arXiv:2004.07942v1 [math.AP] 14 Apr 2020 nta ucini qa ozr n h ih-adsd eog o[ to belongs side right-hand the and zero to equal is function initial n eedn nyon only depending and nta aai [ in data initial o ierbt ooeeu n ohmgnoswal ope s coupled weakly layer nonhomogeneous the and in homogeneous system both linear for ftescn re ihcntn offiins esyta h es inequality the the that of part say majorant We the in coefficients. norm the constant [10 of work with front recent in order t our coefficient for second of problem Cauchy continuation the the to a of solutions is of gradient paper the for mathem present estimates of The subjects classical [15]). are [14], systems and equations Parabolic Introduction 1 Classification: Subject AMS field vector a of derivative directional Keywords: obtained. are derivative directional their and problems these α b eateto ahmtclSine,Uiest fLiverp of University Sciences, Mathematical of Department ∈ ∗ wdn UNUiest,6Mkuh-alyS. ocw 11 , St., Miklukho-Maklay 6 University, RUDN ; † 6 B,U;Dprmn fMteais Link¨oping Univers Mathematics, of Department UK; 3BX, L69 -al [email protected] [email protected] E-mail: E-mail: author. Corresponding Abstract. (0 hr onws siae o ouin fweakly of solutions for estimates pointwise Sharp ope eododrprblcsse nalayer a in system parabolic order second coupled , ) xlctfrua o h hr offiinsi onws estimates pointwise in coefficients sharp the for formulas Explicit 1). a acypolm ekyculdprblcsse,sappitiee pointwise sharp system, parabolic coupled weakly problem, Cauchy eateto ahmtc,AilUiest,Ail40700, Ariel University, Ariel Mathematics, of Department L eda with deal We p nmmr fgetmteaiinSL Sobolev S.L. mathematician great of memory In ( R R n )] eso Kresin Gershon T n +1 m t 1 , , = n ≤ R ≥ m n p and 1 × cmoetvco-audsltost h acyproblem Cauchy the to solutions vector-valued -component ∞ ≤ rmr 54,3A3 eodr 47A30 Secondary 35A23; 35K45, Primary (0 T , o h ohmgnossse espoeta the that suppose we system nonhomogeneous the For . < T .W sueta offiinso h ytmaereal are system the of coefficients that assume We ). a ∗ ∞ n ldmrMaz’ya Vladimir and h ooeeu ytmi osdrdwith considered is system homogeneous The . 1 o,M ool, esnl aaoi equation parabolic single he L tclpyis(..[] [2], [1], (e.g. physics atical t,E513Link¨oping,ity,SE-58183 p 18 7198, ( & R b iaei hr fthe if sharp is timate ulig Liverpool, Building, O cn re parabolic order econd antb diminished. be cannot T n † +1 nsappointwise sharp on ] )] m ∩ o ouin of solutions for [ C α stimates, R T n +1  ] m , Sharp pointwise estimates for solutions to the Laplace, modified Helmholtz, Lam´e, Stokes, and heat equations , as well as for the analytic functions were obtained earlier in [5] - [9]. In this paper we study solutions of the Cauchy problem for parabolic weakly coupled system with real coefficients of the form

∂u n ∂2u n ∂u = a (t) + b (t) + C(t)u + f(x, t) (1.1) ∂t jk ∂x ∂x j ∂x j k j=1 j j,kX=1 X Rn+1 Rn in the layer T = (0, T ) with initial condition u t=0 = ϕ, considering separately two cases, f = 0 and ϕ = 0.× Here and henceforth T < , n | 1, u(x, t)=(u (x, t),...,u (x, t)) ∞ ≥ 1 m and f(x, t)=(f1(x, t),...,fm(x, t)). Throughout the article, we assume that A(t) = ((ajk(t))) is a symmetric positive defi- nite (n n)--valued function on [0, T ], which elements satisfy the H¨older condition × with exponent α/2 (0 <α< 1), b1(t),...,bn(t) are continuous functions on [0, T ], C(t) is continuous on [0, T ] matrix-valued function of order m. We obtain sharp pointwise estimates for u , ∂u/∂ℓ and max|ℓ|=1 ∂u/∂ℓ , where u solves | | | Rn+1| | | the Cauchy problem for system (1.1) in the layer T , denotes the Euclidean length of a vector and ℓ is a unit n-dimensional vector. By ∂u|·|/∂ℓ we mean the derivative of a vector-valued function u(x, t) in the direction ℓ:

∂u u(x + λℓ, t) u(x, t) = lim − ∂ℓ λ→0+ λ m ∂u = (ℓ, )u = j e , (1.2) ∇x ∂ℓ j j=1 X where x =(∂/∂x1,...,∂/∂xn) and ej means the unit vector of the j-th coordinate axis. The∇ present paper consists of five sections, including Introduction. Section 2 is auxiliary. Rn+1 Section 3 is devoted to explicit formulas for solutions of the Cauchy problem in T for system (1.1). In Section 4, we consider a solution of the Cauchy problem

u n 2u n u ∂ ∂ ∂ Rn+1 = ajk(t) + bj(t) + C(t)u in T , ∂t ∂xj ∂xk ∂xj  j,k=1 j=1 (1.3)  X X   u ϕ t=0 = ,  where ϕ [Lp(R n)]m, p [1, ]. The norm in the space [Lp(Rn)]m is defined by ∈ ∈ ∞ k·kp 1/p p ϕ p = ϕ(x) dx k k Rn | | Z  for 1 p< , and ≤ ∞ ϕ = ess sup ϕ(x) : x Rn . k k∞ {| | ∈ }

