getdoc8faa. 435KB Jun 04 2011 12:04:40 AM

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El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y}

Vol. 15 (2010), Paper no. 33, pages 1041–1091. Journal URL

http://www.math.washington.edu/~ejpecp/

Stochastic nonlinear wave equations in local Sobolev

spaces

Martin Ondreját

∗†‡

Abstract

Existence of weak solutions of stochastic wave equations with nonlinearities of a critical growth driven by spatially homogeneous Wiener processes is established in local Sobolev spaces and local energy estimates for these solutions are proved. A new method to construct weak solutions is employed.

Key words:stochastic wave equation.

AMS 2000 Subject Classification:Primary 60H15.

Submitted to EJP on March24, 2010, final version accepted June 30, 2010.

∗_{Affiliation: Institute of Information Theory and Automation, Academy of Sciences, Czech Republic}

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1

Introduction

Nonlinear wave equations

u_{t t}=_Au+f(x,u,u_t,_∇_xu) +g(x,u,u_t,_∇_xu)W˙ (1.1) subject to random excitations have been thoroughly studied recently under various sets of hypothe-ses (see e.g. [6], [7], [8], [10], [12], [13], [14], [22], [23], [25], [26],[27], [28], [29], [30], [32],[35],[36],[37]and references therein) with possible applications in physics (e.g. in relativis-tic quantum mechanics or oceanography) in view.

The random perturbation has been usually modelled by a spatially homogeneous Wiener process which corresponds to a centered Gaussian random field(W(t,x):t_≥0, x_∈Rd)satisfying

EW(t,x)W(s,y) = (t_∧s)Γ(x₋y), t,s_≥0, x,y _∈Rd

for some function or even a distributionΓcalledthe spatial correlationofW (see e.g. [37]for de-tails). The operator_A in (1.1) is a second order elliptic differential operator, usually the Laplacian. (More general elliptic operators are considered only in[29]).

Functionsf andgdependent only onuare dealt predominantly and their global Lipschitz continuity is assumed in most of the papers cited above. Then the Nemytskii operators associated with f and

gare also globally Lipschitzian and existence (and uniqueness) of solutions to (1.1) may be proved for rather general spatial correlationsΓ(which may be a distribution, e.g. the standard cylindrical Wiener process is allowed if the space dimension is one, or at least a continuous function unbounded at the origin), the state space (to which the pair(u,u_t)belongs) being L2(Rd)_⊕W−1,2(Rd). IfΓis more regular then solutions live in the so called “energy space”W1,2(Rd)_⊕L2(Rd) (see e.g. [38] for a discussion of the role of the energy space in the deterministic case).

Locally Lipschitz (or even continuous) real functions f and g are considered in the papers [10], [27],[29],[30],[31]and[32].

Various techniques including Lyapunov functions, energy estimates, Sobolev embeddings, Strichartz inequalities or compactness methods - have been employed to show existence of global solutions in this case. These methods require the state space to be the energy spaceW1,2(Rd)_⊕L2(Rd)and the spatial correlationΓto be a bounded function, i.e. thespectral measureµ= (2π)d2Γbis a finite

measure (cf. equality (3.1) below). The assumptions on Γare relaxed in [27]in the case of the planar domain at a price of assuming g bounded and globally Lipschitz while f(u) = ₋u_|u_|p−1, 1_≤p_≤3.

Let us survey the available results in the most important case of a wave equation with polynomial nonlinearities

u_{t t} = ∆u₋u_|u_|p−1+_|u_|qW˙, u(0) =u₀, u_t(0) =v₀ (1.2) according to particular ranges of the exponentsp,q_∈(0,_∞): It is known that global weak solutions (weak both in the probabilistic and in the PDE sense) exist provided that(u₀,v₀)is an_F₀-measurable

[W1,2(Rd)_∩Lp+1(Rd)]_⊕ L2(Rd)-valued random variable, W is a spatially homogeneous Wiener process with bounded spectral correlationΓ(i.e. µ= (2π)d2Γbmust be a finite measure) and

1_≤q< p+1

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Under (1.3), paths of(u,u_t)take values in[W1,2(Rd)_∩Lp+1(Rd)]_⊕L2(Rd)and are weakly contin-uous inW1,2(Rd)_⊕L2(Rd)(see[29]). In the critical caseq= p+1

2 , existence of solutions was shown if d∈ {1, 2}ord ≥3 and p≤ _d₋d₂ (see[30]). Pathwise uniqueness and pathwise norm continuity of solutions inW1,2(Rd)_⊕L2(Rd)are known to hold ifd∈ {1, 2}or ifd≥3 andp≤ d_d+₋2₂,q≤ d_d+₋1₂, irrespective of (1.3). These results were proved in[29], [30], [31] and [32]in a more general setting (that is, for more general non-linearities). In case d

d−2 < p≤ d+2 d−2 or

d

d−2 < q≤ d+1 d−2 some small additional assumptions are needed. In the subcritical case p < d_d+₋2₂, q < d_d+₋1₂, these results correspond to the present state of art for the deterministic Cauchy problem

u_{t t}= ∆u₋u_|u_|p−1, u(0) =u₀, u_t(0) =v₀ (1.4) on Rd (see [19], [41], [43]and [44]) exactly, whereas there are still some open problems in the (stochastic) critical casep= d_d+2

−2,q= d+1

d−2 (see the discussion in[31]). The aim of the present paper is fivefold. We want to prove

1. existence of weak solutions up to the critical caseq= p+1

2 independently of the dimension of the spatial domainRd,

2. global weak solutions exist for data in thelocalSobolev spaceW_{l oc}1,2(Rd)_⊕L2_{l oc}(Rd)and have trajectories weakly continuous in this local space,

3. solutions in the local Sobolev space satisfy a local energy inequality (Theorem 5.2),

4. include dependence on first derivatives in the non-linearities in the equation (1.1)

5. study systems of stochastic wave equations, i.e. when f and gareRn-valued.

Let us briefly comment on the issues.

Ad (1): As mentioned above, existence of weak solutions in the critical case q = p+1

2 is known to hold only in particular cases depending on the dimension d. We will prove that no additional assumption is, in fact, necessary.

