and V is a further complex Hilbert space that is densely and continuously embedded into H. Let V ′ denote the antidual of V with respect to the pivot space H; the duality between V ′ and V is denoted by ., . . We also denote by (· | ·) V and · V the scalar product and the norm on V , respectively; and by (· | ·) and · the corresponding quantities in H.
We fix T ∈]0, ∞[ and consider a time-dependent family (a(t)) t∈[0,T ] of mappings such that a(t; ·, ·) : and such that furthermore there exist constants M, α > 0 and ω ≥ 0 such that the boundedness and H-ellipticity estimates for a.e t ∈ [0, T ] and u ∈ V, (2.3) hold. In what follows we call such a family a := (a(t)) t∈[0,T ] bounded H-elliptic non-autonomous form: following [AD17] we denote by Form([0, T ]; V, H) the class of all such forms.
By the Lax-Milgram theorem, for each t ∈ [0, T ] there exists an operator associated with a(t, ·, ·), i.e., an isomorphism A(t) : V → V ′ such that accordingly we refer to the family (A(t)) t∈[0,T ] as the operator family associated with a := (a(t)) t∈[0,T ] . Regarded as an unbounded operator with domain V , −A(t) generates a holomorphic semigroup on V ′ , and in fact by [Are06, Thm. 7.1.5] on H too, since a(t) is for all t a bounded, H-elliptic sesquilinear form: with an abuse of notation we denote its generator -the part of −A(t) in Hagain by −A(t), and the semigroup by Hence, for each fixed t, s ∈ [0, T ] the Cauchy probleṁ is well-posed, its solution being given by u(r, x) := e −(r−s)A(t) x. However, we are rather going to focus on the non-autonomous Cauchy probleṁ The Hilbert space setting discussed above is rather benign and we can combine a few known results to observe the following non-autonomous counterpart of the Lumer-Phillips Theorem. Throughout this article we adopt the notation ∆ := {(t, s) ∈ (0, T ) 2 : s < t}.
The following result has already been proved under slightly stronger assumptions in [Laa18, Prop. 2.1]. We re-formulate it for the sake of self-containedness.
defines a strongly continuous evolution family on H, i.e., the following properties hold: In the following we will refer to U := (U (t, s)) (t,s)∈∆ as the evolution family associated with the non-autonomous form a or with the operator family (A(t)) t∈[0,T ] .
where u is the unique solution of the Cauchy problem (2.4) in M R (V, V ′ ). The proof of (i)− ( has for all s > 0 and all x ∈ H a unique solution u ∈ L 2 this defines a strongly continuous evolution family (U (t, s)) 0≤s≤t<∞ .
Remark 2.3. It follows from the above mentioned result by Lions that U (t, s) maps H into V for all s ∈ [0, T ] and almost all t ∈ [0, T ] with (t, s) ∈ ∆. Thus using the evolution family law (ii) we deduce that U (t, s) is a compact operator on H for all (t, s) ∈ ∆ provided V is compactly embedded in H. Likewise, if the embedding of V in H is of p-Schatten class for some p ∈ [1, ∞[, then U (t, s) is of p-Schatten class -hence by (ii) of trace class -for all (t, s) ∈ ∆.
Remark 2.4. A non-autonomous form is called coercive if (2.3) is satisfied with ω = 0. Now observe that a satisfies (2.3) if and only if the form a ω given by is coercive: because a ω ∈ Form([0, T ]; V, H) in its own right, it is associated with an evolution family. Moreover, u is a solution of class M R(V, V ′ ) of (2.4) if and only if v := e −ω(.−s) u is a solution of class M R(V, V ′ ) ofv Thus, the evolution family associated with a ω is simply obtained by rescaling, i.e., The earliest well-posedness results for (2.4) were obtained by Kato based on an approximation method based on the theory of product integrals under strong regularity assumptions on the dependence t → a(t). Kato's approach has been extended to non-autonomous form of class Form([0, T ]; V, H) in [LE13,EL16]; this has, in turn, allowed for a new proof of Lions' Theorem 2.1 in [SL15]. Let us sketch now his approach here for the sake of self-containedness, since we are going to use it repeatedly in the next sections.
All these forms lie in Form([0, T ]; V, H) with constants M , α, and ω. The associated operators A k ∈ L(V, V ′ ) are given by The mapping A(·) : [0, T ] → L(V, V ′ ) is strongly measurable by Pettis' Theorem [ABHN01, Thm. 1.1.1] since t → A(t)u is weakly measurable and V ′ is assumed to be separable. On the other hand, Hence the integrals in (2.7) and (2.8) are well defined.

Its associated time dependent operator family
For each k = 0, 1, . . . , n we denote by we define the operator families U Λ := (U Λ (t, s)) (t,s)∈∆ ⊂ L(V ′ ) by and for λ l−1 ≤ a ≤ b < λ l by Remark that U Λ defines an evolution family on H (as well as on V ′ and V ), since all semigroups T k consist of bounded linear operators on H. Additionally, one sees that the conditions (2.1)-(2.3) are satisfied by the forms a Λ , too: hence for all x ∈ H also the function u Λ (·) := U Λ (·, s)x is the unique solution of class M R(s, T ; V, V ′ ) of the probleṁ More precisely, let x ∈ H. On the interval [λ 0 , λ 1 [, the function u Λ coincides with the solution of the autonomous problemu which belongs to M R(λ 0 , λ 1 ; V, V ′ ) ֒→ C([λ 0 , λ 1 ]; H).
