Hardy type unique continuation properties for abstract Schrödinger equations and applications

Hardy type unique continuation properties for abstract Schrödinger equations and applications

Veli Shakhmurov

1Department of Mechanical Engineering, Okan University, Akfirat, Tuzla 34959 Istanbul, Turkey

2Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, AZ1141, Baku, F. Agaev 9

Received 22 November 2019, appeared 21 December 2019 Communicated by Maria Alessandra Ragusa

Abstract. In this paper, Hardy’s uncertainty principle and unique continuation proper- ties of Schrödinger equations with operator potentials in Hilbert space-valuedL²classes are obtained. Since the Hilbert space Hand linear operators are arbitrary, by choosing the appropriate spaces and operators we obtain numerous classes of Schrödinger type equations and its finite and infinite many systems which occur in a wide variety of physical systems.

Keywords:Schrödinger equations, positive operators, groups of operators, unique con- tinuation, Hardy’s uncertainty principle

2010 Mathematics Subject Classification: 35Q41, 35K15, 47B25, 47Dxx, 46E40.

1 Introduction

Here, the unique continuation properties of the following abstract Schrödinger equation i∂tu+_∆u+A(x)u+V(x,t)u=0, x∈_Rⁿ, t∈ [0,T], (1.1) are studied, where A = A(x)is a linear and V(x,t) is a given potential operator functions in a Hilbert space H; ∆ denotes the Laplace operator inRⁿ and u = u(x,t) is the H-valued unknown function. This linear result was then applied to show that two regular solutionsu₁, u2of non-linear abstract Schrödinger equations

i∂_tu+_∆u+A((x))u=F(u, ¯u), x∈_Rⁿ, t ∈[0,T] (1.2) for general non-linearities F must agree in Rⁿ×[_0,T]_{, when} u₁−u₂ and its gradient decay faster than any quadratic exponential at times 0 andT.

Hardy’s uncertainty principle and unique continuation properties for Schrödinger equa- tions studied e.g in [4–7] and the references therein. Abstract differential equations studied e.g. in [2,12–15,17–19,23,25]. However, there seems to be no such abstract setting for nonlin- ear Schrödinger equations except the local existence of weak solution (cf. [15]). In contrast to

BEmail: veli.sahmurov@okan.edu.tr

these results we will study the unique continuation properties of abstract Schrödinger equa- tions with the operator potentials. Since the Hilbert space H is arbitrary and A is a possible linear operator, by choosing H and A we can obtain numerous classes of Schrödinger type equations and its systems which occur in the different processes. Our main goal is to obtain sufficient conditions on a solutionu, the operator A, potentialV and the behavior of the so- lution at two different timest₀andt₁ which guarantee thatu(x,t)≡0 for x ∈ _Rⁿ,t ∈ [0,T]. If we choose Hto be a concrete Hilbert space, for example H = L²(_Ω), A = L, where Ωis a domain inR^m with sufficiently smooth boundary and Lis a regular elliptic operator then, we obtain the unique continuation properties of the anisotropic Schrödinger equation

∂tu=i(_∆u+Lu) +V(x,t)u, x∈_Rⁿ, y∈ _Ω, t ∈[0,T]. (1.3) Moreover, let we choose H= L²(0, 1)and Ato be differential operator with Wentzell–Robin boundary condition defined by

D(A) =u∈W^2,2(0, 1), Au(j) =0, j=_{0, 1 , (1.4)} A(x)u=a(x,y)u⁽²⁾+b(x,y)u⁽¹⁾,

where a, bare sufficiently smooth functions on Rⁿ×(0, 1) andV(x,t) is a integral operator so that

V(x,t)u=

Z ₁

0 K(x,y,t)u(x,y,t)dy,

where,K=K(x,τ,t)is a complex valued bounded function. From our general results we ob- tain the unique continuation properties of the Wentzell–Robin type boundary value problem (BVP) for the following Schrödinger equation

∂tu=i

∆u+a∂²u

∂y² +b∂u

∂y

+

Z ₁

0 K(x,y,t)u(x,y,t)dy, x∈_Rⁿ_, y∈(0, 1), t∈[0,T],

(1.5)

a∂²_yu(x,j,t) +b∂yu(x,j,t) =0, j=0, 1. (1.6) Note that, the regularity properties of Wentzell–Robin type BVP for elliptic equations were studied e.g. in[10, 11]and the references therein. Moreover, if putH=l₂ and chooseAto be a infinite matrix

a_mj

,m,j=1, 2, . . . ,∞, then we derive the unique continuation properties of the following system of Schrödinger equation

∂_tu_m =i

"

∆u_m+

∑

a_mj(x) +b_mj(x,t)u_j

#

, x∈_Rⁿ, t ∈(0,T), (1.7) wherea_mj are continuous andb_mj are bounded functions.