Ad (2): To our knowledge, stochastic equations with polynomially growing non-linearities have not been studied in local spaces yet despite it is well known that solutions of wave equations propagate at finite speed and the commonly used restriction to global spaces is therefore unimportant. Nota bene, existence of solutions in global spaces follows trivially from existence of solutions in local spaces by the energy estimate in Theorem 5.2, as demonstrated e.g. in Example 5.5.

As a consequence of the “local” approach to the wave equation (1.1), the second order differential operatorA in (1.1) need not beuniformlyelliptic (as is usually assumed) and mere ellipticity ofA

is sufficient (see (2.1)). In particular,_A may even decay or explode at infinity, cf. Example 5.4 and 5.5.

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Ad (3): Energy inequalities are a sort of a twin result to any existence theorem in the theory of wave equations as the solutions of the wave equation are, in fact, stationary points of certain Lagrangians and the energy functionals represent their conservation laws (see e.g. Chapter 2 in[42]). On the other hand, energy inequalities also describe basic behaviour of the solution such as the finite speed of propagation mentioned above, long time behaviour of the paths or the conditional dependence on the initial condition (see Theorem 5.2).

Ad (4) & (5): We are not aware of existence results for stochastic wave equations with non-linearities depending on first derivatives of the solution (the velocity and the spatial gradient). This issue is closely related to the fact that we aim at studying systems of stochastic wave equations (1.1). Such generality is not very substantial for the present paper, however, the corresponding results are essential in the newly started research in the field of stochastic wave equations in Riemannian manifolds with possible applications in physical theories and models such as harmonic gauges in general relativity, non-linear σ-models in particle systems, electro-vacuum Einstein equations or Yang-Mills field theory. These models require the target space of the solutions to be a Riemannian manifold (see[18],[42]for deterministic systems and[3],[4]for stochastic ones). For instance, if the unit sphereSn−1is the considered Riemannian manifold, the stochastic geometric wave equation has the form

u_{t t} = ∆u+ (_|∇_xu_|2_{− |}u_t_|2)u+g(u,u_t,_∇_xu)W˙, _|u_|_Rn=1

where g(p,_·,_·)_∈TpSn−1, p∈Sn−1, see e.g. [4]. We do not cover these particular equations here but the present paper is partly intended as a preparation for further applications and as a cita-tion/reference paper for a companion paper on stochastic wave equations in compact Riemannian homogeneous space by Z. Brze´zniak and the author.

Finally, we remark that our proof of the main theorem is based on a new general method of con-structing weak solutions of SPDEs, that does not rely on any kind of martingale representation theorem and that might be of interest itself. First applications were done already in[4]and, in the finite-dimensional case, also in[20].

The author wishes to thank Jan Seidler and the referee for a kind helping on the redaction of the paper.

2

Notation and Conventions

We consider complete filtrations in this paper. We say that a filtration(_F_t)on a probability space

(Ω,F,P)is complete provided thatF0contains allP-negligible sets ofF. We denote by

• R₊the set of all nonnegative real numbers, i.e. R₊= [0,∞),

• B(X)the Borelσ-algebra on a topological spaceX,

• B_Rthe open ball inRd with center at the origin and of radiusR,

• Lp=Lp(Rd,Rn),Wk,p=Wk,p(Rd,Rn),

• L_locp =L_locp (Rd,Rn)andW_lock,p=W_lock,p(Rd,Rn)equipped with the metrics

(u,v)_7→ ∞

X

j=1 1

2j min{1,ku−vkLp(Bj)}, (u,v)7→ ∞

X

j=1 1

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• HRk=W k+1,2₍_B

R)⊕Wk,2(BR),HR:=HR0,

• Hk=Wk+1,2_⊕Wk,2,_H :=_H0,

• Hloc=W 1,2 loc ⊕L

2 loc,

• D=_D(Rd,Rn)is the space ofRn-valued compactly supportedC∞-functions,

• S is the Schwartz spaces of complex rapidly decreasing C∞-functions onRd ([40]),

• S′is the space of tempered distributions on_S, i.e. the real dual space to_S,

• ξ7→ξbthe Fourier transformation on_S′,

• Ck_b the space of k-times continuously differentiable functions onRd with bounded derivatives up to orderkequipped with the supremum norm of all derivatives up to orderk,

• Cγ([a,b],X)the Banach space ofX-valuedγ-Hölder continuous functions on[a,b]with the norm

kh_k=sup_{kh(t)_k_X :t_∈[a,b]_}+sup

kh(t)₋h(s)_k_X

(t₋s)γ :a≤s<t≤b

whereX is a Banach space,

• A a second order elliptic operator

A =

d

X

k=1 d

X

l=1 ∂ ∂x_k

akl(x)

∂ ∂x_l

wherea(x)is a symmetric, strictly positive,(d×d)-real-matrix for every x∈Rd andais a continu-ous,W_{l oc}1,∞(Rd)-valued function such that

inf¦t−2sup¦a(x)y_Rd:|x|_Rd<t,|y|_Rd=1

©

:t>0©=0, (2.1)

Spatially homogeneous Wiener process

Given a stochastic basis(Ω,_F,(_F_t)_t_≥₀,P), an_S′-valued processW = W_t_t_≥₀ is called a spatially homogeneous Wiener process with a spectral measureµthat we assume to be positive, symmetric and to satisfyµ(Rd)<∞throughout the paper, provided that

• Wϕ:= W_tϕ_t_≥₀is a real(_F_t)-Wiener process, for everyϕ∈ S,

• W_t(aϕ+ψ) =aW_t(ϕ) +W_t(ψ)almost surely for alla_∈R, t_∈R₊andϕ,ψ∈ S,

• E_{W_tϕ1Wtϕ2}=t〈ϕb1,ϕb2〉L2₍_µ₎for allt≥0 andϕ₁,ϕ₂∈ S.