2 In this section we obtain two groups of sharp estimates. First of them concerns modulus to solution u of problem (1.3). Namely, we derive the inequality

u(x, t) (t) ϕ (1.4) | | ≤ Hp k kp with the sharp coefficient

|||eIC∗ (t)||| p(t)= 1/p , (1.5) H 1/2 ′ (2√π)n/p det (t) (p′)n/(2p ) IA   Rn+1 t where (x, t) is an arbitrary point in the layer T . Here and henceforth F (t)= 0 F (t)dt, where F can be a vector-valued or matrix-valued function, p−1 + p′−1 =I 1, the symbol ∗ R denotes passage to the transposed matrix and |||B||| = max z Bz means the spectral norm | |=1 | | Rl 1/2 of the l l real-valued matrix B, z . It is known (e.g. [11], sect. 6.3) that |||B||| = λB , × ∈ ∗ where λB is the spectral radius of the matrix B B. As a special case of (1.5) one has

(t)= |||eIC∗ (t)||| . (1.6) H∞ For the single parabolic equation

∂u n ∂2u n ∂u = a (t) + b (t) + c(t)u + f(x, t) (1.7) ∂t jk ∂x ∂x j ∂x j k j=1 j j,kX=1 X with f = 0, formula (1.6) becomes

t (t) = exp c(t)dt . H∞ Z0  The second group of sharp estimates concerns modulus of ∂u/∂ℓ, where u is solution of problem (1.3). Namely, the formula for the sharp coefficient

′ ′ 1/p −1/2 p +1 IC∗ (t) Γ A (t)ℓ ||| e ||| 2 ℓ(t)= I (1.8) p, 1/2 1/p ′(n+p′)/2 K 2nπ (n+p−1)/2 det (t)  p  IA   in the inequality    ∂u (x, t) ℓ(t) ϕ (1.9) ∂ℓ ≤Kp, k kp

is obtained, where (x, t) is an arbitrary point in the layer Rn+1. T As a consequence of (1.8), the sharp coefficient

p(t) = max p,ℓ(t) (1.10) K |ℓ|=1 K

3 in the inequality ∂u max (x, t) p(t) ϕ p (1.11) |ℓ|=1 ∂ℓ ≤K k k

is found. In particular, 1 − 1/2 (t)= ||| (t)|||||| eIC∗ (t)||| . (1.12) K∞ √π IA For the single parabolic equation (1.7) with f = 0, the previous formula takes the form 1 t (t)= ||| −1/2(t)||| exp c(t)dt . (1.13) K∞ √π IA Z0  As a special case of (1.13) one has

t 1 exp 0 c(t)dt ∞(t)= 1/2 K √π tnR o 0 a(t)dt n o for A(t)= a(t)I, where I is the unit matrix of orderR n. In Section 5 we consider a solution of the Cauchy problem u n 2u n u ∂ ∂ ∂ Rn+1 = ajk(t) + bj(t) + C(t)u + f(x, t) in T , ∂t ∂xj ∂xk ∂xj  j,k=1 j=1 (1.14)  X X   u 0 t=0 = ,  where f [ Lp(Rn+1)]m [Cα Rn+1 ]m, α (0, 1). By [Cα Rn+1 ]m we denote the space of ∈ T ∩ T ∈ T f Rn+1 m-component vector-valued functions  (x, t) which are continuous  and bounded in T and locally H¨older continuous with exponent α in x Rn, uniformly with respect to t [0, T ]. p Rn+1 m ∈ ∈ The space [L T ] is endowed with the norm

 T 1/p p f p,T = (f(x, τ) dxdτ k k Rn Z0 Z 

for 1 p< , and ≤ ∞ f = ess sup f(x, τ) : x Rn, τ (0, T ) . k k∞,T {| | ∈ ∈ } In this section we also obtain two groups of sharp estimates. As before, first of them concerns modulus to solution u of problem (1.14). Namely, we prove the inequality u(x, t) (t) f (1.15) | |≤Np k kp,t with the sharp coefficient

′ p′ 1/p 1 t eIC∗ (t,τ)z (t)= max dτ , (1.16) p n/p ′ n/(2p′) 1/2 p′−1 N (2√π) p |z|=1 ( 0 det (t, τ ) ) Z IA 4  Rn+1 t where (x, t) is an arbitrary point in the layer T . Here and henceforth F (t, τ)= τ F (t)dt, where F can be a vector-valued or matrix-valued function. Formula (1.16)I is obtained under assumption that the integral in (1.16) converges. For instance, integral in (1.16) is convergentR for p> (n + 2)/2 in the case of single parabolic equation with constant coefficients. As a special case of (1.16) one has

t IC∗ (t,τ) ∞(t) = max e z dτ. (1.17) N |z|=1 Z0

For the single parabolic equation (1.7) the previous formula takes the form

t t (t)= exp c(t)dt dτ . N∞ Z0 Zτ  The second group of sharp estimates concerns ∂u/∂ℓ. The explicit formula for the sharp coefficient