Let E be a Banach space. L^p(_Ω;E) denotes the space of strongly measurable E-valued functions that are defined on the measurable subsetΩ⊂_Rⁿwith the norm

kfk_Lp =kfk_Lp(_Ω;E) = _Z

Ωkf(x)k^p_Edx ¹_p

, 1≤ p<_∞. Let Hbe a Hilbert space and

kuk= kuk_H = (u,u)

1 2

H = (u,u)¹² foru∈ H.

For p=2 andE= H, L^p(_Ω;E)becomes a H-valued function space with inner product:

(f,g)_L2(_Ω;H) =

Z

Ω(f(x),g(x))_Hdx, f,g ∈L²(_Ω;H).

Here,W^s,2(_Rⁿ;H), −_∞< s <_∞ denotes the H-valued Sobolev space of orders which is defined as:

W^s,2=W^s,2(_Rⁿ;H) = (I−_∆)^−2s L²(_Rⁿ;H) with the norm

kuk_Ws,2 =(I−_∆)^2s u

L²(_Rⁿ;H)< _∞.

It clear that W^0,2(_Rⁿ;E) = L²(_Rⁿ;H). Let H₀ and H be two Hilbert spaces and H₀ is con- tinuously and densely embedded into H. LetW^s,2(_Rⁿ;H0,H)denote the Sobolev–Lions type space, i.e.,

W^s,2(_Rⁿ;H₀,H) =ⁿu∈W^s,2(_Rⁿ;H)∩L²(_Rⁿ;H₀),

kuk_Ws,2(_Rⁿ;H0,H)=kuk_L2(_Rⁿ;H0)+kuk_Ws,2(_Rⁿ;H) <_∞^o. LetC(_Ω;E)denote the space ofE-valued uniformly bounded continuous functions onΩ with norm

kuk_C(Ω;E) =sup

x∈_Ω

ku(x)k_E.

C^m(_Ω;E)will denote the spaces ofE-valued uniformly bounded strongly continuous and m-times continuously differentiable functions onΩwith norm

kuk_Cm(_Ω;E)= max

0≤|α|≤msup

x∈_Ω

kD^αu(x)k_E.

Here,Or = {x∈_Rⁿ, |x|< r} forr > 0. Let N denote the set of all natural numbers, C denote the set of all complex numbers. Let E₁ and E₂ be two Banach spaces. B(E₁,E₂) will denote the space of all bounded linear operators from E₁ to E2. For E₁ = E2 = E it will be denoted by B(E). By (E₁,E₂)_θ,p, 0 < θ < 1, 1 ≤ p ≤ _∞ we will denote the interpolation spaces obtained from {E₁,E₂}by the K-method [24, §1.3.2]. Here, S = S(_Rⁿ;E) denotes the E-valued Schwartz class, i.e. the space of E-valued rapidly decreasing smooth functions on Rⁿ, equipped with its usual topology generated by seminorms. S(_Rⁿ;C)will be denoted by justS. LetS⁰(_Rⁿ;E)denote the space of all continuous linear operators,L :S →E, equipped with topology of bounded convergence.

LetA= A(x),x∈_Rⁿbe closed linear operator inEwith independent onx∈_Rⁿdomain D(A)that is dense onE. The Fourier transformation of A(x), i.e. ˆA= FA = A^ˆ(ξ)is a linear operator defined as

Aˆ(ξ)u(ϕ) =A(x)u(ϕˆ) foru∈S⁰(_Rⁿ;E), ϕ∈ S(_Rⁿ). (For details see e.g. [1, Section 3]).

For linear operators AandB,[A,B]-denotes a commutator operator, i.e.

[A,B] =AB−BA.

These kind of operators are of fundamental importance in real analysis, potential theory and in the study of elliptic and parabolic differential equations (see e.g. [9,16,22] and refer- ences therein).

Sometimes we use one and the same symbol C without distinction in order to denote positive constants which may differ from each other even in a single context. When we want to specify the dependence of such a constant on a parameter, sayα, we write Cα.

2 Main results

Let A = A(x), x ∈ _Rⁿ be closed linear operator in a Hilbert space H with independent on x∈_RⁿdomainD(A)that is dense on H. Let

H(A) =ⁿu∈D(A), kuk_H(A) =kAuk_H+kuk_H <_∞^o, X= L²(_Rⁿ;H), X(A) = L²(_Rⁿ;H(A)), X(A) =L²(_Rⁿ;H(A)) X^∞(A) =L^∞(_Rⁿ;H(A)), Y^s=W^s,2(_Rⁿ;H), Y^s(A) =W^s,2(_Rⁿ;H(A)),

B= L^∞(_Rⁿ;B(H)) and µ(t) =αt+β(1−t).