Remark 3.1. “Spatial homogeneity” refers to the fact that the process W can be represented as a centered(_F_t)-adapted Gaussian random field(_W(t,x):t≥0,x∈Rd)so that

P

Wtϕ=

Z

Rd

ϕ(x)_W(t,x)d x

=1, ϕ∈ S

and

E[_W(t,x)_W(s,y)] =min_{t,s}Γ(x− y), t,s≥0, x,y ∈Rd

whereΓ = (2π)−d2µbis a bounded continuous function. The reader is referred to[5],[12],[36]and

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Let us denote by H_µ _{⊆ S}′ the reproducing kernel Hilbert space of the _S′-valued random vector

W(1), see e.g. [11]. Then W is an Hµ-cylindrical Wiener process. Moreover, see [5]and[37], if we denote byL2₍_s₎(Rd,µ)the subspace ofL2(Rd,µ;C)consisting of allψsuch thatψ=ψ(s), where

ψ(s)(·) =ψ(−·), then we have the following result:

Proposition 3.2.

Hµ = {dψµ: ψ∈L(2s)(R

d_,_µ₎

},

〈dψµ,ϕµÓ〉H_µ =

Z

Rd

ψ(x)ϕ(x)dµ(x), ψ,ϕ∈L₍2_s₎(Rd,µ).

The following lemma states that, under some assumptions, Hµ is a function space and that multi-plication operators are Hilbert-Schmidt from Hµ to L2 (see[30]for a proof). In that case, we can calculate the Hilbert-Schmidt norm explicitly.

Lemma 3.3. Assume thatµ(Rd)<∞. Then the reproducing kernel Hilbert space Hµ is continuously

embedded in C_b(Rd), the multiplication operator m_g=_{H_µ _∋ξ7→ g_·ξ∈ L2(D)_}is Hilbert-Schmidt and there exists a constantcsuch that

km_g_k_L

2(Hµ,L2(D))=ckgkL2(D) (3.1)

whenever D_⊆Rd is Borel and g_∈L2(D).

Remark3.4. A stochastic integral with respect to a spatially homogeneous Wiener process is under-stood in the classical way, see e.g.[11],[36]or[37].

4

Solution

Definition 4.1. Let f1, . . . ,fd :Rd_×Rn _→Rn×n be Borel matrix-valued functions, let fd+1,gd+1 :

Rd_×Rn_→Rnbe Borel functions,µa given finite spectral measure onRd,(Ω,_F,(_F_t),P)a completely filtered probability space with a spatially homogeneous(_F_t)-Wiener process W with spectral measure

µ. An (_F_t)-adapted process z = (u,v) with weakly continuous paths in Hl oc is a solution of (1.1)

where, for(x,y,z) = (x,y,z₀, . . . ,z_d)_∈Rd_×Rn_×Qd_i₌₀Rn,

f(x,y,z) =

d

X

i=0

fi(x,y)zi+fd+1(x,y), g(x,y,z) = d

X

i=0

gi(x,y)zi+gd+1(x,y) (4.1)

provided that

P

 

Z T

0

n

kf(_·,u(s),v(s),_∇_xu(s))_k_L1₍_B

T)+kg(·,u(s),v(s),∇xu(s))k

2 L2₍_B

T)

o ds



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holds for every T>0and

〈u(t),ϕ〉 = _〈u(0),ϕ〉+

Z t

0

〈v(s),ϕ〉ds (4.3)

〈v(t),ϕ〉 = _〈v(0),ϕ〉+

Z t

0

〈u(s),_Aϕ〉ds+

Z t

0

f(_·,u(s),v(s),_∇_xu(s)),ϕds

+

Z t

0

〈g(_·,u(s),v(s),_∇u(s))dW_s,ϕ〉

holds for every t≥0a.s. wheneverϕ∈ D.

Remark 4.2. The assumption (4.2) guarantees existence of integrals in (4.3). Let us verify the convergence of the stochastic integral in (4.3). For, let us denoteρ=g(_·,u,v,_∇u), letξj=ψÔjµbe an ONB inHµ (i.e. by Proposition 3.2,(ψj)is an ONB in L2₍_s₎(Rd,µ)) and letT >0 be such that the support ofϕ is contained inB_T. Then

k〈ρ(s)_·,ϕ〉k2_L₂(H_µ,R) = X

j

_〈_ρ(s)ξj,ϕ〉

2

= (2π)−dX

j

Z

Rd

Z

Rd

e−i〈x,y〉_〈ρ(s,x),ϕ(x)_〉_Rnψ_j(x)d xµ(d y)

2

= (2π)−d

Z

Rd

Z

Rd

e−i〈x,y〉_〈ρ(s,x),ϕ(x)_〉_Rnd x

2 µ(d y)

≤ c_◦_〈ρ(s,x),ϕ〉Rn

2

L1≤c_◦ _ϕ2

L2₍_B

T)

_ρ(s,x)2_L2₍_B

T)

wherec_◦= (2π)−d_µ₍_Rd₎_{and, by (4.2),}Rt

0kξ7→ 〈ρ(s)ξ,ϕ〉k 2

L2(Hµ,R)ds<∞for allt>0 a.s. Hence,

the stochastic integral in (4.3) is well defined e.g. by[11].

5

The main result

Assume that

i) fi,gi:Rd×Rn→Rn×n,i∈ {0, . . . ,d}, ii) fd+1,gd+1:Rd_×Rn_→Rn,

iii) F:Rd_×Rn_→R₊

are measurable functions,κ∈R₊,(αr,R)r,R>0 are real numbers such that limR→∞αr,R=0 for every

r>0, and, for every y_∈Rn,r>0 andR>0,

iv) F(w,_·)_∈C1(Rn),

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vi) _|f0(w,y)_|2+_|g0(w,y)_|2_≤κ,

vii)

n

X

j=1 n

X

k=1

    

a−12(w)    

f_jk1(w,y)

.. .

f_jkd(w,y)

   

2

Rd

+

a−12(w)    

g1_jk(w,y)

.. .

gd_jk(w,y)

   

2

Rd

    ≤κ,

viii) gd+1(w,y)2+_∇_yF(w,y) +fd+1(w,y)2_≤κF(w,y), ix) _|fd+1(w,y)_{| ≤}κF(w,y),

x) 1[|w|≤r]∩[|y|≥R]|fd+1(w,y)| ≤αr,RF(w,y), xi) kFmax(·,r)kL1₍_B

r)<∞where

F_max(x,r) = sup

|y|≤r

F(x,y) (5.1)

hold for almost everyw∈Rd.