′ ′ ′ 1/p p ′ 1/p p′+1 −1/2 p t ℓ IC∗ (t,τ)z 1 Γ 2 A (t, τ) e (t)= max I dτ (1.18) p,ℓ 1/p ′ (n+p′)/2  p′−1  C n (n+p−1)/2 p   |z|=1 0 1/2 2 π Z det (t, τ)     IA     in the inequality     ∂u (x, t) ℓ(t) f (1.19) ∂ℓ ≤ Cp, k kp,t

for solutions of the Cauchy problem (1.14) is found, where (x, t) Rn+1. T Formula (1.18) is obtained under assumption that the integral∈ in (1.18) converges. We note that this integral is convergent for p > n + 2 in the case of single parabolic equation with constant coefficients (see [10]). As a consequence of (1.18), we arrive at the formula for the sharp coefficient

p(t) = max p,ℓ(t) (1.20) C |ℓ|=1 C in the inequality ∂u max (x, t) p(t) f p,t . (1.21) |ℓ|=1 ∂ℓ ≤ C k k

For instance, 1 t −1/2 IC∗ (t,τ) ∞(t)= max max A (t, τ)ℓ e z dτ . (1.22) C √π |ℓ|=1 |z|=1 0 I Z

For the single parabolic equation (1.7) the previous formula takes th e form t t 1 −1/2 ∞(t)= max A (t, τ)ℓ exp c(t)dt dτ . (1.23) C √π |ℓ|=1 0 I τ Z Z 

5 In the particular case A(t)= a(t)I, formula (1.23) becomes

t t 1 exp τ c(t)dt ∞(t)= 1/2 dτ . C √π 0 tnR o Z τ a(t)dt n o Note that the sharp coefficients (1.8) and (1.18)R do not depend on the coefficient vector b(t)=(b1(t),...,bn(t)).

2 The norm of a certain integral operator

Let ( , ,µ) be a measure space and let 1 p . We introduce the space [Lp( , ,µ)]n of realX vector-valuedA functions endowed with≤ the≤∞ norm X A 1/p f = f(x) pdµ(x) (2.1) k kp | | ZX  for 1 p< , and ≤ ∞ f = ess sup f(x) : x . k k∞ {| | ∈X} By (η,ζ) we denote the inner product of the vectors η and ζ in a . The following assertion was proved in [6] (Proposition 1.2). It contains a representation of the norm S of the integral operator S defined on [Lp( , ,µ)]n and acting into Rm. k kp X A Proposition 1. Let G = ((gij)) be an (m n)-matrix-valued function with the elements p′ × gij L ( , ,µ) whose values gij are everywhere finite. The norm of the linear continuous operator∈ SX:[ALp( , ,µ)]n Rm defined by X A → S(f)= G(x)f(x)dµ(x) (2.2) ZX is equal to ∗ S p = sup G z p′ , (2.3) k k |z|=1k k where G∗ stands for the transposed matrix of G, z Rm and p′ is defined by 1/p +1/p′ =1. ∈ Proof. 1. Upper estimate for S . For any vector z Rm, k kp ∈ (S(f), z)= (G(x)f(x), z)dµ(x)= (f(x),G∗(x)z)dµ(x). (2.4) ZX ZX Hence by H¨older’s inequality

∗ ∗ ∗ (S(f), z) (f(x),G (x)z) dµ(x) G (x)z f(x) dµ(x) G z ′ f . | | ≤ | | ≤ | || | ≤k kp k kp ZX ZX Therefore, taking into account that S(f) = sup (S(f), z) : z = 1 we arrive at the estimate | | {| | | | } ∗ S p sup G z p′ . (2.5) k k ≤ |z|=1k k

6 m−1 m 2. Lower estimate for S p. Let us fix z S = z R : z = 1 . We introduce the vector-valued functionk withk n components∈ { ∈ | | }

hz(x)= gz(x)h(x), (2.6) where h L ( , ,µ), h 1, and ∈ p X A k kp ≤ G∗(x)z G∗(x)z −1 for G∗(x)z =0, | | | | 6 gz(x)=  0 for G∗(x)z =0.  | | p n Note that hz [L ( , ,µ)] and hz 1. Setting (2.6) as f in (2.4) we find ∈ X A  k kp ≤ ∗ ∗ (S(hz), z)=(S(g h), z)= (g (x),G (x)z)h(x)dµ(x)= G (x)z h(x)dµ(x). z z | | ZX ZX Hence

S p = sup S(f) sup S(gzh) sup (S(gzh), z) k k kfkp≤1 | | ≥ khkp≤1 | | ≥ khkp≤1 | |

∗ ∗ = sup G (x)z h(x)dµ(x) = G z ′ . | | k kp khkp≤1 ZX

By the arbitrariness of z Sm− 1, ∈ ∗ S p sup G z p′ , (2.7) k k ≥ |z|=1k k which together with (2.5) leads to (2.3). Remark 1. Let 1

∗ −p′/p ∗ p′ S = sup S(f) (S(hz), z)= G z ′ G (x)z dµ(x) k kp | | ≥ k kp | | kfkp≤1 ZX ∗ −p′/p ∗ p′ ∗ = G z ′ G z ′ = G z ′ , k kp k kp k kp which implies (2.7), because z Sm−1 is arbitrary. ∈ 7 3 Formulas for solutions of weakly coupled parabolic systems