Definition 2.1. A functionu∈ L^∞(0,T;X(A))is called a local weak solution to (1.1) on(0,T) if u satisfies (1.1). In particular, if (0,T) coincides with R, then u is called a global weak solution to (1.1). If the solution of (1.1) belongs to C [0,T];X(A)∩Y²

, then it is called a strong solution.

Our main result in this paper is the following.

Theorem 2.2. Assume that the following condition are satisfied:

(1) A = A(x) and _∂x^∂A

k are symmetric operators in a Hilbert space H with independent on x ∈ _Rⁿ domain D _∂x^∂A

k

= D(A) that is dense on H. Moreover, (A(x)u,u) ≥ 0 and (A(x)u,u) ∈ L²(_Rⁿ) for u∈D(A);

(2) n

k

∑

x_k

A∂f

∂x_k − ^∂A

∂x_k f

,f

X

≥₀ for f ∈ L^∞

0,T;Y¹(A)

;

(3) A(x)A⁻¹(x0)∈ L¹(_Rⁿ;B(H))for some x0∈_Rⁿand V(x,t)∈B(H)for(x,t)∈_Rⁿ×[0, 1]; (4) either, V(x,t) =V₁(x) +V₂(x,t), where V₁(x)∈ B(H)for x ∈_Rⁿ and

M₁= sup

x∈_Rⁿ

kV₁(x)k_B(H) <_∞, sup

t∈[0,1]

e^{|x|2µ−2(t)}V₂(·,t) B <_∞ or

rlim→_∞kVk_L1(0,1;L^∞(_Rⁿ/Or);B(H))=0;

(5) u∈C([0, 1];X(A))is a solution of the equation(1.1)and

e^β−2|x|2u(·_{, 0})

X< _∞, e^α−2|x|2u(·_{, 1})

X <_∞.

where

α, β>0, αβ <2.

Then u(x,t)≡_0.

As a result of Theorem2.2 we get the following Hardy’s uncertainty principle result for the non linear equation (1.2).

Theorem 2.3. Suppose that the assumptions (1)–(2) of Theorem 2.2 are satisfied. Let u₁,u₂ ∈ C [0, 1];Y^k(A), k∈_Z⁺be strong solutions of the equation(1.2)with k> ⁿ₂. Moreover, assume:

(1) F∈ C^k C²,C

and F(0) =∂uF(0) =∂u¯F(0) =0.There areα,β>0withαβ<2such that e^{−β−2|x|2}(u₁(·, 0)−u₂(·, 0))∈X, e^{−α−2|x|2}(u₁(·, 1)−u₂(·, 1))∈ X

(2) there exists a constant B₀>0such that

kF(u, ¯u)k_H ≤ B₀kuk_(H(A),H)

1p,p

for all u∈ (H(A),H)1 p,p. Then u₁≡ u₂.

One of the results we get is the following one.

Theorem 2.4. Assume that the all conditions of Theorem 2.2 are satisfied. Suppose ∆+A+V₁ generates a bounded continuous group. Let u ∈ C([0, 1];X(A)) be a solution of (1.1). Then

e^{|x|2µ−2(t)}u(·,t)

1 µ(t)

X is logarithmically convex in[0, 1]and there is N= N(α,β)such that

e^{|x|2µ−2(t)}u(·,t)

1 µ(t)

X ≤e^N(M1+M2+M²₁+M²₂)

e^β−2|x|2u(·, 0)

β(1−t)µ(t) X

e^α−2|x|2u(·, 1)

αtµ(t)

X ,

when

M₂ =e^2B(V2) sup

t∈[0,1]

e^{|x|2µ−2(t)}V₂(·_,t)

B, B(V₂) = sup

t∈[0,1]

kReV₂(·_,t)k_B. Moreover,

q

t(1−t)

e^{|x|2µ−2(t)}∇u

L²(_Rⁿ×[0,1];H)

≤e^N(M₁+M2+M²₁+M²₂)^h

e^β−2|x|2u(·, 0)

X+e^α−2|x|2u(·, 1)

X

i .

Consider the Cauchy problem for abstract parabolic equations with variable operator co- efficients

∂_tu=_∆u+A(x)u+V(x,t)u, (2.1) u(x, 0) = f(x), x ∈_Rⁿ, t∈[0, 1],

where A(x)is a linear andV(x,t)is the given potential operator functions in H. By employ- ing Theorem2.2we obtain the following result for the abstract parabolic equation (2.1):

Theorem 2.5. Assume the assumptions (1)–(3) of Theorem 2.2 are satisfied. Suppose that u ∈ L^∞(0, 1;X(A))∩L² 0, 1;Y¹

is a solution of (2.1)and kfk_X <_∞,

e^δ−2|x|2u(·, 1) X <_∞ for some δ<1. Then, f(x)≡₀for x∈_Rⁿ_.