Next, for each k = 1, 2, . . . n, the restriction u Λ |[λ k ,λ k+1 [ coincides with the solution of the autonomous problemu which belongs to M R(λ k , λ k+1 ; V, V ′ ). We conclude that U Λ (·, s) ∈ M R(s, T ; V, V ′ ) and is the unique solution of (2.14). A similar approximation scheme was introduced in [EL16] in the more general context of inhomogeneous non-autonomous problems; several convergence results could be deduced there, depending on conditions satisfied by the non-autonomous form. In the language of evolution families, we can paraphrase the main result in [EL16] and state the following.
If y ∈ MR (V, V ′ ) then y(·) 2 H ∈ W 1,1 (s, T ; V ′ ) and (2.15) d dt y(·) 2 H = 2 Re ẏ(·), y(·) by [Sho97, Prop. III.1.2]: accordingly, for a.e. t ∈ [s, T ]. Integrating this equality between s and t and using (2.3) we obtain This estimate and Young's inequality yield the estimate . The term of the right-hand side of this inequality converges to 0 as |Λ| → 0. It follows that u Λ → u in L 2 (s, T ; V ). Again by [LE13, Lemma 2.3 and Lemma 3.1], A Λ u Λ → Au in L 2 (s, T ; V ′ ). Letting While all strongly continuous semigroups are exponentially bounded, this is not the case for general evolution families, cf. [EN00, § VI.9]. Evolution families associated with non-autonomous forms are rather special, though. Let us show how to apply the product integral method to deduce two known results about long-time behavior of evolution families: the assertion about quasicontractivity is [Laa18, Prop. 2.1], whereas strong stability was proved by similar means in a special case in [ADKF14,Thm. 5.4].
Proposition 2.6. Let a ∈ Form([0, T ]; V, H). Then the associated evolution family U is quasicontractive, i.e., whereω is the mean value over [s, t] of some ω ∈ L 1 (0, T ) such that If in particular a ∈ Form([0, ∞[; V, H) and t t 0 ω(s) ds < 0 for some t 0 and all t > t 0 , then U is uniformly exponentially stable.
In view of Proposition 2.6.(1) and Remark 2.4, we will often assume without loss of generality that ω ≡ 0, and thus that an evolution family is contractive up to a scalar perturbation of the family (A(t)) t∈[0,T ] .
Proof. Let [a, b] ⊂ [0, T ], Λ be a partition of [a, b] as in (2.11) and consider the discretized evolution family U Λ . Let ω k ∈ R be defined by Then by definition of a k in (2.7), (2.16) implies Re a k (u, u) + ω k u 2 H ≥ 0 for all u ∈ V and k = 0, 1, · · · , n, hence the associated semigroup satisfies The claim now follows from Theorem 2.5 and Fatou's Lemma.
We conclude this section by formulating a perturbation lemma that will prove useful will discussing concrete examples.
Lemma 2.7. Let α ∈ (0, 1) and let H α be some normed space such that V ֒→ H α ֒→ H and such that additionally This extends [Mug14, Lemma 6.22] to the case of non-autonomous forms and can be proved likewise.

Invariance Properties
Let us discuss invariance of a given subset C of H under U , i.e., whether u(s) ∈ C implies that the solution u(t) of (2.4) lies in C for any (t, s) ∈ ∆. The following criterion is known: it combines [SL15,Thm. 4 Proposition 3.1. Let a ∈ Form([0, T ]; V, H). Let C be a closed convex subset of H and denote by P the projector of H onto C. Consider the following assertions: (ii) C is invariant under the evolution family U .
(iii) P u ∈ V and Re a(t; P u, u − P u) ≥ 0 for all u ∈ V and a.e. t ≥ 0; Then (i) is equivalent to (iii) and both imply (ii).
The implication (iii) ⇒ (ii) has been proved in [ADO14, Thm. 2.2] in the more general case of inhomogeneous equations. Special instances of the same assertion have been obtained in [Tho03, § 3.5.5]. The implication (i) ⇒ (ii) allows us to deduce invariance properties for U even if P is not explicitly known, e.g., when T t is known to preserve convexity for a.e. t [BB12].
is strongly measurable under our standing assumptions. It is unclear whether the same assumptions also imply strong measurability of its resolvents, apart from the somewhat trivial case where the operators commute and are therefore simultaneously diagonalizable.
for each λ is imposed, then in view of the representation of the holomorphic semigroups T t as inverse Laplace transforms of R t (·) [ABHN01, Thm. 3.7.11] we can deduce that [0, T ] ∋ t → T t (r) ∈ L(V ′ ) is strongly measurable, too, for each r. Given a partition Λ of [0, T ] we can define the "averaged operator family " and show that, again, their product integrals converge towards the evolution family U as the partition becomes finer. In this case, if a closed (but not necessarily convex!) subset C of H is invariant under all semigroups T t , then we can apply the strategy in Remark 3.2 and introduce U Λ based on the semigroups in (3.1); accordingly, C is also invariant under U Λ (b, a) for all (b, a) ∈ ∆ and all partitions Λ, hence by Theorem 2.5 also under U .
It is known that under (rather strong) conditions on the dependence on t of the resolvent operators of A(t), U can be showed to be immediately differentiable and even holomorphic (i.e., continuously differentiable, resp. holomorphic from ∆ to L(H)), and to map H into D(A(t)), see [Fat83, Thm. 7.2.5 and Thm. 7.4.1].
For our purposes, a particularly interesting instance of closed convex sets are order intervals in Hilbert lattices: we hence assume in the following H to be a Hilbert lattice. It is known that each separable Hilbert lattice is isometrically lattice isomorphic to a Lebesgue space L 2 (X) for some σ-finite measure space (X, Σ, µ), see e.g. [MN91, Cor. 2.7.5]. Accordingly, we can consider the set H R := L 2 (X; R) of real-valued functions. Let a, b ∈ R ∪ {±∞}: we introduce the (bounded or unbounded) order intervals or, more generally, where φ, ψ : X → [−∞, ∞] are measurable functions: they are closed convex subsets of H. Many qualitative properties of solutions to evolution equations can be described by means of invariance of order intervals under the flow that governs the associated Cauchy problems.
