On mild and weak solutions for stochastic heat equations with piecewise-constant conductivity
aa r X i v : . [ m a t h . P R ] S e p On mild and weak solutions for stochastic heatequations with piecewise-constant conductivity
Yuliya Mishura a , Kostiantyn Ralchenko a , Mounir Zili b a Taras Shevchenko National University of Kyiv, Department of Probability Theory,Statistics and Actuarial Mathematics, Volodymyrska 64/13, 01601 Kyiv, Ukraine b University of Monastir, Faculty of sciences of Monastir, Department of Mathematics,Avenue de l’environnement, 5019 Monastir, Tunisia
Abstract
We investigate a stochastic partial differential equation with second order ellipticoperator in divergence form, having a piecewise constant diffusion coefficient,and driven by a space–time white noise. We introduce a notion of weak solutionof this equation and prove its equivalence to the already known notion of mildsolution.
Keywords: stochastic partial differential equation, discontinuity ofcoefficients, fundamental solution, weak solution, mild solution
1. Introduction
Since the pioneering work of Walsh (1986), investigation of solutions ofstochastic partial differential equations (SPDE) has raised the interest of manyresearchers, especially for their numerous applications (see Dalang et al. (2009);Khoshnevisan (2014) and references therein). In fact, there are two main classesof such solutions; classical or generalized ones. A classical solution is a function,sufficiently smooth, satisfying the equation and its initial condition point-wiseon the same set of probability one. Every solution that is not classical is usuallycalled generalized. A generalized solution extends certain properties of a clas-sical solution without requiring existence of partial derivatives. That is why,there exist various notions of generalized solutions, including mild and weakones. The comparison between such notions has been dealt with only in veryfew cases, such as in the case of the standard stochastic heat equation (see,e. g., Khoshnevisan, 2014). This can be explained by the fact that, in general,there is no obvious reason to claim that the mild and weak solutions define thesame object. In this paper we address this question in the case of the following
Email addresses: [email protected] (Yuliya Mishura), [email protected] (Kostiantyn Ralchenko),