First of all, we generalize the result of G. H. Hardy (see e.g. [20, p. 131]) about uncertainty principle for Fourier transform:

Lemma 2.6. Let f (x)be H-valued function for x∈_Rⁿand kf(x)k=O

e⁻

|x|² β2

,

fˆ(ξ) =O

e⁻⁴

|ξ|² α2

, x,ξ ∈_Rⁿ forαβ<4.

Then f(x)≡0. Also, ifαβ=4, thenkf(x)kis a constant multiple of e⁻

|x|² β2

.

Proof. Indeed, by employing Phragmén–Lindelöf theorem to the classes of Hilbert-valued an- alytic functions and by reasoning as in[8]we obtain the assertion.

Consider the Cauchy problem for free abstract Schrödinger equation

i∂tu+_∆u+Au=0, x∈_Rⁿ, t∈[0, 1], (2.2) u(x, 0) = f(x),

where A = A(x) is a linear operator in a Hilbert space H with independent on x domain D(A).

The above result can be rewritten for solution of the (2.2) onRⁿ×(0,∞). Indeed, assume ku(x, 0)k=O

e⁻

|x|² β2

, ku(x,T)k=O

e^−|x|

2 α2

forαβ<4T.

Then u(x,t) ≡ 0. Also, if αβ = 4T, then u has as a initial data a constant multiple of e⁻

1

β2+_4Tⁱ |x|²

.

Lemma 2.7. Assume that A is a symmetric operator in H with independent on x ∈ _Rⁿ domain D(A)that is dense on H. Moreover, A(x)A⁻¹(x₀) ∈ L¹(_Rⁿ;B(H))for some x₀ ∈ _Rⁿ. Then for

f ∈W^s,2(_Rⁿ;H),s≥0there is a generalized solution of (2.2)expressing as u(x,t) = F⁻¹h

e^iAˆξtfˆ(ξ)ⁱ, Aˆ_ξ = A^ˆ(ξ)− |ξ|², (2.3) where F⁻¹is the inverse Fourier transform and Aˆ(ξ)denotes the Fourier transform of A(x).

Proof. By applying the Fourier transform to the problem (2.2) we get

i∂_tuˆ(ξ,t) +A^ˆ_ξuˆ(ξ,t) =0, x ∈_Rⁿ, t∈[0, 1], (2.4)

ˆ

u(ξ, 0) = f^ˆ(ξ), ξ ∈ _Rⁿ. It is clear to see that the solution of (2.4) can be exspressed as

ˆ

u(ξ,t) =e^iAˆξtfˆ(ξ). Hence, we obtain (2.3)

3 Estimates for solutions

We need the following lemmas for proving the main results. Consider the abstract Schrödinger equation

∂_tu= (a+ib) [_∆u+Au+Vu+F(x,t)], x∈_Rⁿ, t∈ [0, 1], (3.1) where a, bare real numbers, A = A(x)is a linear operator, V = V(x,t)is a given potential operator function in Hand F(x,t)is a given H-valued function.

Condition 3.1. Assume that:

(1) A = A(x) is a symmetric operator in Hilbert space H with independent on x ∈ _Rⁿ domain D(A) that is dense on H; moreover, (A(x)u,u) ≥ _{0 and} (A(x)u,u) ∈ L²(_Rⁿ) foru∈ D(A);

(2) ^∂A_∂x

k are symmetric operators in H with independent onx ∈ _Rⁿ domain D _∂x^∂A

k

= D(A). Moreover,

∑

x_k

A∂f

∂x_k − ^∂A

∂x_k f

,f

X

≥0, for f ∈ L^∞

0,T;Y¹(A); (3.2)

(3) a>0,b∈_R;V =V(x,t)∈B(H). Let

|∇υ|²_H =

∑

∂υ

∂x_k

2

forυ∈W^1,2(_Rⁿ;H).

Lemma 3.2. Assume that the Condition 3.1 holds. Then the solution u of (3.1) belonging to L^∞(0, 1;X(A))∩L² 0, 1;Y¹

satisfies the following estimate e^MT

e^φ(·,T)u(·,T)

X≤e^γ|x|2u(·, 0)

X+κ e^φ(t)F

L¹(0,T;X)+ak(Au,u)k_X, (3.3) where

φ(x,t) = ^γa|x|²

a+_4γ(a²+b²)t, M_T = kaReV−bImVk_B, κ =^pa²+b², γ≥0.

Proof. Letυ=e^ϕu, whereϕis a real-valued function to be chosen later. The functionυverifies

∂_tυ=Sυ+Kυ+ (a+ib)e^ϕF, (x,t)∈_Rⁿ×[0, 1],

whereS,Kare symmetric and skew-symmetric operators respectively given by S= aA₁−ibB₁+ϕ_t+aReV−bImV, K=ibA₁−aB₁+i(bReυ+aImυ), here

A₁ =_∆+A(x) +|∇ϕ|², B₁ =2∇ϕ.∇+_∆ϕ.