Theorem 5.1(Existence). Letµbe a finite spectral measure and letΘbe a Borel probability measure on_H_{l oc} such that

Θ_{(u,v):_{∈ H}_{l oc}:_kF_max(_·,_|u(_·)_|+1)_k_L1₍_B

r)<∞}=1, r>0. (5.2)

Then there exists a completely filtered stochastic basis (Ω,_F,(_F_t),P) with a spatially homogeneous

(_F_t)-Wiener process W with spectral measure µand an (_F_t)-adapted process z= (u,v)with weakly continuous paths in Hl oc which is a solution of the equation (1.1) in the sense of Definition 4.1 and P[z(0)_∈A] = Θ(A)for every A_{∈ B}(_H_{l oc}).

Theorem 5.2(Energy estimate). Letµ,Θ,κand(Ω,_F,(_F_t),P,W,z)be the same as in Theorem 5.1, let (5.2) hold and letκ˜∈R+. Let G:Rd×Rn→R+ be a measurable function such that

• G(w,_·)_∈C1(Rn),

• |gd+1(w,y)_|2+_|∇_yG(w,y) + fd+1(w,y)_|2_≤κ˜G(w,y), y_∈Rn

hold for almost every w∈Rd and let

Θ_{(u,v):∈ Hl oc:kGmax(·,|u(·)|+1)kL1₍_B

r)<∞}=1, r>0

where

G_max(w,r) = sup

|y|≤r

G(w,y), r>0.

Then there exists a constantρ∈R₊ depending only on the numbersµ(Rd)andmax_{κ, ˜κ}such that, given x_∈Rd, T >0,

λ≥ sup w∈B(x,T)k

a(w)_k12, (5.3)

a non-decreasing function L∈C(R+)∩C2(0,∞)such that

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the estimate

E

¨

1_A(z(0)) sup r∈[0,t]

L(G(r,z(r)))

«

≤4eρtE1_A(z(0))L(G(0,z(0))) (5.5)

holds with the convention0_{· ∞}=0for every A_{∈ B}(_H_{l oc})and t_∈[0,T/λ]whereG=G_λ_,_x_,_T is the conic energy function for G defined as in (2.2).

Remark 5.3. Let us observe that Theorem 5.2 is a sort of extension of Theorem 5.1 that claims

that the particular solution constructed in Theorem 5.1 satisfies the infinite number of qualitative properties (i.e. given whichever entriesG, L etc.) in Theorem 5.2. We however cannot exclude, at this moment, the possibility of existence of a weak solution of (1.1) for which (5.5) is not satisfied.

5.1

Examples

Example 5.4 (Local space). Let α ∈ (_−∞, 2), let a_i,b_i : Rd+1 _→ R, i _{∈ {}0, . . . ,d_} be bounded measurable functions continuous in the last variable, letA:Rd →R+,B:Rd+1→Rbe measurable

functions such thatBis continuous in the last variable, letB2_≤ρAfor someρ≥0, letAbe locally integrable, let

0<p, 0≤q≤ p+1

2 , βi≤ α

2, γi≤ α

2, i∈ {1, . . . ,d}, letΘbe a Borel probability measure on_H_{l oc} such that

Z

Br

A(x)_|u(x)_|p+1d x<∞, r>0

holds forΘ-almost every(u,v)_{∈ H}_{l oc}and letµbe a finite spectral measure. Then the equation

u_{t t} = (1+_|x_|)α∆u+



a₀(x,u)u_t+

d

X

i=1

(1+_|x_|)βi_a

i(x,u)

∂u

∂xi −

A(x)u_|u_|p−1  

+



b₀(x,u)ut+ d

X

i=1

(1+_|x|)γi_b

i(x,u)

∂u

∂x_i +B(x,u)|u|

q



_W˙ _(5.6)

has a weak solution z = (u,v) with weakly Hl oc-continuous paths, with the initial distribution Θ and

sup t∈[0,r]

Z

Br

A(x)_|u(t,x,ω)_|p+1d x<∞, r >0 (5.7)

holds a.s. where W is a spatially homogeneous Wiener process with spectral measure µ. It is straightforward to verify that the hypotheses i) - xi) at the beginning of Section 5 hold if we put

F(x,y) =A(x)|y|p+1

p+1 +1

andαr,R=supt≥Rtp( tp+1

p+1+1)

−1_.

Example5.5 (Global space). The equation

u_{t t} = ∆u+a₀(x,u)u_t+

d

X

i=1

a_i(x,u)u_x_i ₋u_|u_|p−1

+



b₀(x,u)ut+ d

X

i=1

bi(x,u)uxi+bd+1(x,u)|u|

q



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is a particular case of (5.6) withα=βi=γi =0,A=1 andB=bd+1, so we know, by Example 5.4, which we develop here further, that ifai,bi :Rd+1→R, i∈ {0, . . . ,d+1}are bounded measurable functions continuous in the last variable,

1≤q≤ p+1

2 ,

Θis a Borel probability measure on_H such thatu_∈Lp+1holds forΘ-almost every(u,v)_{∈ H} and µ is a finite spectral measure, that the equation (5.8) has a weak solutionz = (u,v) with weakly

Hl oc-continuous paths, with the initial distribution Θ and W is a spatially homogeneous Wiener process with spectral measureµ. Notwithstanding, the estimate (5.7) can be further strengthened to

sup t∈[0,r] k

z(t)_k_H +_ku(t)_k_Lp+1<∞, r>0, a.s. (5.9)

by applying Theorem 5.2 on the function G(x,y) =_|y_|p+1/(p+1) +_|y_|2/2, L(x) = x andλ=1 (which satisfy the assumptions of Section 5). In particular, paths of the solution z are not only weaklyHl oc-continuous, but weaklyH-continuous a.s.