[email protected] (Mounir Zili) ∂u ( t, x ) ∂t = L u ( t, x ) + ˙ W ( t, x ); t ∈ (0 , T ] , x ∈ R ,u (0 , · ) := 0 , x ∈ R . (1)Here ˙ W denotes a “derivative” of a centered Gaussian field W = { W ( t, C ); t ∈ [0 , T ] , C ∈ B b ( R ) } with covariance E ( W ( t, C ) W ( s, D )) = ( t ∧ s ) λ ( C ∩ D ) , (2)where λ is the Lebesgue measure on R , and L is the operator defined by L = 12 ρ ( x ) ddx (cid:18) ρ ( x ) A ( x ) ddx (cid:19) ,A ( x ) = a { x ≤ } + a { 2) are strictly positive constants, and dfdx denotes the derivativeof f in the distributional sense. Equation (1) represents a natural extension ofthe stochastic heat equation driven by space-time white noise, which has beenwidely studied in the literature (see, e. g., Dalang et al. (2009); Khoshnevisan(2014); Walsh (1986) and references therein). Equation (1) has been intro-duced in Zili and Zougar (2018) because of its interest in modeling diffusionphenomena in medium consisting of two kinds of materials, undergoing stochas-tic perturbations. In Zili and Zougar (2018), the authors proved the existenceof the mild solution to (1), they presented explicit expressions of its covari-ance and variance functions, and they analyzed some regularity properties of itssample paths. Then, Zili and Zougar (2019a) presented an estimation methodof the parameters a and a appearing in (3) and, after that, in Zili and Zougar(2019b), they made a deep study of the spatial quadratic variations of the mildsolution process.We make here an interesting new step in the study of SPDE (1), by in-troducing a notion of weak solution of this equation and then, by showing itsequivalence with the already known notion of mild solution. Our proofs requirea stochastic Fubini theorem version and some non-random partial differentialequations characteristics; they are particularly based on the use of the knownexpression of the fundamental solution related to the operator L and some ofits characteristics.The paper is organized as follows. In the first part of the next section weintroduce and explain the notion of weak solution to SPDE (1) that we will dealwithin this paper. Then, the rest of the paper is devoted to the proof of theequivalence between the notions of mild and weak solutions for Equation (1). 2. Weak and mild solutions Let us suppose formally for the moment that equation (1) admits a classicalsolution u : Ω × [0 , ∞ ) × R → R that is a function belonging to C ([0 , ∞ ) × R )2nd satisfying equation (1) on a set Ω ′ ∈ Ω of probability 1.Then, for every ϕ ∈ C ∞ c ([0 , T ] × R ) and for every ω ∈ Ω ′ Z [0 ,T ] × R ∂u ( s, x ) ∂s ϕ ( s, x ) ρ ( x ) dx ds | {z } I − Z [0 ,T ] × R L u ( s, x ) ϕ ( s, x ) ρ ( x ) dx ds | {z } J = Z [0 ,T ] × R ϕ ( s, x ) ρ ( x ) W ( ds, dx ) . An integration by parts allows us to obtain: I = Z R h u ( s, x ) ϕ ( s, x ) i T ρ ( x ) dx − Z [0 ,T ] × R u ( s, x ) ∂ϕ ( s, x ) ∂s ρ ( x ) dx ds = − Z [0 ,T ] × R u ( s, x ) ∂ϕ ( s, x ) ∂s ρ ( x ) dx ds. As for the integral