(e) completely contractive if it is both L 1 -contractive and L ∞ -contractive; (f ) completely quasi-contractive if there is some constantω such that the rescaled evolution family Uω defined by Remark 3.4. (i) Let U be a completely quasi-contractive evolution family on L 2 (X). Then by Riesz-Thorin the rescaled evolution family Uω is L p -contractive for all p ∈ [1, ∞]. Hence each It is clear that the extended family U p satisfies conditions (i)−(ii) in Theorem 2.1. Moreover, by the interpolation inequality (Hölder inequality) we obtain that U p satisfies (iii) in For future reference let us note explicitly the following consequence of Proposition 3.1.
Proposition 3.6. Let ψ : X → [0, ∞) and φ : X → (−∞, 0] be measurable. The evolution family U associated with a ∈ Form([0, T ]; V, H) leaves the order interval Let us state a further consequence of Proposition 3.1 concerning irreducibility of evolution families on L 2 (X) on a given σ-finite measure space (X, Σ, µ). We denote by 1 Ξ the characteristic function of any given Ξ ∈ Σ.
for all Ξ ∈ Σ, then U is positivity improving.
This enables us to provide sufficient conditions for the evolution family to converge towards a rank-one projector, thus extending to the non-autonomous setting one of the main results of the classical Perron-Frobenius theory for semigroups. Proof. Under our assumptions the semigroups T t are for a.e. t ∈]0, ∞[ holomorphic, positivity improving, and compact: hence by classical Perron-Frobenius theory there is a spectral gap of sizẽ s(A(t)) > 0 between their common dominant eigenvalue 0 and the bottom of the strictly positive part of the spectrum of their generators A(t): accordingly, the partÃ(t) in the Hilbert space Ker(P ) is associated with a formã ∈ Form(0, ∞; V ∩ Ker(P ), Ker(P )) that satisfies Given a compact interval [a, b] and a partition Λ = (λ 0 , . . . , λ n+1 ) of [a, b], we can hence define in the usual way the averaged formsã k , which satisfy Therefore, the associated semigroup (e −rÃ k ) r≥0 is uniformly exponentially stable: more precisely Accordingly, and for all x ∈ H by Theorem 2.5 Now the claim follows, since ε(r) > 0 for a.e. r.
In the following sections we will often need to discuss complete contractivity. In order to find sufficient conditions therefor, observe that U is L 1 -contractive if and only if U (t, s) * is L ∞ -contractive for all (t, s) ∈ ∆. How to prove L ∞ -contractivity of all U (t, s) * ? Consider the non-autonomous adjoint form a * : , too, and hence a * is associated with an evolution family (U * (t, s)) (t,s)∈∆ , one has in general U * (t, s) = U (t, s) * . However, it was observed in [Dan00, Thm. 2.6] that the returned adjoint form which clearly belongs to Form([0, T ]; V, H), too, is associated with an evolution family ←− U * that satisfies In particular, U is L 1 -contractive if and only if ←− U * is L ∞ -contractive; U is completely contractive if so is ←− U * ; and by Proposition 3.5 we conclude the following.
Proposition 3.9. The evolution family U associated with a ∈ Form (2) completely contractive provided (1 ∧ |u|) sgn u ∈ V and Re a(t; Recall that a C 0 -semigroup S = (S(r)) r≥0 on L 2 (X) is said to be L p -quasi-contractive for some In this case, S extends by continuity to a bounded linear operator on L p (X) and, by the Riesz-Thorin Theorem, on L q (X) for all q between 2 and p.
We can now give a sufficient condition for L p -quasi-contractivity of the evolution family U that governs the Cauchy problem (2.4).
Thus the desired estimate (3.6) follows from Theorem 2.5 and Fatou's Lemma.
Let us now discuss stochasticity, another feature that cannot be easily interpreted as an invariance property.
Proof. We will again use the approximation techniques described in Section 2. Let Λ = (λ 0 , λ 1 , . . . , λ n ) be a partition of [0, T ], a k : V ×V → C be given by (2.7) and T k be the C 0 -semigroup associated with a k in H for k = 0, 1, . . . , n: under our assumptions it is easy to check that T k is positive, 1 ∈ V , and a k (Re v, 1) = 0 for all v ∈ V and k = 0, 1, . . . , n which is equivalent to the fact the all semigroups T k are stochastic. Now the claim follows from Theorem 2.5.
Our last result in this section is devoted to the issue of domination of evolution families.
Proposition 3.12. Let a ∈ Form([0, T ]; V, H) and denote as usual by U the associated evolution family. Let furthermore W be a separable Hilbert space that is densely and continuously embedded in H and b ∈ Form([0, T ]; W, H): we denote by V the associated evolution family. Assume that u ∈ W implies |u| ∈ W and u 1 ∈ V and u 2 ∈ W are such that |u 2 | ≤ |u 1 |, then u 2 sgn u 1 ∈ V ; Then U is dominated by V, i.e., , domination holds if and only if V is a generalized ideal of W and Re a(t; u, v) ≥ b(t; |u|, |v|) for all u, v ∈ V such that uv ≥ 0; in this case, the latter property holds also for the averaged forms b Λ and hence, again by [MVV05,Thm. 4.1], also the associated semigroups dominate T t . It follows from (2.12) that U Λ is dominated by V Λ : letting |Λ| → 0 and in view of Theorem 2.5 we conclude that U is dominated by V.