By (x, y) we mean the inner product of the vectors x and y in Rn. The Schwartz class of m-component vector-valued rapidly decreasing C∞-functions on Rn will be denoted by [ (Rn)]m. S t Since the matrix A(t) is positive definite for any t [0, T ], the matrix A(t)= 0 A(t)dt ∈ I 1/2 is positive definite for any t (0, T ]. So, there exist positive definite matricesR A (t) −1/2 ∈ 1/2 2 −1/2 2 −1 I and A (t) of order n such that A (t) = A(t) and A (t) = A (t) for any t (0I, T ](e.g. [11], sect. 2.14). I I I I ∈We start with the Cauchy problem (1.3) for the homogeneous equation. The assertion be- low can be proved analogously to the corresponding statement for the heat equation (e.g.[4], Th. 5.4; [12], Sect. 4.8; [13], Sect. 7.4). We restrict ourselves to a formal argument. Lemma 1. Let ϕ [ (Rn)]m. A solution u of problem (1.3) is given by ∈ S u(x, t)= G(x y, t)ϕ(y)dy , (3.1) Rn − Z where 2 IC (t) −1/2 e − IA (t)(x+Ib(t)) 4 G(x, t)= e . (3.2) (2√π)n det 1/2(t) IA  Proof. Let u be solution of the Cauchy problem (1.3). We introduce the function

v = e−IC (t)u. (3.3)

Then v is solution of the problem

v n 2v n v ∂ ∂ ∂ Rn+1 = ajk(t) + bj(t) in T , ∂t ∂xj ∂xk ∂xj  j,k=1 j=1 (3.4)  X X   v ϕ t=0 = ,   Applying the Fourier transform 1 v −i(x,ξ)v ˆ(ξ, t)= n/2 e (x, t)dx (3.5) (2π) Rn Z to the Cauchy problem (3.4), we obtain dvˆ = (A(t)ξ,ξ)+ i(b(t),ξ) vˆ , vˆ(ξ, 0) = ϕˆ (ξ) . (3.6) dt −  The solution of problem (3.6) is

t vˆ(ξ, t)= ϕˆ (ξ)eR0 {−(A(t)ξ,ξ)+i(b(t),ξ)}dt = ϕˆ (ξ)e−(IA(t)ξ,ξ)+i(Ib(t),ξ). (3.7)

8 By the inverse Fourier transform 1 v i(x,ξ)v (x, t)= n/2 e ˆ(ξ, t)dξ, (2π) Rn Z we deduce from (3.7) 1 v i(x,ξ)ϕ −(IA(t)ξ,ξ)+i(Ib(t),ξ) (x, t)= n/2 e ˆ (ξ)e dξ (2π) Rn Z 1 i(x,ξ) −(IA(t)ξ,ξ)+i(Ib(t),ξ) −i(y,ξ)ϕ = n e e e (y)dy dξ (2π) Rn Rn Z Z  1 i(x−y+Ib(t),ξ) −(IA(t)ξ,ξ) ϕ = n e e dξ (y)dy . (3.8) (2π) Rn Rn Z Z  Let us denote 1 −(IA(t)ξ,ξ)+i(x+Ib(t),ξ) G0(x, t)= n e dξ . (3.9) (2π) Rn Z The known formula (e.g.[16], Ch. 2, Sect. 9.7)

n/2 π −1 e−(Mξ,ξ)+i(ζ,ξ)dξ = e−(M ζ,ζ)/4, (3.10) Rn √det M Z where M is a symmetric positive definite matrix of order n, together with (3.9) leads to

1 −1 −(I (t)(x+Ib(t)),x+Ib(t)) 4 G0(x, t)= e A . (3.11) (2√π)n det (t) IA  Since (B−1ζ,ζ)=(B−1/2B−1/2ζ,ζp)=(B−1/2ζ, B−1/2ζ) = B−1/2ζ 2 as well as det B = 2 | | det B1/2 for any symmetric positive definite matrix B and every vector ζ Rn, we can write (3.11) as ∈ 2  −1/2 1 − IA (t)(x+Ib(t)) 4 G0(x, t)= e . (3.12) (2√π)n det 1/2(t) IA  It follows from (3.3), (3.8), (3.9) and (3.12) that the solution of problem (1.3) can be repre- sented as (3.1), where G(x, t) is given by (3.2).

The next assertion can be proved in view of (3.3) and (3.4) on the base of Lemma 1 similarly to the analogous statement for the heat equation (e.g.[4], Th. 5.5; [13], Sect. 7.4). p Rn m Rn+1 Rm Proposition 2. Suppose that 1 p and ϕ [L ( )] . Define u : T by (3.1), where G is given by (3.2).≤ Then≤u ∞(x, t) is solution∈ of the system →

∂u n ∂2u n ∂u = a (t) + b (t) + C(t)u ∂t jk ∂x ∂x j ∂x j k j=1 j j,kX=1 X in Rn+1. If 1 p< , then u( , t) ϕ in [Lp]m as t 0+. T ≤ ∞ · → → 9 Remark 2. It is known (e.g. [2], Ch.1, Sect. 6, 7 and 9) that

v(x, t)= G0(x y, t)ϕ(y)dy , Rn − Z where G0 is given by (3.11), represents a unique bounded solution of the Cauchy problem (3.4) and v(x, t) ϕ(x) as t 0+ at any x Rn under assumption that ϕ belongs to [C(Rn)]m [L∞(R→n)]m. This fact→ together with (3.3)∈ and the Lusin’s theorem (see [17], Ch. VI, Sect.∩ 6) implies that u( , t) ϕ almost everywhere in Rn as t 0+, where u is a solution of problem (1.3) with· ϕ →[L∞(Rn)]m. → ∈ Further, let us consider solution u of the Cauchy problem (1.14) for the nonhomogeneous system. We introduce the function (3.3). Then v is solution of the problem

∂v n ∂2v n ∂v −IC (t) Rn+1 = ajk(t) + bj(t) + e f(x, t) in T , ∂t ∂xj ∂xk ∂xj  j,k=1 j=1 (3.13)  X X   v 0 t=0 = .   In view of (3.3) and (3.13), the next statement can be proved analogously to the similar assertion for the heat equation (e.g. [12], Sect. 4.8). We restrict ourselves to a formal argument.