By differentiating the inner product in X, we get

∂_tkυk²_X =2 Re(Sυ,υ)_X+2 Re(Kυ,υ)_X

+2 Re((a+ib)e^υF,υ)_X+2 Re(a+ib) (Vυ,υ)_X, t ≥0. (3.4)

A formal integration by parts gives that Re(Sυ,υ)_X=−a

Z

Rⁿ|∇υ|²_Hdx+

Z

Rⁿ

a|∇ϕ|²+ϕt

kυk²dx+a Z

Rⁿ(Aυ,υ)dx, Re(Kυ,υ)_X =−aγ

Z

Rⁿ

h

(2∇ϕ.∇υ,υ) +_∆ϕkυk²ⁱdx. (3.5) It is clear that

Re(a+ib) (Vυ,υ)_X= a Z

Rⁿ

(ReVυ,υ)dx−b Z

Rⁿ

(ImVυ,υ)dx,

Re((a+ib)e^ϕF,υ)_X =aRe Z

Rⁿ

(e^ϕF,υ)dx−bIm Z

Rⁿ

(e^ϕF,υ)dx

=ae^ϕRe(F,υ)_X−be^ϕIm(F,υ)_X.

Then by using the Cauchy–Schwarz inequality, by assumptions (2), (3), in view of (3.4) and (3.5) we obtain

∂_tkυk²_X ≤₂kaReV−bImVk_Bkυk²_X+₂κke^ϕF(t,·)k_Xkυk_X+ak(Aυ,υ)k_X, wherea,band ϕare such that

a+ ^b

2

a

|∇ϕ|²+ϕ_t ≤0 inRⁿ+⁺¹. (3.6) The remaining part of the proof is obtained by reasoning as in [7, Lemma 1] .

When ϕ(x,t) =q(t)ψ(x), it suffices that

a+^b

2

a

q²(t)|∇ψ|²+q⁰(t)ψ(x)≤0. (3.7) If we putψ(x) =|x|²then (3.6) holds, when

q⁰(t) =−4

a+ ^b

2

a

q²(t), q(0) =γ, γ≥0. (3.8) Let

ψ_r(x) =

(|x|², |x|<r,

∞, |x|>r.

Regularizeψ_rwith a radial mollifierθ_ρ and set

ϕ_ρ,r(x,t) =q(t)θ_ρ∗ψ_r(x), υ_ρ,r(x,t) =e^ϕρ,ru, whereq(t) =γa

a+4γ a²+b² t−1

is the solution to (3.7). Because the right hand side of (3.5) only involves the first derivatives of ϕ, ψ_ris Lipschitz and bounded at infinity,

θ_ρ∗ψ_r(x)≤θ_ρ∗ |x|² =C(n)ρ²

and (3.6) holds uniformly in ρ and r, when ϕ is replaced by ϕρ,r. Hence, it follows that the estimate

e^MT

e^φ(T)u(T)

X≤ M_T

e^γ|x|2u(0)

X+^pa²+b²ke^ϕρ,rFk_L1(0,T;X)

holds uniformly in ρ and r. The assertion is obtained after letting ρ tend to zero and r to infinity.

Remark 3.3. It should be noted that if H=_C,A=0 andV(x,t)is a complex valued function, then the abstract condition (3.3) can be replaced by

M_T =kaReV−bImVk_L1(0,T;L^∞(_Rⁿ)) <_∞.

Let

Q(t) = (f,f)_X, D(t) = (S f,f)_X, N(t) =D(t)Q⁻¹(t), ∂_tS=S_t. In a similar way as in [7, Lemma 2] we have the following result.

Lemma 3.4. Assume that S = S(t)is a symmetric, K = K(t)is a skew-symmetric operators in H, G(x,t)is a positive and f(x,t)is an X-valued reasonable function. Then,

Q⁰⁰(t) =2∂tRe(∂tf−S f −K f, f)_X+2(Stf + [S,K]f, f)_X

+k∂_tf−S f +K fk²_X− k∂_tf −S f −K fk²_X (3.9) and

∂_tN(t)≥ Q⁻¹(t)

(S_tf + [S,K]f,f)_X− ¹

2k∂_tf−S f −K fk²_X

.

Moreover, if

k∂_tf−S f −K fk_H ≤M₁kfk_H+G(x,t), S_t+ [S,K]≥ −M₀ for x ∈_Rⁿ, t∈[0, 1]and

M₂= sup

t∈[0,1]

kG(.,t)k_L2(_Rⁿ)(kf(·,t)k_X)⁻¹ <_∞.

Then Q(t)is logarithmically convex in[0, 1]and there is a constant M such that Q(t)≤ e^M(M0+M1+M2+M²₁+M₂²)_Q1−t(0)Q^t(1), 0≤t ≤1.