Example 3.13. To illustrate our results obtained so far, let us briefly dip into the topic of diffusion equations on dynamic graphs that appear in different applications, like flocking models [VZ12].
Consider a (finite or infinite) simple graph G with vertex set V and edge set E, with V vertices and E edges (i.e., V = |V| and E = |E|). Fix an orientation of G and introduce the V × E (signed) incidence matrix I = (ι ve ) of G by otherwise.
Let m ∈ ℓ ∞ (E) be a family of edge weights and consider the (weighted) Laplacian L := IMI T on ℓ 2 (V), where 0 ≤ M := diag(m(e)) e∈E . (L can be shown to be independent of the orientation.) We assume that G is uniformly locally finite, i.e., there is M < ∞ such that e∈E |ι ve | ≤ M for all v ∈ V: in this case I is a bounded linear operator from ℓ 2 (E) to ℓ 2 (V) [Mug14,Lemma. 4.3], hence L is a positive semi-definite, bounded self-adjoint operator on ℓ 2 (V): we can thus take V = H = V ′ = ℓ 2 (V). It is well-known that the semigroup generated by −L is sub-Markovian, see e.g. [Mug14, § 6.4.1]; if the graph is finite, then it is Markovian and stochastic, too.
Let us now regard G as a reference graph (one may e.g. think of a complete graph, or else of a lattice graph Z d ) and consider a family (G(t)) t∈[0,T ] of modifications of G -in other word, a graphvalued dynamical system, or dynamic graph [Šil08]. We describe the dependence of G(t) on t by introducing a measurable function [0, T ] ∋ t → m(t) ∈ ℓ ∞ (E): this allows e.g. for sudden switching of edges is allowed (as in the case of adjacency driven by a Poisson process). In particular, we consider the non-autonomous form a defined by It is easy to see that a ∈ Form([0, T ]; ℓ 2 (V), ℓ 2 (V)) and the associated operators are the Laplacians (L G(t) ) t∈[0,T ] . (We are not assuming boundedness from below on m: this is made unnecessary by the boundedness of the operator L G(t) for all t; in fact, even negative weights and hence signed graphs are allowed.) We deduce by Proposition 3.1 that the non-autonomous Cauchy probleṁ is governed by an evolution family on ℓ 2 (V); in fact, for all x ∈ ℓ 2 (V) the above equation enjoys backward well-posedness, too, and the unique solution u is of class H 1 (R; ℓ 2 (V)): the corresponding evolution family (U (t, s)) (t,s)∈R 2 can be defined via product integrals.
Because D t is for all t a generalized ideal of V = ℓ 2 (V), the associated Laplacian L |Dt generates for all t a semigroup (e −rL |D t ) r≥0 each of which is -again by [Ouh05, Cor. 2.22] -indeed dominated by (e −rL ) r≥0 . Therefore, by Proposition 3.1 and Proposition 3.12 the evolution family (U (t, s)) (t,s)∈∆ satisfies We have seen in the introduction that if A(t) ≡ A, then the evolution family that governs the non-autonomous problem is nothing but hence U satisfies Gaussian bounds if and only if so does (e rA ) r≥0 . We are already in the position to shows a less trivial instance of Gaussian-type estimate.
Gaussian-type kernel bounds on (e −tL G ) t≥0 have been proved in [Del99] for certain classes of G. Thus, if (G t ) t∈[0,T ] is a family of subgraphs of a reference graph G with measurable t → m(t, e) for all e ∈ E, and if U is the evolution family associated with the corresponding Laplacians −L |Dt , then (3.13) yields a Gaussian-type kernel estimate. If we e.g. take G to be Z, then 0 ≤ Γ(t, s; n 1 , n 2 ) ≤ G(t − s; n 1 , n 2 ), (t, s) ∈ ∆, n 1 , n 2 ∈ Z, where G(r; n 1 , n 2 ) := 1 2π

Ultracontractivity
In this and the next section we are going to restrict to the case of H = L 2 (X), where X an σ-finite measurable space. Recall that a C 0 -semigroup S on L 2 (X) is said to be ultracontractive if there exist constants c 0 , n > 0, andω ∈ R such that for all r ≥ 0 and all f ∈ L 2 (X) ∩ L 1 (X).
In this section we are going to develop a theory of ultracontractive evolution families.
We observe in passing that if U is ultracontractive and Ω has finite measure, then Γ(t, s) ∈ L 2 (Ω×Ω) for all (t, s) ∈ ∆, hence U consists of Hilbert-Schmidt operators; in fact even of trace class operators, by the operator equation that defines evolution families ((ii) in Theorem 2.1). Different sufficient conditions for the trace class property of U have been presented in [Laa18, § 3]. It is well-known that ultracontractivity of semigroups can be deduced from the Nash or Gagliardo-Nirenberg inequalities for the domain of the associated form, see [Ouh05,Chapt. 6]. We are going to extend this result to the non-autonomous setting.
Definition 4.2. Let V be a subspace of L 2 (X). The space V is said to satisfy (i) a Nash inequality if there exist constants C N , µ > 0 such that for all u ∈ L 1 (X) ∩ V ; (ii) a Gagliardo-Nirenberg inequality if there exist constants C G , N > 0 such that holds for all q ∈]2, ∞[ such that N q−2 2q ≤ 1.
Sobolev spaces H 1 (I) on intervals I ⊂ R satisfy e.g. the Nash inequality, see e.g. Here and in the following, we are adopting the usual notations introduced in (2.1)-(2.3).