Proof. The assumptions of Theorem 5.2 are satisfied by a direct verification so if we define

HR={(u,v)∈ H :k(u,v)kH +kukLp+1≤R}, R>0

then, for a fixedρ >0 independent ofz,RandT,

E1_H

R(z(0))sup

r≤t

  k

z(r)_k2_HT−t

2 +

ku(r)_kp+1

Lp+1₍_B

T−t)

p+1

 

 ≤ 4eρt Z

HR

  k

(u,v)_k2

HT

2 +

kukp_L+p+11₍_B

T)

p+1

  dΘ

≤ 4eρt

R2

2 +

Rp+1 p+1

=:C_t_,_R

holds for everyT >0 andt∈[0,T)by Theorem 5.2. LettingT→ ∞, we obtain

E1_H_R(z(0))sup r≤t



kz(r)k2H

2 +

ku(r)_kp+1

Lp+1 p+1

 _≤C_t_,_R

by the Fatou lemma. Thus

P



1HR(z(0))sup

r≤t

 kz(r)k

2

H

2 +

ku(r)_kp+1

Lp+1 p+1



<∞, R>0

 =1

whence we get (5.9).

6

Guideline through the paper

The present work generalizes the state of art in the five directions mentioned in the Introduction. Let us, though, illustrate the progress with respect to the most related paper[29]on the example of the equation

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in the global space W1,2(Rd)_⊕L2(Rd). Here, the nonnegative function F is just the potential of

−f, i.e. f +_∇F = 0 (provided it exists) and its purpose is to control the growth of the norm of the solutions in the energy space - see the apriori estimates (8.4) and (10.4) which generalize the conservation of energy law in the theory of deterministic wave equations (see e.g.[38]).

If the equation (6.1) is scalar, existence of a weak solution of (6.1) was proved, independently of the dimension d, in [29] provided that, roughly speaking, f and g are continuous, the primitive functionF to−f is positive and

lim

|t|→∞

|f(t)_|+_|g(t)_|2

F(t) =0, (6.2)

i.e. only the case of subcritically growing polynomial nonlinearities q < (p+1)/2 was covered for the equation (1.2) (cf. the hypothesis (1.3)). The condition (6.2) was induced in[29]by the method of proof in global spaces and it was not surprising because it just accompanied the Strauss hypothesis lim_|_t_|→∞_|f(t)_|/F(t) = 0 on the drift in the deterministic equation (see [44]) by the expected hypothesis lim_|_t_|→∞_|g(t)_|2/F(t) =0 on the diffusion. However, the subtle arguments in Section 11 based on the local nature of the equation show that mere eventual boundedness of|g|2_/_F is sufficient for the proofs to go through and, consequently, the critical caseq = (p+1)/2 for the equation (1.2) is covered.

The proofs of both Theorem 5.1 and Theorem 5.2 are based on a refined stochastic compactness method (adapted from [17]) which consists in the following: a sequence of solutions of suitably constructed approximating equations is shown to be tight in the path space of weakly continuous vector-valued functions _Z = C_w(R₊,W_{l oc}1,2)_×C_w(R₊,L2_{l oc}). This space is not metrizable hence the Jakubowski-Skorokhod theorem[21]is applied (instead of the Skorokhod representation theorem) to model the Prokhorov weak convergence of laws as an almost sure convergence of processes on a fixed probability space to a limit process which is, eventually, proved to be the desired weak solution. Apart from the Jakubowski-Skorokhod theorem, another novelty of the method relies in the fact that the identification of the limit process with the solution is not done via a martingale representation theorem (which is not available in our setting anyway) but by a few tricks with quadratic variations (see Sections 9 and 10).

Let us briefly comment on the structure of the proofs: The stochastic Cauchy problem (1.1) is first reduced (by a localization in Section 7) to the global Hilbert spaceW1,2(Rd)_⊕L2(Rd)where an apriori estimate (8.4) independent of the localization is proved. The stochastic compactness method is then applied in two steps:

First, existence of a weak solution withW1,2(Rd)_⊕L2(Rd)-continuous paths is established in Section 9 for sub-linearly growing Lipschitz functions f = (fi), g = (gi) using the apriori estimate (8.4), where the nonlinearities are simply mollified by smooth densities. This step is not trivial since the Nemytski operators associated to f and gare not “locally Lipschitz” on the state spaceW1,2(Rd)_⊕

L2(Rd)- yet, the stochastic compactness method is employed in its standard form (see e.g. [2],[17] or[29]).

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7

Localization of the operator

A

Our differential operator _A must be localized first in order general theorems on generating C0 -semigroups can be applied. For, let us define theC1-functionh:R→[0, 1]by

h(t) =1, t_∈(_−∞, 1], h(t) = (2t₋1)(t₋2)2, t_∈[1, 2], h(t) =0, t_∈[2,_∞),

φm(x):=h(_|x_|/m)x, x_∈Rd, m_∈N. (7.1) We may verify quite easily thatφm is diffeomorphic on_{x :_|x_|< 11+

p

57

16 }and also on{x : 11+p57

16 <

|x_| < 2_} hence the set B₂_m _∩[φm ∈ C] is negligible for the Lebesgue measure whenever so is

C ∈ B(Rd), hencea◦φm _{is uniformly continuous on}_Rd_{, belongs to}_W1,∞₍_Rd₎_where

∂(a_◦φm)

∂x_j =1B2m

d

X

l=1 ∂a

∂x_l(φ

m₎∂ φ m l

∂x_j , j∈ {1, . . . ,d}, m∈N

and if we define

Am=

d

X

i=1 d

X

j=1 ∂ ∂x_i

(a_{i j}_◦φm) ∂

∂x_j

, m_∈N (7.2)

then we have the following result which concerns a realization of the differential operator_AminL2

on the Sobolev spaceW2,2, states its funcional-analytic properties and allows to introduce a matrix infinitesimal generator of a waveC0-group.

Proposition 7.1. For m_∈N, it holds that

• the operatorAm_with_Dom_Am₌_W2,2_{is uniformly elliptic, selfadjoint and negative on L}2_,

• Dom(I_{− A}m)12 =W1,2with equivalence of the graph norm and the W1,2-norm,

• the graph norm onDom_Am is equivalent with the W2,2-norm,

• the operator

Gm= 0 I

Am 0

, Dom_Gm=W2,2_⊕W1,2

generates a C₀-group(Sm_t )on the space W1,2_⊕L2,

• the graph norm onDomGm _{is equivalent with the W}2,2_⊕_W1,2_-norm.