Lemma 3.5. Assume that the Condition3.1holds. Moreover, suppose sup

t∈[0,1]

kV(·,t)k_B ≤ M₁,

e^γ|x|2u(·, 0)

X<_∞,

e^γ|x|2u(·_{, 1})

X< _∞, M₂= _sup

t∈[0,1]

e^γ|x|2F(·,t) X

kuk_X <_∞.

Then, for solution u ∈ L^∞(0, 1;X(A))∩ L² 0, 1;Y¹

of (3.1), e^γ|x|2u(·,t) is logarithmically convex in[_{0, 1}]and there is a constant N such that

e^γ|x|2u(·_,t)

X ≤e^{N M(a,b)}

e^γ|x|2u(·_{, 0})

1−t X

e^γ|x|2u(·_{, 1})

t

X, (3.10)

where

M(a,b) =κ^{2 γM}₁²+M²₂

+κ(M₁+M₂) when0≤t ≤_1.

Proof. Let f =e^γϕu, whereϕis a real-valued function to be chosen. The function f(x)verifies

∂_tf = S f +K f + (a+ib) (V f+e^γϕF) inRⁿ×[0, 1], (3.11) whereS,Kare symmetric and skew-symmetric operator, respectively given by

S= aA₁−ibγB₁+ϕ_t+aReV−bImV, K=ibA₁−aγB₁+i(bReυ+aImυ), where

A₁=_∆+A(x) +γ²|∇ϕ|², B₁=2∇ϕ.∇+_∆ϕ.

A calculation shows that,

S_t+ [S,K] =γ∂²_tϕ+2γ²a∇ϕ.∇ϕ_t−2ibγ(2∇ϕ_t.∇+_∆ϕ_t)

−γκ²⁴∇. D²ϕ∇−4γ²D²ϕ∇ϕ+_∆²ϕ

+2[A(x)∇ϕ.∇ − ∇ϕ.∇A]. (3.12) If we put ϕ= |x|², then (3.12) reduces to the following

S_t+ [S,K] =−γκ²

h8∆−32γ²|x|²ⁱ+ ^d

dtA+2[A(x)∇ϕ.∇ − ∇ϕ.∇A].

Moreover by assumption (2), (S_tf+ [S,K]f,f) =γκ

Z

Rⁿ

8|∇f|²_H+32γ²|x|²kfk²dx +2

Z

Rⁿ([A(x)∇ϕ.∇f− ∇ϕ.∇A f],f)dx≥0. (3.13) This identity, the condition onV and (3.13) imply that

k∂tf−S f −K fk_X≤κ²(M₁kfk_X+e^γϕkFk_X). (3.14) If we knew that the quantities and calculations involved in the proof of Lemma 3.4 (similar as in [7, Lemma 2]) were finite and correct, when f = e^γ|x|2u we would have the logarithmic convexity ofQ(t) =e^γ|x|2u(·,t)

X and the estimate (3.10) from Lemma3.4. But this fact is verifying by reasoning as in [7, Lemma 3].

Let

σ= q

t(1−t)e^γ|x|2, Y= L²([0, 1]×_Rⁿ;H). Lemma 3.6. Assume that a, b, u, A and V are as in Lemma3.5andγ>0. Then,

kσ∇uk_Y+kσ|x|uk_Y ≤N[(1+M₁)]

"

sup

t∈[0,1]

e^γ|x|2u(.,t)

X+ sup

t∈[0,1]

e^γ|x|2F(.,t)

Y

# ,

where N is bounded number, whenγandκ are bounded below.

Proof. The integration by parts shows that Z

Rⁿ

|∇f|²_H+4γ²|x|²kfkdx=

Z

Rⁿ

h

e^2γ|x|2

|∇u|²_H−2nγ

kuk²ⁱdx,

when f = e^γ|x|2u, while integration by parts, the Cauchy–Schwarz inequality and the identity, n=∇·x, give that

Z

Rⁿ

|∇f|²_H+4γ²|x|²kfkdx≥2γnkfk²_X. The sum of the last two formulae gives the inequality

2 Z

Rⁿ

|∇f|²_H+_4γ²|x|²kfkdx≥

Z

Rⁿe^γ|x|2|∇f|²_Hdx. (3.15) Integration over [0, 1]of t(1−t)times the formula (3.6) for Q⁰⁰(t) and integration by parts, shows that

2 Z ₁

0 t(1−t) (Stf+ [S,K]f,f)_Xdt+

Z ₁

0 Q(t)dt

≤Q(1) +Q(0) +2 Z ₁

0

(1−2t)Re(∂_tf−S f −K f,f)_Xdt +

Z ₁

0 t(1−t)k∂_tf−S f −K fk²_Xdt. (3.16) Assuming again that the last two calculations are justified for f =e^γ|x|2. Then (3.13)–(3.16) imply the assertion.