Theorem 4.3. Let a ∈ Form([0, T ]; V, H) such that the associated evolution family U is completely quasi-contractive with constantω ∈ R. If V satisfies a Nash inequality (4.3) for some constants µ, C N > 0, then U is ultracontractive and Definition 4.4. An evolution family U on L 2 (X) is called linearly quasi-contractive if for some constants α 1 , α 2 independent of p (4.6) Assume that U and ←− U * are both linearly quasicontractive with constants α 1 , α 2 , α * 1 , α * 2 . If V satisfies a Gagliardo-Nirenberg inequality for some C G , N > 0, then U is ultracontractive and we have for all (t, s) ∈ ∆, Proof of Theorem 4.3. Upon rescaling U (t, s) by e − max{ω,ω}(t−s) we can without loss of generality assume both a to be coercive and the evolution family U to be completely contractive. The first part of the proof is similar to that of [AtE97, Prop. 3.8]. Let f ∈ L 1 (X) ∩ V and let s ∈ [0, T ) be fixed. Using (2.15), (2.3) and since t → U (t, s)f ∈ M R(s, T ; V, V ′ ) we obtain that for all f ∈ V ∩ L 1 (X) and a.e. (t, s) ∈ ∆ Integrating this inequality between s and t we find for all (t, s) ∈ ∆.
In order to obtain the L 2 − L ∞ -bound and thus prove the claimed ultracontractivity we will use the returned adjoint form ← − a * introduced in Section 3. In fact, arguing as in the first part of the proof we find that the evolution family ← − U * associated with ← − a * satisfies (4.8) with the same bound. Then using the identity (3.3) we conclude that for all (t, s) ∈ ∆.
Finally, the evolution law satisfied by U completes the proof.
Linear L p -quasi-contractivity turns out to be a key notion when it comes to checking ultracontractivity. In the proof of Theorem 4.5 we will need the following lemma.
Step 1. We will first prove that , where the positive constants C and µ depend only on N and κ 1 . For some r > 2 that will be fixed later we can combine (4.10) with the linear L r -quasi-contractivity of U and obtain by a version of Riesz-Thorin interpolation theorem [Dav07, Thm. 2.2.14] that for any θ ∈ [0, 1] where 1 p 1 := 1−θ r + θ 2 , 1 q 1 := 1−θ r + θ N 2 . Let now p ∈]2, ∞[. Choosing θ := 1 p and r = 2(p − 1) in the above equation we obtain that (4.14) U (t, s) L(L p (X),L Np (X)) ≤ κ Next, set R = N N −1 , p k = 2R k and t k = N +1 2N (2R) −k = N +1 N p k 2 −k for all k ∈ N. Moreover, let s 0 = s and s k+1 = s k + t k (t − s) for each integer k > 0. Then we have k t k = 1, k 1 p k = N 2 . Furthermore, s k+1 < s k for all k ∈ N and t = lim k→∞ s k . Thus, applying (4.14) for p = p k , using (4.6) and the evolution law satisfied by U we deduce that where the positive constants C and µ depend only on N and κ 1 .
Step 2. It remains to estimate U (t, s) in L(L 1 (X), L 2 (X)). To this end, we will follow an idea in [Dan00, Corollary 5.3] and use the returned adjoint form ← − a * . Indeed, by assumption ← − a * is linearly contractive. Thus one can just repeat the argument in Step 1 and obtain for each (t, s) ∈ ∆ and some constantsC,μ that depend only on N, κ 1 . This yields, in turn, an estimate of U (t, s) from L 2 (X) to L 1 (X), thanks to (4.15). Finally, using again the evolution law satisfied by U we conclude that U is ultracontractive and (4.12) holds.
Remark 4.7. Theorem 4.5 holds in particular for N q−2 2q = 1: in this case the Gagliardo-Nirenberg inequality becomes i.e., (4.4) reduces to the elementary assumption that V is continuously embedded in some L q (X): a classical Sobolev inequality. More precisely, if there exists N > 2 such that then U is ultracontractive and (4.7) holds.

Gaussian bounds
The existence of integral kernels of the evolution family, established in the previous section, paves the way to the discussion of kernel estimates. We have already seen a simple class of evolution families satisfying Gaussian-type estimates in Example 3.13. In the following we are going to study this issue more systematically.
Definition 5.1. Let U be an evolution family on L 2 (R d ) with an integral kernel Γ. Then U is said to satisfy Gaussian bounds if there exist b, c > 0, n > 0, and ω ∈ R such that for all (t, s) ∈ ∆ and a.e. x, y ∈ R d .
We regard L 2 (Ω) as a closed subspace of L 2 (R d ), extending operators on L 2 (Ω) to L 2 (R d ) by 0. In this way we can naturally define Gaussian bounds for operators on L 2 (Ω).
Gaussian bounds for evolution equations can be characterized by ultracontractivity. This characterisation is well-known for autonomous closed forms. It is based on the so-called Davies' trick, first appeared in [Dav87], see also [AtE97,Thm. 3.3] and [Are06, Thm. 13.1.4] for more general versions. Davies' trick is essentially an algorithm centered around an auxiliary result, whose non-autonomous counterpart is Theorem 5.2 below.
To begin with we introduce a suitable space . . , n of smooth functions. By [Rob91,, the function d : is a metric equivalent to the Euclidean one: there exists β > 0 such that Let U be an evolution family on L 2 (Ω) and, as usual, extend it if needed to L 2 (R d ). For a fixed ψ ∈ W we define perturbed evolution families U ρ on L 2 (R d ) by Gaussian bounds for U can now be derived from uniform ultracontractivity of the perturbed evolution families U ρ with respect to ρ and ψ. The proof of this fact is very similar to that of the autonomous case studied in [AtE97, Prop. 3.3]: our result contains [Dan00, Thm. 6.1] as a special case.