n m Lemma 2. Let f( , t) [ (R )] for any t [0, T ] and let the quantities Cα,m in the estimates · ∈ S ∈ (1 + x m) ∂αf(x, t) C | | | x | ≤ α,m are independent of t for any integer m 0 and multiindex α. The solution of problem (1.14) is given≥ by

t u(x, t)= P (x y, t, τ)f(y, τ)dydτ , (3.14) Rn − Z0 Z where P is defined by

2 eIC (t,τ) −1/2 − IA (t,τ)(x+Ib(t,τ)) 4 P (x, t, τ)= e . (3.15) (2√π)n det 1/2(t, τ) IA  Proof. Applying the Fourier transform to the Cauchy problem (3.13), we obtain dvˆ = (A(t)ξ,ξ)+ i(b(t),ξ) vˆ + e−IC (t)fˆ(ξ, t) , vˆ(ξ, 0)=0 . (3.16) dt − The solution of problem (3.16) is

t vˆ(ξ, t) = e−(IA(t)ξ,ξ)+i(Ib(t),ξ) e−IC (τ)fˆ(ξ, τ)e(IA(τ)ξ,ξ)−i(Ib(τ),ξ)dτ 0 t Z −I (τ) R t{−(A(s)ξ,ξ)+i(b(s),ξ)}ds = e C fˆ(ξ, τ)e τ dτ . (3.17) Z0 10 By the inverse Fourier transform in (3.17), we have

t 1 −I (τ) R t{−(A(s)ξ,ξ)+i(b(s),ξ)}ds i(x,ξ) v C fˆ τ (x, t) = n/2 e (ξ, τ)e dτ e dξ (2π) Rn 0 Z t Z  1 i(x,ξ) −I (τ) R t{−(A(s)ξ,ξ)+i(b(s),ξ)}ds C fˆ τ = n/2 e e (ξ, τ)e dξ dτ (2π) 0 Rn Zt Z  1 −I (τ) i(x,ξ) −i(y,ξ) R t{−(A(s)ξ,ξ)+i(b(s),ξ)}ds C f τ = n e e e (y, τ)dy e dξ dτ (2π) 0 Rn Rn t Z Z Z   1 i(x−y+R t b(s)ds,ξ) − R t(A(s)ξ,ξ)ds −I (τ) τ τ C f = n e e dξ e (y, τ)dydτ. Rn (2π) Rn Z0 Z  Z  Multiplying the last equality by eIC (t), in view of (3.3), we arrive at

t IC (t,τ) e i(x−y+R t b(s)ds,ξ) − R t(A(s)ξ,ξ)ds u τ τ f (x, t)= n e e dξ (y, τ)dydτ. (3.18) Rn (2π) Rn Z0 Z  Z  By (3.10), expression inside of the braces in the right-hand side of (3.18) is equal to

IC (t,τ) e − I−1(t,τ)(x−y+I (t,τ)),x−y+I (t,τ) /4 e ( A b b ) . (2√π)n det (t, τ) IA Applying the same argumentsp as in the proof of Lemma 1, we can rewrite (3.18) as (3.14), where P (x, t, τ) is given by (3.15).

4 Sharp estimates for solutions to the homogeneous weakly cou- pled parabolic system

In this section we obtain estimates for solution of the Cauchy problem (1.3). First, we prove the sharp pointwise estimate for u with ϕ [Lp(Rn)]m, where p [1, ]. | | ∈ ∈ ∞ Rn+1 Theorem 1. Let (x, t) be an arbitrary point in T and u be solution of problem (1.3). The sharp coefficient p(t) in inequality (1.4) is given by (1.5). As a special case of (1.5) with p = one has (1H.6). ∞ Proof. By Proposition 1 and (3.1), (3.2), the sharp coefficient in inequality (1.4) is given by

′ I (t) ∗ 2 1/p e C z ′ −1/2 −p IA (t)(x−y+Ib(t)) 4 p(t) = max 1/2 e dy . (4.1) H |z|=1 n Rn (2√ π) det A (t) I Z   Since ∗ eIC (t) = eIC∗ (t), (4.2)  11 it follows from (4.1) that

′ I ∗ (t) 2 1/p C ′ −1/2 |||e ||| −p IA (t)(x−y+Ib(t)) 4 p(t)= 1/2 e dy . (4.3) H (2√π)n det (t) Rn IA Z   −1/2 1/2 Further, we introduce the new variable ξ = A (t)(x y + b(t)). Since y = A (t)ξ + 1/2 I − I −I x + b(t), we have dy = det A (t)dξ, which together with (4.3) leads to the following representationI I

′ IC∗ (t) 1/p |||e ||| −p′|ξ|2/4 p(t)= 1/p e dξ . (4.4) H 1/2 Rn (2√π)n det (t) Z  IA   Passing to the spherical coordinates in (4.4), we obtain

′ IC∗ (t) ∞ 1/p |||e ||| n−1 −p′ρ2/4 p(t) = 1/p dσ ρ e dρ H 1/2 Sn−1 (2√π)n det (t) Z Z0  IA 1/p′  IC∗ (t)  ∞ |||e ||| n−1 −p′ρ2/4 = 1/p ωn ρ e dρ , (4.5) 1/2 (2√π)n det (t)  Z0  IA   n/2 n−1 n where ωn = 2π /Γ(n/2) is the area of the unit sphere S in R . Further, making the change of variable ρ = √u in the integral