4 Appell transformation in abstract function spaces

Let

ρ(t) =α(1−t) +βt, ϕ(x,t) = (α−β)x² 4(a+ib)ρ(t)^, ν(s) =

γαβρ²(s) + (α−β)a 4(a²+b²)^ρ(s)

.

Lemma 4.1. Assume A and V are as in Lemma3.5and u =u(x,s)is a solution of the equation

∂_su= (a+ib) [_∆u+Au+V(y,s)u+F(y,s)], y ∈_Rⁿ, s∈ [0, 1]. Let a+ib6=0,γ∈_Randα,β∈_R+. Set

˜

u(x,t) =^pαβρ⁻¹(t)

n

2 up

αβxρ⁻¹(t),βtρ⁻¹(t)e^ϕ. (4.1)

Then,u˜(x,t)verifies the equation

∂_tu˜ = (a+ib)_∆u˜+Au˜+V^˜ (x,t)u+F^˜(x,t), x∈_Rⁿ, t∈[0, 1] with

V˜ (x,t) =αβρ⁻²(t)Vp

αβxρ⁻¹(t),βtρ⁻¹(t),

F˜(x,t) =^pαβρ⁻¹(t)

n

2+2p

αβxρ⁻¹(t),βtρ⁻¹(t). Moreover,

e^γ|x|2F˜(·_,t)

X =αβρ⁻²(t)e^ν|y|2kF(s)k_X ,

e^γ|x|2u˜(·,t)

X =_e^ν|y|2ku(s)k_X, when s=µ(t)andγ∈_R.

Proof. Ifuis a solution of the equation

∂_su= (a+ib) [_∆u+Au+Q(y,s)], y∈_Rⁿ, s∈[0, 1] (4.2) then, the functionu₁(x,t) =u √

rx,rt+τ

verifies

∂_tu₁ = (a+ib)_∆u1+Au₁+rQ √

rx,rt+τ

, y∈ _Rⁿ, s ∈[0, 1] andu₂(x,t) =t⁻ⁿ²u ^x_t,¹_t

e

|x|²

4(a+ib)t is a solution to

∂tu2 =−(a+ib)

∆u2+Au+t⁻(2+ⁿ₂)_Q^x t,1

t

e

|x|² 4(a+ib)t

, y ∈_Rⁿ, s ∈[0, 1]. These two facts and the sequel of changes of variables below verify the lemma, whenα> β, i.e.

u s

αβ

α−βx, αβ

α−βt− ^β α−β

!

is a solution to the same non-homogeneous equation but with right-hand side αβ

α−βQ s

αβ

α−βx, αβ

α−βt− ^β α−β

! . The function,

1 (α−t)^n2u

pαβx pα−β(α−t)^,

αβ

(α−β) (α−t)− ^β α−β

! e

|x|² 4(a+ib)(α−t)

verifies (4.2) with right-hand side αβ

(α−β) (α−t)^n2+2Q

pαβx p

α−β(α−t)^,

αβ

(α−β) (α−t)− ^β α−β

! e

|x|² 4(a+ib)(α−t). Replacing(x,t)by p

α−βx,(α−β)t

we get that ρ⁻

n 2 (t)u

p

αβρ⁻¹(t)x,αβρ⁻¹(t)−β α−β

e^(α−β)

|x|²ρ(t)

4(a+ib) (4.3) is a solution of (4.2) but with right-hand

ρ⁻(ⁿ₂⁺2) (t)Q

p

αβρ⁻¹(t)x,αβρ⁻¹(t)−β α−β

e^(α−β)

|x|²ρ(t)

4(a+ib). (4.4) Finally, observe that

s =βtρ(t) = ^αβρ

−1(t)−β α−β

and multiply (4.3) and (4.4) we obtain the assertion for α > β. The case β > α follows by reversing by changes of variables,s⁰ =₁−s andt⁰ =₁−t.

5 Variable coefficients. Proof of Theorem 2.4

We are ready to prove Theorem2.4. Let

B= L¹(0, 1;L^∞(_Rⁿ;B(H))).

Proof of Theorem2.4. We may assume that α 6= β. The case α = β follows from the latter by replacing β by β+δ, δ > 0, and letting δ tend to zero. We may also assume that α < β.

