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∂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)-matrix-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