Proof. The operator_Am is selfadjoint e.g. by a consequence of Theorem 4 in Section 1.6 in[24], the operator

₀ _I

Am−I 0

is skew-adjoint in Dom(I_{− A}m)12 ⊕L2 hence generates a unitary C₀-group by the Stone theorem

(see e.g. Theorem 10.8 in Chapter 1.10 in[34]). The operator

_{0 0}

I 0

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is bounded on Dom(I_{− A}m)12 ⊕L2 soGm generates aC

0-group by Theorem 1.1 in Chapter 3.1 in [34].

The equivalence of the graph norm on Dom_Am with the W2,2-norm follows from Theorem 5 in Section 1.6 in[24].

We close this section by introducing the localized conic energy function relative to F and to the operator_Am analogously to (2.2). Toward this end, given x _∈Rd, T>0,λ >0 and a measurable functionF:Rd×Rn→R+, we define

F_m,λ,x,T(t,u,v) =

Z

B(x,T−λt) (

1 2

d

X

k=1 d

X

l=1

(a_kl◦φm)¬u_x_k,u_x_l¶

Rn+ 1 2|v|

2

Rn+F(y,u)

)

d y (7.3)

fort_∈[0,T_λ]and(u,v)_{∈ H}_{l oc}.

Remark7.2. Observe that the integrand in (7.3) coincides with the integrand of (2.2) on the cen-tered ball B_m, hence the localized conic energy functionF_m,λ,x,T relative to F and to the operator

Amis really a “localization” of the conic energy functionF_λ_,_x_,_T relative toF and to the operator_A.

8

A local energy inequality

In this technical section we shall establish a backward cone energy estimate that, on one hand, makes it possible to find uniform bounds for a suitable approximating sequences of processes that will later on yield a solution by invoking a compactness argument, and on the other hand, imply finite propagation property of solutions of (1.1).

Proposition 8.1. Let m_∈N, T >0, let U be a separable Hilbert space and W a U-cylindrical Wiener process. Letαandβbe progressively measurable processes with values in L2andL2(U,L2)respectively

such that

P

 

Z T

0

n

kα(s)_k_L2+kβ(s)k2

L2(U,L2) o

ds<∞



=1. (8.1)

Assume that z= (u,v)is an adapted process with continuous paths inH such that

〈u(t),ϕ〉L2 = 〈u(0),ϕ〉_L2+ Z t

0

〈v(s),ϕ〉L2ds (8.2)

〈v(t),ϕ〉L2 = 〈v(0),ϕ〉_L2+ Z t

0

〈u(s),_Amϕ〉L2ds+ Z t

0

〈α(s),ϕ〉L2ds+ Z t

0

〈β(s)dW_s,ϕ〉L2

holds a.s. for every t≥0and everyϕ∈ D. Assume that F:Rd_×Rn_→R₊is such that

(a) F(w,·)_∈C1(Rn)for every w∈Rd, (b) F(_·,y)measurable for every y∈Rn

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sup

(

F(w,y)

1+_|y_|2 +

_∇_yF(w,y)

1+_|y_| :|w| ≤r,y∈R

n

)

<∞, r>0.

Fix x∈Rd andλso that

λ≥ sup w∈B(x,T)k

a(φm(w))_k12

whereφm _{was defined in (7.1), and consider the conic energy function}_F₌_F

m,λ,x,T for F (see (7.3)).

Assume also that a nondecreasing function L:R₊_→Ris of C2-class on[0,_∞)and put

V(t,z) = 1

2L

′′₍_F₍_t_,_z₎₎X

l

v,β(t)el

2

L2₍_B₍_x_,_T₋_λt₎₎+

1 2L

′₍_F₍_t_,_z₎₎_k_β₍_t₎_k2

L2(U,L2(B(x,T−λt)))

+ L′(F(t,z))_〈v,∇yF(·,u) +α(t)〉L2₍_B₍_x_,_T₋_λt₎₎ (8.3)

for t∈[0,T_λ]and z= (u,v)_{∈ H} where(el)is any ONB in U. Then

L(F(t,z(t))) _≤ L(F(r,z(r))) +

Z t

r

V(s,z(s))ds

+

Z t

r

L′(F(s,z(s)))_〈v(s),β(s)dW_s〉L2₍_B₍_x_,_T₋_λs₎₎ (8.4)

is satisfied for every0_≤r<t_≤ T_λ almost surely.

Proof. By density, (8.2) holds for everyϕ∈W2,2and if we test withϕ=ǫ2(ǫ− Am)−2ψforǫ >0 andψ∈L2then

uǫ(t) = uǫ(0) +

Z t

0

vǫ(s)ds

vǫ(t) = vǫ(0) +

Z t

0

Amuǫ(s) +αǫ(s)

ds+

Z t

0

βǫ(s)dWs

holds for everyt_≥0 a.s. where

uǫ= ǫ2(ǫ− Am)−2u, vǫ= ǫ2(ǫ− Am)−2v, zǫ= (uǫ,vǫ)

αǫ= ǫ2(ǫ− Am)−2α, βǫξ= ǫ2(ǫ− Am)−2[βξ], ξ∈ U

and the integrals converge in Dom_Am =W2,2 whose norms are equivalent by Proposition 7.1. If

Fj:Rd×Rn→R+satisfies

i) Fj(w,y) =0 for everyw_∈Rd and_|y_|> j, ii) Fj(w,_·)_∈C∞(Rn)for everyw_∈Rd, iii) sup¦|DγyFj(w,y)|:|z| ≤r,y∈Rn

©

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then

Fj=F_mj _,_λ_,_x_,_T _∈C1,2[0,T/λ]_×W2,2⊕W2,2

as

kϕkL2₍_∂_B₍_y_,_r₎₎≤2kϕk_L2k∇ϕk_L2, ϕ∈W1,2

holds for every y _∈Rd and r > 0. We may thus apply the Ito formula (see[11]) on L(Fj(z_ǫ))to obtain