J we have: J = 12 Z [0 ,T ] × R ∂∂x (cid:18) ρ ( x ) A ( x ) ∂u ( s, x ) ∂x (cid:19) ϕ ( s, x ) ds dx = − Z [0 ,T ] × R ρ ( x ) A ( x ) ∂u ( s, x ) ∂x ∂∂x ( ϕ ( s, x )) ds dx where in the last equality we used the definition of the derivative in the distri-bution sense of the locally integrable function x ρ ( x ) A ( x ) ∂u ( s,x ) ∂x .Consequently, J = − Z [0 ,T ] × R ⋆ ρ ( x ) A ( x ) ∂u ( s, x ) ∂x ∂∂x ( ϕ ( s, x )) ds dx = − Z [0 ,T ] × (0 , + ∞ ) ρ a ∂u ( s, x ) ∂x ∂∂x ( ϕ ( s, x )) ds dx − Z [0 ,T ] × ( −∞ , ρ a ∂u ( s, x ) ∂x ∂∂x ( ϕ ( s, x )) ds dx By an integration by parts we get: Z (0 , + ∞ ) ∂u ( s, x ) ∂x ∂ϕ ( s, x ) ∂x dx = − u ( s, ∂ϕ ( s, ∂x − Z (0 , + ∞ ) u ( s, x ) ∂ ϕ ( s, x ) ∂ x dx, and Z ( −∞ , ∂u ( s, x ) ∂x ∂ϕ ( s, x ) ∂x dx = u ( s, ∂ϕ ( s, ∂x − Z ( −∞ , u ( s, x ) ∂ ϕ ( s, x ) ∂ x dx. Thus, J = 12 Z [0 ,T ] ( ρ a − ρ a ) u ( s, ∂ϕ ( s, ∂x ds + 12 Z [0 ,T ] × R ⋆ u ( s, x ) ρ ( x ) A ( x ) ∂ ϕ ( s, x ) ∂ x dxds Z [0 ,T ] ( ρ a − ρ a ) u ( s, ∂ϕ ( s, ∂x ds + Z [0 ,T ] × R ⋆ u ( s, x ) L ( ϕ ( s, x )) ρ ( x ) dxds Therefore, − Z [0 ,T ] × R ⋆ u ( s, x ) " ∂ϕ ( s, x ) ∂s + L ϕ ( s, x ) ρ ( x ) dx ds = Z [0 ,T ] × R ϕ ( s, x ) ρ ( x ) W ( ds, dx )+ 12 Z T ( ρ (0 + ) A (0 + ) − ρ (0 − ) A (0 − )) u ( s, ∂ϕ ( s, ∂x ds, (4)This leads to the first definition of the generalized solution to the SPDE (1).It is called a weak solution. Definition 1. A stochastic process u := { u ( t, x ) } { t ≥ ,x ∈ R } is a weak solutionto the stochastic partial differential equation (1) if u ∈ L Loc ( R + × R , dx dt ), andfor all T > ϕ ∈ C ∞ c ([0 , T ] × R ) , u satisfies equation (4).In the following proposition we present the expression of the fundamentalsolution associated to the SPDE (2). For a proof see, e.g., Zili (1995, 1999) andChen and Zili (2015). Proposition 2. The fundamental solution G of the partial differential equation(6) is given by G ( t, x, y ) = (cid:20) √ πt (cid:18) { y ≤ } √ a + { y> } √ a (cid:19) × (cid:26) exp (cid:18) − ( f ( x ) − f ( y )) t (cid:19) + √ a + √ a ( α − √ a − √ a ( α − 1) sign( y ) exp (cid:18) − ( | f ( x ) | + | f ( y ) | ) t (cid:19)(cid:27)(cid:21) Definition 3. A stochastic process u := { u ( t, x ) } { t ≥ ,x ∈ R } is a mild solution tothe stochastic partial differential equation (1) if it can be written in the followingintegral form: u ( t, x ) := Z t Z R G ( t − s, x, y ) W ( ds, dy ) , where G denotes the fundamental solution to the non random PDE ∂u ( t, x ) ∂t := L u ( t, x ) . (6)The purpose of this paper is to investigate the equivalence between mild andweak solutions. 