∞ ′ 2 ρn−1e−p ρ /4dρ Z0 and using the formula (e.g. [3], 3.381, item 4)

∞ xα−1e−βxdx = β−αΓ(α) (4.6) Z0 with positive α and β, we obtain

n ∞ ∞ 2 ′ 2 1 n ′ 1 4 n n−1 −p ρ /4 2 −1 −p u/4 ρ e dρ = u e du = ′ Γ , 0 2 0 2 p 2 Z Z     which leads to

n ∞ n/2 2 n n/2 n−1 −p′ρ2/4 2π 1 4 n 2 π ωn ρ e dρ = n ′ Γ = ′ n/2 . (4.7) 0 Γ 2 p 2 p Z 2     Substituting (4.7) into (4.5), we arrive at (1.5).  As a particular case of (1.5) with p = , we obtain (1.6). ∞

12 In the next assertion we prove the sharp pointwise estimate for ∂u/∂ℓ with ϕ [Lp(Rn)]m, p [1, ]. | | ∈ ∈ ∞ Rn+1 Theorem 2. Let (x, t) be an arbitrary point in T and u be solution of problem (1.3). The sharp coefficient p,ℓ(t) in inequality (1.9) is given by (1.8). As a consequence,K the sharp coefficient (t) in inequality (1.11) is given by Kp ′ ′ 1/p −1/2 p +1 IC∗ (t) Γ ||| A (t)|||||| e ||| 2 p(t)= I ′ . (4.8) K 2nπ(n+p−1)/2 det 1/2(t) 1/p  p′(n+p )/2  IA   As a special case of (4.8) with p = one has (1 .12). ∞   Proof. Differentiating in (3.2) with respect to xj, j =1,...,n, we obtain

2 ∂ eIC (t) −1/2 −1 − IA (t)(x+Ib(t)) 4 G(x, t)= A (t)(x + b(t)) e , (4.9) ∂x − n 1/2 I I j j 2(2√π) det A (t)  I  which together with (1.2) and (3.1), leads to ∂u n ∂ 1 =(ℓ, x)u = ℓj G(x y, t)ϕ(y)dy = 1/2 ∂ℓ ∇ Rn ∂xj − −2(2√π)n det (t) j=1 Z IA X 2 −1/2 −1 − IA (t)(x−y+Ib(t)) 4 IC (t) A (t)(x y + b(t)), ℓ e e ϕ(y)dy. (4.10) × Rn I − I Z   Applying Proposition 1 to (4.10), we conclude that the sharp coefficient in estimate (1.9) is given by ∗ eIC (t) z p,ℓ(t) = max K |z|=1 n 1/2 2(2√ π) det A (t) I ′ 2 1/p ′ ′ −1/2 −1 p −p IA (t)(x−y+Ib(t)) 4 A (t)(x y + b(t)), ℓ e dy , × Rn I − I Z    which, in view of (4.2), implies

|||eIC∗ (t)||| p,ℓ(t) = K n 1/2 2(2√π) det A (t) I ′ 2 1/p ′ ′ −1/2 −1 p −p IA (t)(x−y+Ib(t)) 4 A (t)(x y + b(t)), ℓ e dy . × Rn I − I Z    Changing the variable ξ = −1/2(t)(x y + (t )) in the last integral in view of dy = IA − Ib det 1/2(t) dξ, we arrive at the following representation IA   1/p′ ∗ 1/2 ′ IC (t) ′ 1/p |||e ||| det A (t) p ′ 2 I −1/2 ℓ −p |ξ| /4 p,ℓ(t)= 1/2 A (t)ξ, e dξ . K 2(2√π)n det (t) Rn I IA Z   

13 By the symmetricity of −1/2(t), we have IA ′ I ∗ (t) ′ 1/p |||e C ||| p ′ 2 −1/2 ℓ −p |ξ| /4 p,ℓ(t)= 1/p ξ, A (t) e dξ . (4.11) K n 1/2 Rn I 2(2√π) det A (t) Z    I   Passing to the spherical coordinates in (4.11), we obtain

′ I ∗ (t) ∞ ′ 1/p |||e C ||| ′ ′ 2 p p +n−1 −p ρ /4 e −1/2 ℓ p,ℓ(t)= 1/p ρ e dρ σ, A (t) dσ , (4.12) K n 1/2 0 Sn−1 I 2(2√π) det A (t) Z Z  I    n−1 where eσ is the n-dimensional unit vector joining the origin to a point σ of the sphere S . Let ϑ be the angle between e and −1/2(t)ℓ. We have σ IA

′ ′ π/2 −1/2 p −1/2 p p′ n−2 eσ, A (t)ℓ dσ =2ωn−1 A (t)ℓ cos ϑ sin ϑdϑ Sn−1 I I Z Z0  (n−1)/2 p′+1 ′ 2π Γ −1/2 p′ p +1 n 1 −1/2 p′ 2 = ωn−1 (t)ℓ B , − = (t)ℓ ′ . (4.13) IA 2 2 IA Γ n+p    2

Further, making the change of variable ρ = √u in the integral 

∞ ′ ′ 2 ρp +n−1e−p ρ /4dρ Z0 and applying (4.6), we obtain

p′+n ∞ ∞ ′ 2 ′ ′ ′ 2 1 p +n ′ 1 4 n + p ρp +n−1e−p ρ /4dρ = u 2 −1e−p u/4du = Γ . (4.14) 2 2 p′ 2 Z0 Z0     Combining (4.13) and (4.14) with (4.12), we arrive at (1.8). Formula (4.8) follows from (1.8) and (1.10). As a particular case of (4.8) with p = , we obtain (1.12). ∞