L(Fj(t,zǫ(t)))−L(Fj(r,zǫ(r))) =

− λ

Z t

r

L′(Fj(s,zǫ(s)))kFj(·,uǫ(s))kL1₍_∂_B₍_x_,_T₋_λs₎₎ds

− λ

2

Z t

r

L′(Fj(s,zǫ(s)))

d X

i=1 d

X

k=1

(a_ik_◦φm)

∂uǫ(s)

∂x_i ,

∂uǫ(s)

∂x_k Rn

L1₍_∂_B₍_x_,_T₋_λs₎₎ ds

− λ

2

Z t

r

L′(Fj(s,z_ǫ(s)))v_ǫ(s)2_L2₍_∂_B₍_x_,_T₋_λs₎₎ ds

+

Z t

r

L′(Fj(s,zǫ(s)))

(Z

B(x,T−λs)

d

X

i=1 d

X

k=1

(aik◦φm)

∂u_ǫ(s)

∂x_i ,

∂v_ǫ(s)

∂x_k Rn ) ds + Z t r

L′(Fj(s,z_ǫ(s)))n¬v_ǫ(s),_Amu_ǫ(s) +αǫ(s) +∇yFj(·,uǫ(s))

¶

L2₍_B₍_x_,_T₋_λs₎₎ o

ds

+

Z t

r

L′(Fj(s,zǫ(s)))

¦

vǫ(s),βǫ(s)dWs

L2₍_B₍_x_,_T₋_λs₎₎ © ds + 1 2 Z t r

L′(Fj(s,zǫ(s)))kβǫ(s)k2_L

2(U,L2(B(x,T−λs))ds

+ 1 2 X l Z t r

L′′(Fj(s,z_ǫ(s)))v_ǫ(s),βǫ(s)el

2

L2₍_B₍_x_,_T₋_λs₎₎ds (8.5)

for every 0_≤r<t_≤T a.s. We may, in fact, find the functionsFj satisfying i)-iii) even so that

iv) Fj(w,_·)_→F(w,_·)uniformly on compacts inRn, for everyw∈Rd, v) _∇_yFj(w,_·)_{→ ∇}_yF(w,_·)uniformly on compacts inRn, for everyw_∈Rd, vi) and

sup

(

Fj(w,y) 1+_|y|2 +

_∇_yFj(w,y)

1+_|y_| :|w| ≤r,y∈R

n_,_j

∈N

)

<∞, r>0.

Thus

lim_j_→∞_k∇_yFj(_·,u_ǫ(t,ω))_{− ∇}_yF(_·,u_ǫ(t,ω))_k_L2₍_B₍_x_,_T₋_λt₎₎=0, t∈[0,T/λ],

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for everyω∈Ω. Moreover, by the Gauss theorem, Z

B(x,T−λs)  

d

X

i=1 d

X

k=1

(aik◦φm)

∂u_ǫ(s)

∂x_i ,

∂v_ǫ(s)

∂x_k

Rn

+vǫ(s),Amuǫ(s)

Rn  ds = = d X

i=1 d

X

k=1

Z

∂B(x,T−λs)

(a_ik_◦φm)

vǫ,

∂uǫ

∂xi

Rn

yk−xk

T−λs d y ≤ Z

∂B(x,T−λs)

|vǫ|

 

n

X

l=1

k(a_◦φm)_∇ul_ǫ_k2   1 2 d y ≤ λ Z

∂B(x,T−λs)

|vǫ|

 

n

X

l=1

k(a12◦φm)∇ul

ǫk 2   1 2 d y = λ Z

∂B(x,T−λs)

|vǫ|

d X

i=1 d

X

k=1

(a_ik_◦φm)

_∂_u

ǫ

∂xi ,∂uǫ

∂xk

Rn 1 2 d y ≤ λ

2kvǫk 2

L2₍_∂_B₍_x_,_T₋_λs₎₎+

λ 2 d X

i=1 d

X

k=1

(a_ik_◦φm)

_∂_u

ǫ

∂xi ,∂uǫ

∂xk

Rn

L1₍_∂_B₍_x_,_T₋_λs₎₎

(8.6)

so, after applying (8.6) and letting j→ ∞in (8.5),

L(F(t,zǫ(t)))−L(F(r,zǫ(r)))≤

+

Z t

r

L′(F(s,zǫ(s)))

n¬

vǫ(s),αǫ(s) +∇yF(·,uǫ(s))

¶

L2₍_B₍_x_,_T₋_λs₎₎ o

ds

+

Z t

r

L′(F(s,z_ǫ(s)))¦v_ǫ(s),βǫ(s)dWs

L′(F(s,zǫ(s)))kβǫ(s)k2_L

2(U,L2(B(x,T−λs))ds

+ 1 2 X l Z t r

L′′(F(s,zǫ(s)))

vǫ(s),βǫ(s)el

2

L2₍_B₍_x_,_T₋_λs₎₎ ds (8.7)

for every 0_≤r<t_≤T a.s. by the Lebesgue dominated convergence theorem and the convergence theorem for stochastic integrals (see e.g. Proposition 4.1 in[33]).

Now, since

sup ǫ>1k

ǫ2(ǫ− Am)−2_k_L₍_L2₎<∞ and lim

ǫ→∞kǫ

2₍_ǫ

− Am)−2ϕ−ϕkL2=0, ϕ∈L2,

there is

lim ǫ→∞

sup t∈[0,T/λ]k

zǫ(t,ω)−z(t,ω)kW1,2_⊕_L2+ sup

t∈[0,T/λ]|

F(t,zǫ(t,ω))−F(t,z(t,ω))|

=0

sup_{kzǫ(t,ω)kW1,2_⊕_L2+F(t,z_ǫ(t,ω)):ǫ >0,t ∈[0,T/λ]}<∞

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9

Linear growth

+

Global space case

We first prove existence of weak solutions for a localized equation with regular nonlinearities. The proof is based on a compactness method: local energy estimates yield tightness of an approximating sequence of solutions. This sequence converges, on another probability space, to a limit due to the Jakubowski-Skorokhod theorem and finally, it is shown, that this limit is the desired weak solution of the localized equation (9.1).