4 .2. Mild solution implies weak solution In this subsection we prove that every mild solution is also a weak solution. Theorem 4. Every mild solution to equation (1) is also its weak solution.Proof. Let u be a mild solution and ϕ ∈ C ∞ c ([0 , T ] × R ). Introduce the operator H := − ∂∂s − L . With this notation, and for any fixed T > Z [0 ,T ] × R ⋆ u ( s, x ) H ( ϕ )( s, x ) ρ ( x ) ds dx = Z [0 ,T ] × R ⋆ Z [0 ,T ] × R G ( s − u, x, y ) W ( du, dy ) H ( ϕ )( s, x ) ρ ( x ) ds dx. (7)Now we apply the stochastic Fubini theorem for worthy martingale measures(Walsh, 1986, Th. 2.6) (see also Dalang et al., 2009, Th. 5.30). According toDalang et al. (2009, p. 20), the white noise W is a worthy martingale measurewith dominating measure K defined by K ( dx, dy, ds ) := dx dy ds. In this case,we can apply the stochastic Fubini theorem for worthy martingale measures tothe functions integrable with respect to the product of measures. The functionthat will be integrated, equals g ( s, u, x, y ) = G ( s − u, x, y ) H ( ϕ )( s, x ) ρ ( x ) R \{ } ( x ) , ( s, u, x, y ) ∈ [0 , T ] × R . This function is measurable, integrable with respectto the product of measures, and therefore Z [0 ,T ] × R | g ( s, u, x, y ) g ( s, u, z, y ) | K ( ds, dx, dz ) du dy = Z [0 ,T ] × R (cid:12)(cid:12) G ( s − u, x, y ) H ( ϕ )( s, x ) ρ ( x ) R \{ } ( x ) G ( s − u, z, y ) × H ( ϕ )( s, z ) ρ ( z ) R \{ } ( z ) (cid:12)(cid:12) ds du dx dy dz ≤ [max( ρ , ρ )] Z [0 ,T ] × R (cid:12)(cid:12) G ( s − u, x, y ) H ( ϕ )( s, x )) R \{ } ( x ) G ( s − u, z, y ) × H ( ϕ )( s, z ) R \{ } ( z ) (cid:12)(cid:12) ds du dx dz dy. Moreover, on the one hand, from Expression (5) of the fundamental solution G and the fact that the terms exp (cid:16) − ( f ( x ) − f ( y )) t (cid:17) and exp (cid:16) − ( | f ( x ) | + | f ( y ) | ) t (cid:17) arebounded by 1, we have | G ( s − u, z, y ) | ≤ C √ s − u u 6s for J ,ǫ we have: J ,ǫ = Z (0 ,ǫ ) × R ⋆ G ( s, x, y ) H ( ϕ )( u + s, x ) ρ ( x ) ds dx = − Z (0 ,ǫ ) × [0 , + ∞ ) G ( s, x, y ) (cid:16) ∂∂s ϕ ( u + s, x ) ρ ( x ) + ρ a ∂ ∂x ϕ ( u + s, x ) (cid:17) ds dx − Z (0 ,ǫ ) × ( −∞ , G ( s, x, y ) (cid:16) ∂∂s ϕ ( u + s, x ) ρ ( x ) + ρ a ∂ ∂x ϕ ( u + s, x ) (cid:17) ds dx. Thus, | J ,ǫ | ≤ C Z (0 ,ǫ ) Z R (cid:12)(cid:12)(cid:12) G ( s, x, y ) (cid:12)(cid:12)(cid:12) ds dx with C = max( ρ , ρ ) sup ( t,x ) ∈ R + × R (cid:12)(cid:12)(cid:12)(cid:12) ∂ϕ ( · , x ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) + max( a ρ , a ρ ) sup ( t,x ) ∈ R + × R (cid:12)(cid:12)(cid:12)(cid:12) ∂ ϕ ( t, · ) ∂x (cid:12)(cid:12)(cid:12)(cid:12) . This with (8) leads to | J ,ǫ | ≤ Cǫ, and consequently, lim ǫ → J ,ǫ = 0.Concerning J ,ǫ we have: J ,ǫ = Z [ ǫ,T − u ] × R ⋆ G ( s, x, y ) H ( ϕ )( u + s, x ) ρ ( x ) ds