5 Sharp estimates for solutions to the nonhomogeneous weakly coupled parabolic system

In this section we derive estimates for solution of the Cauchy problem (1.14). Here we suppose p Rn+1 m α Rn+1 m that f [L ( T )] [C T ] , α (0, 1). It follows from the known assertions for the single parabolic∈ equation∩ (e.g. [2], Ch.1,∈ Sect. 7 and 9) and (3.3), (3.13) that formula (3.14),  p Rn+1 m α Rn+1 m where P is defined by (3.15), solves problem (1.14) with f [L ( T )] [C T ] . First, we prove the sharp pointwise estimate for u , where∈ u is solution of∩ problem (1.14). | | 

14 Rn+1 Theorem 3. Let (x, t) be an arbitrary point in T and u be solution of problem (1.14). Let us suppose that the integral

p′ t eIC∗ (t,τ)z dτ 1/2 p′−1 0 det (t, τ ) Z IA is convergent. The sharp coefficient p(t) in inequality (1.15) is given by (1.16). As a special case of (1.16) with p = one has (1N.17). ∞ Proof. By Proposition 1 and (3.14), (3.15), the sharp coefficient in inequality (1.15) is given by

′ ′ 1/p t I (t,τ) ∗ p 2 1 e C z ′ −1/2 −p IA (t,τ)(x−y+Ib(t,τ)) 4 (t)= max e dydτ , p n 1/2 p′ N (2√π) |z|=1 Rn ( 0 det A (t, τ) ) Z Z I  which, in view of  ∗ eIC (t,τ) = eIC∗ (t,τ), (5.1) implies 

′ ′ 1/p t I ∗ (t,τ) p 2 e C z ′ −1/2 1 −p IA (t,τ)(x−y+Ib(t,τ)) 4 (t)= max e dydτ . (5.2) p n 1/2 p′ N (2√π) |z|=1 ( 0 Rn det (t, τ ) ) Z Z IA  −1/2  1/2 Now, we introduce the new variable ξ = A (t, τ)(x y + b(t, τ)). Since y = A (t, τ)ξ + 1/2 I − I −I x + b(t, τ), we have dy = det A (t, τ)dξ, which together with (5.2) leads to the following representationI I

′ p′ 1/p t IC∗ (t,τ) 1 e z ′ 2 (t)= max e−p |ξ| /4dξdτ . (5.3) p n 1/2 p′−1 N (2√π) |z|=1 ( 0 Rn det (t, τ ) ) Z Z IA Passing to the spherical coordinates in (5.3), we obtain 

′ p′ 1/p t IC∗ (t,τ) ∞ 1 e z ′ 2 (t) = max dτ dσ ρn−1e−p ρ /4dρ p n 1/2 p′−1 N (2√π) |z|=1 Sn−1 (Z0 det A (t, τ ) Z Z0 ) I ′ p′ 1/p t IC∗ (t,τ) ∞ 1 e z ′ 2 = max ω dτ ρn−1e−p ρ /4dρ . (5.4) n n 1/2 p′−1 (2√π) |z|=1 ( 0 det (t, τ ) 0 ) Z IA Z Substituting (4.7) into (5.4), we arrive at (1.16). As a particular case of (1.16) with p = , we obtain (1.17). ∞ In the next assertion we prove the sharp pointwise estimate for ∂u/∂ℓ , where u is solution of problem (1.14). | |

15 Rn+1 Theorem 4. Let (x, t) be an arbitrary point in T and let u solve problem (1.14). Suppose that the integral p′ −1/2 p′ t (t, τ)ℓ eIC∗ (t,τ)z IA p′−1 dτ 0 1/2 Z det (t, τ) IA   is convergent for every unit n-dimensional vector ℓ. Then the sharp coefficient p,ℓ(t) in inequality (1.19) is given by (1.18). C As a consequence of (1.18), the sharp coefficient (t) in inequality (1.21) is given by Cp ′ ′ ′ 1/p p ′ 1/p p′+1 −1/2 p t ℓ IC∗ (t,τ)z 1 Γ 2 A (t, τ) e (t)= max max I dτ . (5.5) p 1/p ′ (n+p′)/2  p′−1  C n (n+p−1)/2 p   |ℓ|=1 |z|=1 0 1/2 2 π Z det (t, τ)     IA     As a special case of (5.5) with p= one has (1.22).  ∞   Proof. Differentiating in (3.15) with respect to xj, j =1,...,n, we obtain ∂ P (x, t, τ) ∂xj 2 eIC (t,τ) −1/2 −1 − IA (t,τ)(x+Ib(t,τ)) 4 = (t, τ)(x + b(t, τ)) e , (5.6) − n 1/2 IA I j 2(2√π) det A (t, τ)  I  which together with (1.2) and (3.14), leads to ∂u n t ∂ 1 ℓ u f =( , x) = ℓj P (x y, t, τ) (y, τ)dydτ = n ∂ℓ ∇ Rn ∂x − −2(2√π) j=1 0 j X Z Z t IC (t,τ) −1 2 2 e − I / (t,τ)(x−y+I (t,τ)) 4 −1 ℓ A b f 1/2 A (t, τ)(x y + b(t, τ)), e (y, τ)dydτ. × Rn I − I Z0 Z det A (t, τ)  I  Applying Proposition 1 to the last representation, we conclude that the sharp coefficient in estimate (1.19) is given by 1 ℓ(t)= Cp, 2(2√π)n 1 ′ ′ t I (t,τ) ∗ p 2 p e C z ′ ′ −1/2 −1 p −p IA (t,τ)(x−y+Ib(t,τ)) 4 max (t, τ)(x y+ (t, τ)), ℓ e dydτ , 1/2 p′ A b × |z|=1 Rn I − I (Z0 Z det A (t, τ )  ) I  which in view of (5.1), implies 1 ℓ(t)= Cp, 2(2√π)n 1 ′ ′ t I ∗ (t,τ) p 2 p e C z ′ ′ −1/2 −1 p −p IA (t,τ)(x−y+Ib(t,τ)) 4 max (t, τ)(x y+ (t, τ)), ℓ e dydτ . 1/2 p′ A b × |z|=1 Rn I − I (Z0 Z det A (t, τ )  ) I 