Lemma 9.1. Letµbe a finite spectral measure onRd, let m_∈N, letν be a Borel probability measure supported in a ball in_H, let fi,gi :Rd_×Rn _→Rn×n for i_{∈ {}0, . . . ,d_}, fd+1,gd+1:Rd_×Rn_→Rn

be measurable functions such that

sup

¨

|fd+1(w,y)_|+_|gd+1(w,y)_|

1+_|y_| :|w| ≤r, y∈R

n

«

<∞,

sup

¨

|fi(w,y₁)₋fi(w,y₂)_|

|y₁₋y₂_| :|w| ≤r, y16= y2∈R

n_,_i

∈ {0, . . . ,d+1_}

«

<∞

sup

¨

|gi(w,y1)−gi(w,y2)|

|y1−y2|

:_|w_{| ≤}r, y₁₆= y₂_∈Rn,i_{∈ {}0, . . . ,d+1_}

«

<∞

hold for every r > 0. Then there exists a completely filtered stochastic basis (Ω,F,(_F_t),P) with a spatially homogeneous (_F_t)-Wiener process W with the spectral measure µ and an (_F_t)-adapted process z with continuous paths in_H which is a solution of the equation

u_{t t}=_Amu+1_B

mf(·,u,ut,∇u) +1Bmg(·,u,ut,∇u)W˙ (9.1)

in the sense of Section 4 with the notation (4.1), (4.1),ν is the law of z(0)and Bmis the open centered

ball inRd with radius m.

The proof of Lemma 9.1 will be carried out in a sequence of lemmas. For, let us introduce the mappings f_k:_{H → H} andg_k:_{H → L}₂(Hµ,H)defined by

f_k(u,v) = 1_B

m

0

f0(_·,u)(ζk∗v) +

Pd i=1f

i₍

·,u)(ζk∗uxi) + f

d+1₍

·,u)

,

g_k(u,v) = 1_B

m

0

g0(_·,u)(ζk∗v) +

Pd i=1g

i₍

·,u)(ζk∗uxi) +g

d+1₍

·,u)

ξ, ξ∈H_µ.

Lemma 9.2. For every k _∈ N, there exists a completely filtered stochastic basis (Ωk,_Fk,Pk) with a spatially homogeneous(_F_tk)-Wiener process Wk with spectral measureµand an(_F_tk)-adapted process zk= (uk,vk)withH-continuous paths such thatν is the law of zk(0)underPk and

zk(t) =S_tmzk(0) +

Z t

0

Sm_t₋_sfk(zk(s))ds+

Z t

0

S_tm₋_sgk(zk(s))dWsk, t≥0.

Moreover, for every p∈[2,_∞), there exists a constant K_p(1_,_l)=K_l(_,1_m)_,_g_,_f_,_p_,_ν_,_c_,_asuch that

Ek sup s∈[0,l]k

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and, if q ∈ (1,∞) and γ > 0 are such that γ+ 1

q < 1

2 then there exists a constant K

(2)

q,l =

K(2)

l,m,γ,q,f,g,a(2,m)_,_c_,_ν such that

Ekkvk(t)_k2q

Cγ_([_0,_l_]_;_W−1,2₎≤K (2)

q,l, k,l∈N. (9.3)

Proof. The mappings f_k :_{H → H} and g_k : _{H → L}₂(H_µ,_H) are Lipschitz on bounded sets and have at most linear growth hence there exists a completely filtered stochastic basis(Ωk,_Fk,Pk)with a spatially homogeneous(_Fk

t)-Wiener process Wk with spectral measure µ and an (Ftk)-adapted processzkwith_H-continuous paths such thatν is the law ofzk(0)underPkand

zk(t) =S_tmzk(0) +

Z t

0

Sm_t₋_sfk(zk(s))ds+

Z t

0

Sm_t₋_sgk(zk(s))dWsk, t ≥0

by e.g. [11]extended in the sense of Theorem 12.1 in Chapter V.2.12 in[39]whose generalization to SPDE is possible and can be proved in the same way as in[39]) since

〈Gmz,z〉_Dom₍_I

−Am₎1₂_⊕_L2≤

1 2kzk

2

Dom(I−Am₎12_⊕_L2, z∈DomG

m

hence the square norm of the local solutionzk2

Dom(I−Am₎1₂_⊕_L2 cannot explode in finite time and

so zk is a global solution in the sense of (4.3) by the Chojnowska-Michalik theorem (see [9] or Theorem 12 in[33]).

By Proposition 8.1 applied on T > 0, x = 0, l(r) = log(1+rp) for p _∈ [2,_∞), F(y) =_|y_|2/2, λ0=supw∈Rdka(φm(w))k

1

2, with the notationF_T =F_m_,_λ

0,0,T and

F_∞(u,v) =1

2

Z

Rd

( _d X

i=1 d

X

l=1

(ail◦φm)

¬ uxi,uxl

¶

Rn+|v| 2 Rn+|u|

2 Rn

) d y,

there is

l(F_T(t,zk(t)))_≤l(F_T(0,zk(0))) +M_T,k(t)− 1

2〈MT,k〉(t)

+ c

2

Z t

0

p(p₋1)F_Tp−2(s,zk(s))

1+Fp_T(s,zk₍_s₎₎ kv k₍_s₎

k2_L2₍_B

T−λ0s)kg(·,u

k₍_s₎_,_vk₍_s₎_,

∇uk(s))_k2_L2₍_B_m∩_B

T−λ0s)ds

+ c

2

Z t

0

l′(F_T(s,zk(s)))_kg(_·,uk(s),vk(s),_∇uk(s))_k2_L2₍_B_m∩_B

T−λ0s)ds

+

Z t

0

l′(F_T(s,zk(s)))_kvk(s)_k_L2₍_B

T−λ0s)ku

k₍_s₎

kL2₍_B

T−λ0s)ds

+

Z t

0

l′(F_T(s,zk(s)))_kvk(s)_k_L2₍_B

T−λ0s)kf(·,u

k₍_s₎_,_vk₍_s₎_,

∇uk(s))_k_L2₍_B_m∩_B

T