dx = − Z [ ǫ,T − u ] × R ⋆ G ( s, x, y ) ∂ϕ ( u + s, x ) ∂s ρ ( x ) ds dx − Z [ ǫ,T − u ] × R ⋆ G ( s, x, y ) L ϕ ( u + s, x ) ρ ( x ) ds dx = A ǫ + B ǫ . Integrating by parts, we get: A ǫ = Z R ⋆ h − ϕ ( u + s, x ) G ( s, x, y ) ρ ( x ) i T − uǫ dx + Z [ ǫ,T − u ] × R ⋆ ϕ ( u + s, x ) ∂G ( s, x, y ) ∂s ρ ( x ) ds dx = Z R ⋆ ϕ ( ǫ + u, x ) G ( ǫ, x, y ) ρ ( x ) dx − Z R ⋆ ϕ ( T, x ) G ( T − u, x, y ) ρ ( x ) dx + Z [ ǫ,T − u ] × R ⋆ ϕ ( u + s, x ) ∂G ( s, x, y ) ∂s ρ ( x ) ds dx = Z R ⋆ ϕ ( ǫ + u, x ) G ( ǫ, x, y ) ρ ( x ) dx + Z [ ǫ,T − u ] × R ⋆ ϕ ( u + s, x ) ∂G ( s, x, y ) ∂s ρ ( x ) ds dx, since ϕ ( T, · ) = 0.For B ǫ we have, 7 R ⋆ ∂∂x (cid:18) ρ ( x ) A ( x ) ∂ϕ ( s, x ) ∂x (cid:19) G ( s − u, x, y ) dx = Z (0 , + ∞ ) ρ a ∂ ϕ ( s, x ) ∂x G ( s − u, x, y ) dx + Z ( −∞ , ρ a ∂ ϕ ( s, x ) ∂x G ( s − u, x, y ) dx. Therefore B ǫ = − Z [ ǫ + u,T ] × (0 , + ∞ ) ρ a ∂ ϕ ( s, x ) ∂x G ( s − u, x, y ) ds dx − Z [ ǫ + u,T ] × ( −∞ , ρ a ∂ ϕ ( s, x ) ∂x G ( s − u, x, y ) ds dx. Integrating by parts we get Z −∞ ∂ ϕ ( s, x ) ∂x G ( s − u, x, y ) dx = G ( s − u, , y ) ∂ϕ ( s, ∂x − Z −∞ ∂ϕ ( s, x ) ∂x ∂G ( s − u, x, y ) ∂x dx and Z + ∞ ∂ ϕ ( s, x ) ∂x G ( s − u, x, y ) dx = − G ( s − u, , y ) ∂ϕ ( s, ∂x − Z + ∞ ∂ϕ ( s, x ) ∂x ∂G ( s − u, x, y ) ∂x dx Thus, B ǫ = 12 Z Tǫ + u ( ρ a − ρ a ) G ( s − u, , y ) ∂ϕ ( s, ∂x ds + 12 Z [ ǫ + u,T ] × R ⋆ ρ ( x ) A ( x ) ∂ϕ ( s, x ) ∂x ∂G ( s − u, x, y ) ∂x dxds. On another side, we have, Z R ⋆ ϕ ( s, x ) L G ( s − u, x, y ) ρ ( x ) dx = 12 Z R ⋆ ϕ ( s, x ) ∂∂x (cid:18) A ( x ) ρ ( x ) ∂G ( s − u, x, y ) ∂x (cid:19) dx = 12 Z (0 , + ∞ ) ϕ ( s, x ) a ρ ∂ G ( s − u, x, y ) ∂x dx + 12 Z ( −∞ , ϕ ( s, x ) a ρ ∂ G ( s − u, x, y ) ∂x dx = − Z R ⋆ ∂ϕ ( s, x ) ∂x A ( x ) ρ ( x ) ∂G ( s − u, x, y ) ∂x dx + 12 (cid:18) a ρ ∂G ( s − u, − , y ) ∂x − a ρ ∂G ( s − u, + , y ) ∂x (cid:19) ϕ ( s, Z R ⋆ ∂ϕ ( s, x ) ∂x A ( x ) ρ ( x ) ∂G ( s − u, x, y ) ∂x dx = − Z R ⋆ ϕ ( s, x ) L G ( s − u, x, y ) ρ ( x ) dx + 12 (cid:18) a ρ ∂G ( s − u, − , y ) ∂x − a ρ ∂G ( s − u, + , y ) ∂x (cid:19) ϕ ( s, B ǫ = 12 Z Tǫ + u ( ρ a − ρ a ) G ( s − u, , y ) ∂ϕ ( s, ∂x ds − Z [ ǫ + u,T ] × R ⋆ ϕ ( s, x ) L G ( s − u, x, y ) ρ ( x ) dxds + Z Tǫ + u (cid:18) a ρ ∂G ( s − u, − , y ) ∂x − a ρ ∂G ( s − u, + , y ) ∂x (cid:19) ϕ ( s, ds. and consequently, J ,ǫ = A ǫ + B ǫ = Z R ϕ ( ǫ + u, x ) G ( ǫ, x, y ) ρ ( x ) dx + Z ( ǫ,T − u ) × R ⋆ ϕ ( u + s, x ) (cid:20) ∂G ( s, x, y ) ∂s − L G ( s, x, y ) (cid:21) ρ ( x ) dx ds + 12 Z Tǫ + u ( ρ (0 + ) A (0 + ) − ρ (0 − ) A (0 − )) G ( s − u, , y ) ∂ϕ ( s, ∂x ds + Z Tǫ + u (cid:18) A (0 − ) ρ (0 − ) ∂G ( s − u, − , y ) ∂x − A (0 + ) ρ (0 + ) ∂G ( s − u, + , y ) ∂x (cid:19) ϕ ( s, ds. Let