 16 Changing the variable ξ = −1/2(t, τ)(x y + (t, τ)) in the last integral in view of dy = IA − Ib det 1/2(t, τ) dξ, we arrive at the following representation IA   ′ p′ 1/p t IC∗ (t,τ) ′ 1 e z p ′ 2 (t)= max −1/2(t, τ)ξ, ℓ e−p |ξ| /4dξdτ . p,ℓ n 1/2 p′−1 A C 2(2√π) |z|=1 ( 0 Rn det (t, τ ) I ) Z Z IA  

By the symmetricity of −1/2(t, τ), we have  IA ′ p′ 1/p t IC∗ (t,τ) ′ 1 e z p ′ 2 (t)= max ξ, −1/2(t, τ)ℓ e−p |ξ| /4dξdτ . p,ℓ n 1/2 p′−1 A C 2(2√π) |z|=1 ( 0 Rn det (t, τ ) I ) Z Z IA  

Passing to the spherical coordinates in the inner integral, we obtain 1 ℓ(t)= Cp, 2(2√π)n ′ p′ 1/p t IC∗ (t,τ) ∞ ′ e z ′ ′ 2 p max dτ ρp +n−1e−p ρ /4dρ e , −1/2(t, τ)ℓ dσ , (5.7) 1/2 p′−1 σ A × |z|=1 ( 0 det (t, τ ) 0 Sn−1 I ) Z IA Z Z  where eσ is as before, the n-dimensional unit vector joining the origin to a point σ of the sphere Sn−1. Let ϑ be the angle between e and −1/2(t, τ)ℓ. Similarly to (4.13), we have σ IA ′ 2π(n−1)/2Γ p +1 −1/2 p′ −1/2 p′ 2 eσ, A (t, τ)ℓ dσ = A (t, τ)ℓ n+p′ . (5.8) Sn−1 I I Γ   Z 2  Combining (4.14) and (5.8) with (5.7), we arrive at (1.18). Equality (5.5) follows from (1.18) and (1.20). Putting p = in (5.5), we arrive at (1.22). ∞ Acknowledgement. The publication has been prepared with the support of the ”RUDN University Program 5-100”.

References

[1] S.D. Eidel’man, Parabolic Systems, North-Holland and Noordhoff, Amsterdam, 1969. [2] A. Friedman, Partial Differential Equations of Parabolic Type, R.E. Krieger Publ. Comp., Malabar, Florida, 1983. [3] I.S. Gradshtein and I.M. Ryzhik; A. Jeffrey, editor, Table of Integrals, Series and Prod- ucts, Fifth edition, Academic Press, New York, 1994. [4] J.K. Hunter, Notes on Partial Differential Equations, University of California at Davis, 2014 (el. edition).

17 [5] G. Kresin and V. Maz’ya, Sharp Real-Part Theorems. A Unified Approach, Lect. Notes in Math., 1903, Springer, , 2007. [6] G. Kresin and V. Maz’ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, Math. Surveys and Monographs, 183, Amer. Math. Soc., Providence, Rhode Island, 2012. [7] G. Kresin and V. Maz’ya, Optimal estimates for derivatives of solutions to Laplace, Lam´eand Stokes equations, J. Math. Sci., New York, 196:3 (2014), pp. 300–321. [8] G. Kresin and V. Maz’ya, Sharp estimates for the gradient of solutions to the heat equa- tion, Algebra i Analiz, 31:3 (2019) , pp. 136-153 (transl. in St. Petersburg Mathematical Journal). [9] G. Kresin and T. Ben Yaakov, Sharp pointwise estimates for solutions of the modified Helmholtz equation, Pure and Appl. Funct. Anal., 5:2 (2020), pp. 349-367. [10] G. Kresin and V. Maz’ya, Sharp estimates for the gradients of solutions to lin- ear parabolic second order equation in the layer, publ. online in Appl. Anal. (DOI: 10.1080/00036811.2020.1732356). [11] P. Lancaster, Theory of Matrices, Academic Press, New York-, 1969. [12] O.A.Oleinik, Lectures on Partial Differential Equations, 2nd ed., Moscow, BINOM, 2005 (in Russian). [13] M.A. Shubin, Lectures on Equations of Mathematical Physics, 2nd ed., Moscow, MC- NMO, 2003 (in Russian) (Engl. transl. Invitation to Partial Differential Equations, el. edition). [14] S.L. Sobolev, Partial Differential Equations of Mathematical Physics, Dover Publica- tions, New York, 1989. [15] A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, Dover Pub- lications, New York, 1990. [16] V.S. Vladimirov, Equations of Mathematical Physics, 2nd English ed., Mir Publ., Moscow, 1984. [17] B.Z. Vulich, A Short Course in the Theory of Functions of the Real Variable, 2nd edition, , Moscow, 1973 (in Russian).

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