Otherwise, replace u by ¯u(1−t). Assume a > 0. Set W = _∆+A+V₁. By Lemma2.7 the problem

∂_tu= (a+ib) [_∆u+A(x)u+V₁(x)u], x∈_Rⁿ, t∈ [0, 1], (5.1) u(x, 0) =u0(x).

has a solution u=U(t)u₀= e^t(a+ib)Wu₀ ∈C([0, 1];X(A)), where U(t) =F⁻¹h

e^−Q(ξ)i

, Q(ξ) = (a+ib)^h− |ξ|²+A^ˆ(ξ) +V^ˆ₁(ξ)ⁱ,

here,F⁻¹is the inverse Fourier transform, ˆA(ξ), ˆV₁(ξ)respectively denote the Fourier trans- forms of A(x),V₁(x). By reasoning as the Duhamel principle we get that the problem

∂_tu= (a+ib) [_∆u+A(x)u+V(x,t)]u, x∈_Rⁿ_, t∈[0, 1], u(x, 0) =u₀(x)

has a solution expressing as u(x,t) =H(t)u₀+i

Z _t

0 H(t−s)V₂(x,s)u(x,s)ds for x∈_Rⁿ, s∈[0, 1], (5.2) where

e^itW = H(t) =H(t,x) = F⁻¹h e^iQ(ξ)i

. For 0≤ε≤1 set

Fε(x,t) = ⁱ

ε+ie^εtWV2(x,t)u(x,t) (5.3) and

u_ε(x,t) =e^(ε+i)tWu₀+ (ε+i)

Z _t

0

e^{(ε+i)(t−s)W}F_ε(x,s)u(x,s)ds. (5.4) Then,u_ε(x,t)∈ L^∞(0, 1;X(A))∩L² Rⁿ;Y¹

and satisfies

∂_tu_ε = (ε+i) (Wu+F_ε) inRⁿ×[0, 1], uε(·, 0) =u₀(·).

The identities

e^(z1+z2)W =e^z1We^z2W, when Rez₁, Rez2 ≥0, and (5.2)–(5.4) show that

u_ε(x,t) =e^εtWu(x,t), fort∈ [0, 1]. (5.5)

In particular, the equalityu_ε(x, 1) =e^εWu(x, 1), Lemma3.2with a+ib=ε,γ= ¹

β, F≡0 and the fact thatu_ε(₀) =u(0)imply that

e

|x|²

β2+4εu_ε(·, 1) X

≤e^εkV1kB e

|x|² β2 u(·, 1)

X

,

e

|x|² α2

uε(·, 0) X

= e

|x|² α2

u(·, 0) X

.

A second application of Lemma3.2 witha+ib= ε,F ≡ 0, the value ofγ = µ⁻²(t)and (5.2) show that

ε

|x|²

µ2(t)+4εtF_ε(·,t) X

≤e^εkV1kB ε

|x|²

µ2(t)V₂(·,t) B

ku(·,t)k_X

fort ∈[0, 1]. Setting,αε =α+2εandβε = β+2ε, the last three inequalities give that

e

|x|² β2

ε uε(·, 1) X

≤e^εkV1kB e

|x|²

β2 u(·, 1) X

, (5.6)

e

|x|² α2

ε uε(·, 0) X

≤e^εkV1kB e

|x|² α2

u(·, 0) X

,

ε^|x|

2µ⁻²(t)F_ε(·,t)

X≤ e^εkV1kB ε^|x|

2µ⁻²(t)V₂(·,t)

Bku(·,t)k_X, t∈[0, 1]. (5.7) A third application of Lemma3.2witha+ib=b,F≡0,γ=0, and (5.2), (5.5) implies that

kFε(·,t)k_X ≤e^εkV1kBkV₂(·,t)k_Bku(·,t)k_X, (5.8) kuε(·,t)k_X ≤e^εkV1kBku(·,t)k_X, t ∈[0, 1].

Setγε = _α¹

εβ_ε and let

˜

u_ε(x,t) = p

α_εβ_ερ⁻_ε ¹(t) ⁿ₂

up

α_εβ_εxρ⁻_ε ¹(t),β_εtρ⁻_ε ¹(t)

e^ϕε

be the function associated to u_ε in Lemma 4.1, where a+ib = ε+i and α, β are replaced respectively byαε,βε when

ρε(t) =αε(1−t) +βεt, ϕε = ϕε(x,t) = (αε−βε)|x|² 4(a+ib)ρε(t)^. Becauseα< β, ˜uε ∈ L^∞(0, 1;X)∩L² 0, 1;Y¹

and satisfies the equation

∂_tu˜_ε = (ε+i) _∆u˜_ε+A(x)u˜_ε+V^˜₁^ε(x,t)u˜_ε+F^˜_ε(x,t) inRⁿ×[0, 1], where

V˜₁^ε(x,t) =α_εβ_ερ⁻_ε ²(t)V₁p

α_εβ_ερ⁻_ε ¹(t)x

, sup

t∈[0,1]

V˜₁^ε(·,t)

B ≤ ^β αM₁, F˜_ε(x,t) =^hpα_εβ_ερ⁻_ε ¹(t)ⁱ

n 2+2

F_εp

α_εβ_ερ⁻_ε ¹(t)x,β_εtρ⁻_ε ¹(t)e^ϕε, (5.9)