In this case the family of mappings a ρ given by Lemma 5.3. Assume that V is W -invariant and a ρ ∈ Form([0, T ]; V, H) for each ρ ∈ R such that there exist M ρ , α ρ > 0 and ω ρ ∈ R with Then (A ρ (t)) t∈[0,T ] and U ρ are the operator family and the evolution family on H associated with a ρ , respectively.
Proof. Let ρ ∈ R, ψ ∈ W and t ∈ [0, T ] be fixed and denote by C ρ (t) the operator associated with a ρ (t; ·, ·) on H. Then for each f, u ∈ L 2 (Ω) we have, u ∈ D(C ρ (t)) and C ρ (t)u = f if and only if u ∈ V and a(t; M ρ u, M −1 This is equivalent to M ρ u ∈ D(A(t)) and M −1 The last assertion is easy to prove: in fact, U ρ solves the Cauchy problem associated with (A(t)) t∈[0,T ] .
After all these preparatory results we are finally in the position to present our main theorems: given a ∈ Form([0, T ]; V, H) we introduce two sets of assumptions, which impose a Sobolev-like embedding on V and a contractivity condition on the perturbed semigroups T ρ t , and show that each of them imply Gaussian bounds for the evolution family associated with a.
Theorem 5.4. Let a ∈ Form([0, T ]; V, H). Assume that V is W -invariant and that (5.6) holds for a uniform choice of α and for ω ρ such that for some constant ω > 0 that is independent of ρ. Assume V satisfies a Nash inequality and the semigroups (e −ωρr T ρ t (r)) r≥0 are completely contractive for a.e. t ∈ [0, T ] and all ρ ∈ R. Then the evolution family U associated with a satisfies Gaussian bounds.
Proof. The assertion can be proved similarly to Theorem 5.4, based in this case on Theorem 5.2 and Theorem 4.5.
6. Applications 6.1. Time-dependent pageranks. Let us study a model similar to that of Example 3.13: it is based on an idea originally presented in [Chu07] and thoroughly developed ever since, cf. the survey [Gle15], where the connectivity of G describes the internal links of a server network -possibly the whole World Wide Web.
We thus consider an orientation of a finite complete graph (i.e., a graph such that either (v, w) ∈ E or (w, v) ∈ E for any v, w ∈ V with v = w).
As in 3.13, we assign a weight m to each edge: if e.g. m(t, e) ∈ {0, 1} for all e ∈ E and all t ∈ [0, T ], then we are effectively shutting off/switching on certain links in the considered network. We then consider the matrix (6.1) where I is again the incidence matrix of G (see Example 3.13), I − := (ι − ve ) is its negative part, M = diag(m(e)) e∈E , and D out := diag(deg out (v)) v∈V , where deg out (v) := e∈E |ι − ve m(e)|. Then, A defines a so-called heat kernel pagerank e −rA x of G with parameters r and x: here r is a positive time and x a probability distribution on V, i.e., x ∈ R V , x v ≥ 0 for all v ∈ V and x 1 = 1. The rationale behind this definition is that A is a column stochastic matrix, hence (e −rA ) r≥0 is a stochastic semigroup and e −rA f is thus again a probability distribution for all r ≥ 0, which can be used to measure the relevance of a certain node within a network in a way similar to Google's classical PageRank, cf. [Mug14, § 2.1.7.3].
We can now consider a measurable function t → (m(t, e)) e∈E and accordingly a time-dependent matrix family (A(t)) t∈[0,T ] as in (6.1) these matrices will in general not be symmetric, but in view of finiteness of V they are certainly associated with a form a ∈ Form([0, T ]; ℓ 2 (V), ℓ 2 (V)). Accordingly, in view of Proposition 3.11 the associated evolution family (U (t, s)) (t,s)∈∆ consists of stochastic operators and hence U (t, s)f is a probability distribution on V for all (t, s) ∈ ∆ and all probability There is a known correspondence between linear transport differential equations on networks and flows on their underlying graphs [Dor05]: accordingly, our results also extend to the space-continuous case. In this way, the well-posedness result in [Bay12, § 6] -which relies on the assumption that the dependence of the graph on time is absolutely continuous -can be strengthened: we omit the details.
6.2. Black-Scholes equation with time-dependent volatility. The Cauchy problem consisting of the backward parabolic equation x ∈]0, ∞[ was derived in [BS73] and is currently considered among the main mathematical tool in the pricing theory of European options: the positive constants σ, r describe volatility and interest rate of the system, respectively, whereas τ is the maturity time of an option. An effective variational approach to the relevant operator appearing in the Black-Scholes equation has been discussed in [Ein08]: it is based on studying the sesquilinear form defined on the form domain which is a Hilbert space with respect to the inner product Then it was proved in [Ein08, § 7.2] that V is dense in L 2 ]0, ∞[ and that furthermore for all u ∈ V, (6.4) i.e., a is bounded and elliptic; and Re a(u, u) ≥ 0 if 3r ≥ σ 2 .
The original Black-Scholes-theory assumes σ to be time-independent, but is rather unrealistic and has been questioned ever since: we mention the celebrated Heston model [Hes93], which leads to a non-autonomous PDE similar to (6.2), based on the assumption that the volatility evolves following a certain Brownian-like motion. This justifies the study of Furthermore, the semigroup associated with a is quasi-contractive and sub-Markovian: we deduce from (A manifold of financial models exist that display a similar mathematical structure, albeit their meaning is different: the popular Cox-Ingersoll-Ross along with several other so-called short-rate models surveyed in [CKLS92] involve time-dependent σ and/or r and can be discussed with only minor variations to our treatment above.) 6.3. Second-order elliptic operators on networks. With the purpose of introducing a differential operator on a network-like structure, we consider once again a graph G = (V, E) (like in Example 3.13) and identify each edge e ∈ E with an interval [0, 1]. In other words, we are considering a collection of copies of [0, 1] and gluing them in a graph-like fashion: we thus obtain what are often called metric graphs or networks in the literature [BK13,Mug14]. (For the sake of simplicity we are going to assume such a metric graph to be connected.) The history of non-autonomous diffusion equations on networks goes back at least to pioneering investigations by von Below, Lumer, and Schnaubelt: well-posedness results could be proved in [Bel88,LS99], further results on long-time asymptotics have been deduced in [ADKF14].