us calculate the term a ρ ∂G ( t, − ,y ) ∂x − a ρ ∂G ( t, + ,y ) ∂x .If x > y > t > G ( t, x, y ) = 1 √ πt √ a × (cid:26) exp (cid:18) − ( x − y ) a t (cid:19) + β exp (cid:18) − ( x + y ) a t (cid:19)(cid:27) , where β = √ a + √ a ( α − √ a −√ a ( α − .Thus, ∂G ( t, x, y ) ∂x = 1 √ πt √ a (cid:26) − x − ya t exp (cid:18) − ( x − y ) a t (cid:19) − β x + ya t exp (cid:18) − ( x + y ) a t (cid:19)(cid:27) . Therefore, ∂G ( t, + , y ) ∂x = 1 √ πt √ a (cid:26) − − ya t exp (cid:18) − y a t (cid:19) − β ya t exp (cid:18) − y a t (cid:19)(cid:27) = − y √ πta t √ a ( β − 1) exp (cid:18) − y a t (cid:19) . If x < y > t > 0, then G ( t, x, y ) = 1 √ πt √ a (cid:26) exp (cid:18) − ( √ a x − √ a y ) a a t (cid:19) + β exp (cid:18) − ( √ a x − √ a y ) a a t (cid:19)(cid:27) β √ πt √ a exp (cid:18) − ( √ a x − √ a y ) a a t (cid:19) . Thus, ∂G ( t, x, y ) ∂x = 1 + β √ πt √ a −√ a ( √ a x − √ a y ) a a t exp (cid:18) − ( √ a x − √ a y ) a a t (cid:19) . Therefore, ∂G ( t, − , y ) ∂x = 1 + β √ πt √ a √ a √ a ya a t exp (cid:18) − y a t (cid:19) . All this implies that a ρ ∂G ( t, − , y ) ∂x − a ρ ∂G ( t, + , y ) ∂x = a ρ β √ πt √ a √ a √ a ya a t exp (cid:18) − y a t (cid:19) − a ρ − y √ πta t √ a ( β − 1) exp (cid:18) − y a t (cid:19) = ρ (1 + β ) √ a √ a + ρ ( β − ! × y √ a t √ πt exp (cid:18) − y a t (cid:19) = 2 ρ a + 2 a ρ ( α − √ a √ a − a ( α − y √ a t √ πt exp (cid:18) − y a t (cid:19) . Using the fact that α = 1 − ρ a ρ a , we easily see that2 ρ a + 2 a ρ ( α − √ a √ a − a ( α − 1) = 0 . Therefore, a ρ ∂G ( t, − , y ) ∂x − a ρ ∂G ( t, + , y ) ∂x = 0 , for every y > 0. By similar calculation, we get the same result if y ≤ 0. Now,since G is a fundamental solution of PDE (6), we have ∂G ( s,x,y ) ∂s −L G ( s, x, y ) = 0 , for every s ∈ (0 , T ) and x ∈ R \ { y } and consequently, J ,ǫ = Z R ϕ ( ǫ + u, x ) G ( ǫ, x, y ) ρ ( x ) dx + 12 Z Tǫ + u ( ρ (0 + ) A (0 + ) − ρ (0 − ) A (0 − )) G ( s − u, , y ) ∂ϕ ( s, ∂x ds → Z R δ ( x, y ) ϕ ( u, x ) ρ ( x ) dx + 12 Z Tu ( ρ (0 + ) A (0 + ) − ρ (0 − ) A (0 − )) G ( s − u, , y ) ∂ϕ ( s, ∂x ds. Z (0 ,T ) × R u ( s, x ) H ϕ ( s, x ) ρ ( x ) dx ds = Z (0 ,T ) × R ϕ ( s, y ) ρ ( y ) W ( ds dy )+ Z (0 ,T ) × R Z Tu ( ρ (0 + ) A (0 + ) − ρ (0 − ) A (0 − )) G ( s − u, , y ) ∂ϕ ( s, ∂x dsW ( du, dy ) . Applying the Fubini theorem (Walsh, 1986, Th. 2.6) with G = [0 , T ], ν = ds , M = W and g ( u, s, y ) = [0 ,s ] ( u ) G ( s − u, , y ) ∂ϕ ( s, ∂x we get Z (0 ,T ) × R ⋆ u ( s, x ) H ϕ ( s, x ) ρ ( x ) dx ds = Z (0 ,T ) × R ϕ ( s, y ) ρ ( y ) W ( ds dy )+ 12 Z T ( ρ (0 + ) A (0 + ) − ρ (0 − ) A (0 − )) Z (0 ,s ) × R G ( s − u, , y ) W ( du, dy ) ∂ϕ ( s, ∂x ds = Z (0 ,T ) × R ϕ ( s, y ) ρ ( y ) W ( ds dy ) + 12 Z T ( ρ (0 + ) A (0 + ) − ρ (0 − ) A (0 − )) u ( s, ∂ϕ ( s, ∂x ds. In this section we prove that every weak solution is a mild solution. In viewof Theorem 4, it suffices to prove the uniqueness of a weak solution. Theorem 5. The mild solution is a unique weak solution to the stochasticpartial differential equation (1) .Proof. Assume that u and u are two weak solutions to (1). Fix some T > ϕ ∈ C ∞ c ([0 , T ] × R ), − Z [0 ,T ] × R ⋆ (cid:0) u ( s, x ) − u ( s, x ) (cid:1) (cid:20) ∂ϕ ( s, x ) ∂s + L ϕ ( s, x ) (cid:21) ρ ( x ) dx ds = 12 Z T (cid:0) ρ (0 + ) A (0 + ) − ρ (0 − ) A (0 − ) (cid:1)(cid:0) u ( s, − u ( s, (cid:1) ∂ϕ ( s, ∂x ds a. s.(9)First, let us prove that u = u in (0 , T ) × (0 , + ∞ ). Let ψ be an arbitraryfunction from C ∞ c ((0 , T ) × (0 , + ∞ )). We may define it on the entire [0 , T ] × R by putting ψ = 0 outside (0 , T ) × (0 , + ∞ ). We claim that there exists ϕ ∈ C , ([0 , T ] × R ) such that ∂ϕ ( t, x ) ∂t + 12 a ∂ ϕ ( t, x ) ∂x = ψ ( t, x ) , t ∈ [0 , T ] , x ∈ R . (10)11ndeed, it is well known (see, e. g. Evans, 1998, Th. 2, p. 50) that for any f ∈ C , ( R + × R ) with compact support the function v ( t, x ) = Z t p π ( t − s ) Z R e − | x − y | t − s ) f ( s, y ) dy ds belongs to C , ((0 , + ∞ ) × R ) and satisfies the heat equation ∂v∂t − ∂ v∂x = f, t > , x ∈ R . (11)Then the desired solution to (10) can be constructed from the solution to (11)by the time change s = a ( T − t ).Assume for definiteness that supp ψ ⊂ [ σ, τ ] × [ a, b ] ⊂ (0 , T ) × (0 , + ∞ ). Usingregularization (see H¨ormander (2003, Sec. 1.3) or Vladimirov (2002, Sec. 1.2)),we can approximate the function ˜ ϕ = ϕ [ σ,τ ] × [ a,b ] by functions ϕ ε ∈ C ∞ c ([0 , T ] × R ) in such a way that, for sufficiently small ε , supp ϕ ε ⊂ (0 , T ) × (0 , + ∞ ), k ϕ ε k L ∞ ≤ k ˜ ϕ k L ∞ , (cid:13)(cid:13)(cid:13)(cid:13) ∂ϕ ε ∂t (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂ ˜ ϕ∂t (cid:13)(cid:13)(cid:13)(cid:13) L ∞ , (cid:13)(cid:13)(cid:13)(cid:13) ∂ ϕ ε ∂x (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ≤ (cid:13)(cid:13)(cid:13)(cid:13) ∂ ˜ ϕ∂x (cid:13)(cid:13)(cid:13)(cid:13) L ∞ , and the convergences ϕ ε → ϕ [ σ,τ ] × [ a,b ] , ∂ϕ ε ∂t → ∂ϕ∂t [ σ,τ ] × [ a,b ] , ∂ ϕ ε ∂x → ∂ ϕ∂x [ σ,τ ] × [ a,b ] , as ε ↓ , hold almost everywhere in [0 , T ] × R .By inserting this ϕ ε into (9) we get Z (0 ,T ) × (0 , + ∞ ) (cid:0) u ( s, x ) − u ( s, x ) (cid:1) (cid:20) ∂ϕ ε ( t, x ) ∂t + 12 a ∂ ϕ ε ( t, x ) ∂x (cid:21) dx ds = 0 a. s.Note that the expression in square brackets is bounded and compactly sup-ported, and u , u are locally integrable. 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