We are going to apply in this context the theory developed in the previous sections: when introducing operators on metric graphs we avoid to go into full details and refer the interested reader to [Mug14,Chapter 6]. To fix the ideas, consider a possibly infinite, but uniformly locally finite (see Example 3.13) graph G = (V, E) and upon rescaling identify each edge e with an interval (0, 1). On each interval e we consider the operator family We will additionally assume that the operator family is uniformly elliptic, i.e., (6.5) c e (t, x) ≥ γ for all e ∈ E and a.e. x ∈ (0, 1), t ∈ [0, T ], for some γ > 0. In order to reflect the topology of the graph, transmission conditions in the vertices are required: the most common conditions are usually referred to as continuity/Kirchhoff and amount to asking that • u is continuous, i.e., the boundary values of u e and u f agree whenever evaluated at endpoints of the intervals e, f that are glued together in the network G (continuity); • u satisfies a Kirchhoff-type rule, i.e., at any vertex the sum over all neighboring edges of the normal derivatives evaluated at the vertex vanishes. However, many more boundary conditions are conceivable: indeed, a parametrization of a large class of boundary conditions that fits very well the setting of sesquilinear forms has been discussed in [Mug14, § 6.5.1], based on the finite case treated in [Kuc04,Thm. 5].
Fix a closed subspace Y of the Hilbert space ℓ 2 (E) × ℓ 2 (E), let (Σ(t)) t∈[0,T ] be a family of bounded linear operators on Y , and consider the non-autonomous form a defined by Then the conditions in the vertices satisfied by functions in the domain of each operator A(t) associated with a can be written in a compact form as .
We finally assume that for some P, S > 0 Then taking into account Lemma 2.7 it is easy to see that a ∈ Form([0, T ]; V, H).
Our abstract results in the previous sections hence yield the following.
Proposition 6.1. Under the above assumptions on the coefficients c e , p e , the space Y , and the operators Σ, the form a is associated with a strongly continuous evolution family U on H. If p e (t) ≥ 0 and Σ(t) is accretive for a.e. t ∈ [0, T ], then U is contractive. If all these coefficients are defined on the whole interval [0, ∞[, then U extends to an evolution family on {(t, s) : 0 ≤ s ≤ t ≤ ∞}.
Due to the standing assumption that G is uniformly locally finite, P Y is a block operator matrix whose blocks are of the form 1 n J n (J n denoting the n×n all-1-matrix, n the degree of the corresponding vertex). Because P Y leaves invariant the order interval [−1, 1] ℓ 2 ×ℓ 2 , we deduce that if merely (e −rΣ(t) ) r≥0 is L p -contractive for some p ∈]1, ∞[ and a.e. t ∈ [0, T ], then so is (e −rA(t) ) r≥0 and a.e. t ∈ [0, T ] and, by Proposition 3.1, also the evolution family U .
(1) If e −rΣ(t) (for all r ≥ 0 and a.e. t ∈ [0, T ]) and P Y are real and positive, then U is positive. If additionally p e ≡ 0 and 1 ∈ Y , then U is stochastic.
Example 6.4. The continuity/Kirchhoff vertex conditions are special cases of the general conditions in (6.6). Indeed, denote by c V the vector in ℓ 2 (E) × ℓ 2 (E) that consists of vertex-wise constants, i.e., entries of c V agree whenever they correspond to endpoints of edges the same vertex v ∈ V is incident with. Let by Y the subspace of ℓ 2 (E) × ℓ 2 (E) spanned by c V and take Σ = 0: then (6.6) agrees with continuity-Kirchhoff conditions: we denote by Under stronger assumptions on p e , c e , a well-posedness result comparable to Proposition 6.1 has been obtained in [ADKF14, Thm. 3.3].
Under the same assumption that Σ ≡ 0, we can consider two different cases: • if t t 0 ess inf p e (t, x) > 0 for a.e. x ∈ (0, 1) and all e ∈ E, then U is by Proposition 2.6 uniformly exponentially stable; if p e ≡ 0 and the network is finite (i.e., |E| < ∞), then by Corollary 3.8 and [Mug14, Proposition 6.70] A similar result has been obtained for general inhomogeneous diffusion equations on finite networks in [ADKF14, § 5.1].
Proof. It follows from [Mug07, Lemma 6.1] and Corollary 6.3 that U is completely contractive. It has been shown in [Prö,Chapt. 3] that H 1 (G) satisfies a Nash inequality whenever G is a connected, locally finite metric graph with edge lengths uniformly bounded away from 0: accordingly, the non-autonomous form with domain H 1 (G) is associated with an ultracontractive U , owing to Theorem 4.3.
In order to complete the proof, observe that by [CM09, Thm. 6.2] the semigroup associated with a(t) ≡ a with domain H 1 Y (G) is dominated by the semigroup associated with the same form with domain H 1 (G), provided Y is a generalized ideal of c V (in fact by [Nag86, Thm. C.II-5.5] the latter is the modulus semigroup of the former one). It is a direct consequence of Proposition 3.1 that the same holds for the associated evolution families, hence the former heat kernel inherits ultracontractivity from the latter one.
The following result seems to be new even in the autonomous case: in [Mug07] Gaussian bounds for heat kernels on finite networks have been proved only in the special case of Y = c V , see also [Mug14,Chapt. 7] for an abstract approach based on the theory of Dirichlet forms.
Corollary 6.6. Under the assumptions of Proposition 6.5, let Y be c V -invariant (i.e., the entrywise product ψc V lies in Y for all ψ ∈ Y ), where the vector c V is defined as in Example 6.4. If furthermore Σ(t) is diagonal for a.e. t ∈ [0, T ], then U satisfies Gaussian bounds.
We stress that our assumption on P Y and Σ are only enforcing complete contractivity, but the evolution family need not be positive. An example is given by the non-autonomous parabolic equation on a loop with boundary conditions defined by Σ ≡ 0 and Y = 1 −1 , which is c V -invariant (here c V = 1 1 ): this equation is governed by a completely contractive and (in view of the Nash inequality for H 1 (G)) ultracontractive evolution family U , which therefore enjoys Gaussian bounds. However, U is not positive, since neither is P Y .
Proof of Corollary 6.6. We apply Davies' Trick in a slightly different version. Indeed, we adapt the usual setting to our network environment by introducing the space W G := ψ ∈ H 1 (G) ∩ C ∞ (0, 1; ℓ 2 (E)) | ψ ′ ∞ ≤ 1, ψ ′′ ∞ ≤ 1 : then one can check that d(x, y) := sup{|ψ(x) − ψ(y)| | ψ ∈ W G }, x, y ∈ G, defines a metric on G that is equivalent to the canonical one [Mug14, § 3.2]. (Recall that H 1 (G) denotes the space H 1 Y (G), whereỸ = c V is the space spanned by c V : each function in H 1 (G) is by definition continuous on the metric space G. Observe that c V is in fact an algebra with respect to the entry-wise product.) By definition, H 1 Y (G) is W G -invariant if and only if u ∈ Y implies e ρψ u ∈ Y , i.e., if and only if Y is c V -invariant.
If Σ(t) ≡ 0, then the assertion has been proved in [Mug07,Thm. 4.7] by showing that the relevant form (let us denote it by a 0 to stress the absence of boundary terms) induces perturbed forms a ρ 0 that are associated with completely contractive perturbed evolution families (with the form domain being unchanged and still satisfying a Nash inequality). In the general case of a(t; u, v) = a 0 (t; u, v) + (Σ(t)u|v) Y , we find that a ρ (t; u, v) = a ρ 0 (t; u, v) + (Σ(t)u|v) Y . These forms are associated with completely contractive evolution families, hence the claim follows. for all u, v ∈ V where V is a closed subspace of H 1 (Ω) that contains H 1 0 (Ω). We assume that the coefficients a k,j , b k , c k , a 0 lie in L ∞ ([0, T ] × Ω; C). Moreover, we assume that the principal part is uniformly elliptic, i.e., there exist a constant ν > 0 such that (6.8) Re d k,j=1 a kj (t; x)ξ k ξ j ≥ ν|ξ| 2 for a.e. t ∈ [0, T ], a.e. x ∈ Ω, and all ξ ∈ C n .
Then a V defined in (6.7) belongs to Form([0, T ]; V, L 2 (Ω)) [Ouh05, Section 4.1]. In fact, we have for all u, v ∈ V and a.e. t ∈ [0, T ], where M > 0 is a constant depending only on a kj ∞ , b k ∞ , c k ∞ , and a 0 ∞ , and we can choose Here (Re a 0 ) − = max{0, − Re a 0 }. We can then associate an operator A V (t) ∈ L(V, V ′ ) with the form a V which is formally given by (6.9) Let A V (t) be the operator associated with a(t; ·, ·) on L 2 (Ω). Thus A V (t) is the realization of A V (t) in L 2 (Ω) with various boundary condition which are determined by the form domain V. where Γ is a closed subset of the boundary of Ω. In particular, a V is associated with an evolution family U V that governs the non-autonomous problem driven by the operator family (A V (t)) t∈[0,T ] . Each of these evolution families is positive, it dominates the evolution family U H 1 0 and is dominated by U H 1 . Following [Ouh04] we introduce the following notations: Re(a kj )D k uD j vdx + Lemma 6.7. Let a V be given by (6.7) and denote by U V the associated evolution family on L 2 (Ω). Assume that (|v| ∧ 1) sgn v ∈ V for all v ∈ V . Moreover, we assume that a kj (·, ·) are real-valued functions for all k, j = 1, 2, . . . d. Then the evolution family U V is L p -quasi-contractive for all p ∈ (1, ∞[ and we have (6.10) U V (t, s)f L p (Ω) ≤ e (t−s)ωp f L p (Ω) for all (t, s) ∈ ∆, where (6.11) ω p := (Re a 0 ) − ∞ + 1 Now we are going to prove that the evolution family U V governed by the time dependent elliptic operator (6.9) satisfies Gaussian bounds. We known from Theorem 5.2 that U V satisfies Gaussian bounds if and only if there exist a constants c > 0, n > 0 and ω ∈ R such that for all ρ ∈ R, ψ ∈ W and 0 ≤ s < t ≤ T. Let a given by (6.7). Then the non-autonomous form a ρ (t, u, v) := a(t; M ρ u, M −1 ρ v) is given by (6.15) In the following we define for each ρ ∈ R, ψ ∈ W the constants α i,ρ , α * i,ρ , i = 1, 2, via formulas which are analogous to (6.12), (6.13) and (6.14) where Re a 0 is replaced by Re a 0 − m if a k,j are complex-valued functions. Further, we set (6.16) c 0 := max{ a k,j ∞ , b k ∞ , c k ∞ , c 0 ∞ , k, j = 1, 2, . . . , d} and (6.17) ω := 4c 0 d 2 + 4c 0 d 3 ν −1 .